.*r; 23 THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES >' K t 4 \ : i4 • .. .-Urdcul 'i-tL^ Mirk w {Za^^de^^^ '^ '■ ' ^tJ,C OL£JJa ijA-e^ 'Vii'Lii 4-i-t4%f ' Then there was Charles Davies, appointed in 1857 to be Professor of Higher Mathematics. Until Davies began his activity, the algebra and geometry that were tauo-ht in the United States were those that had been taught at Oxford for 200 years, the geometry being based on Todhunter's Euclid. It remained for Davies to bring to America knowledge of the new and strikingly original ideas of the French mathematicians. In this way he gave our people a new mode of approach and a new mode of thinking in reference to mathematics. The textbooks of the Frenchman, Legendre, were translated and edited by Davies and came into use all over the United States. l-t>vM/»v''Vv-c*''*«.4^' 'f<^'; ELEMENTS OF ALGEBRA: INCLUDING STURM S' THEOREM. TRANSLATED FROM THE FRENCH OF M. BOURDON. ADAPTED TO THE COURSE OF MATHEMATICAL INSTRUCTION IN THB DNITED STATES, BY CHARLES DAVIES, LL.D. AUTHOR OF ARITHMETIC, ELEMENTARY ALGEBRA, ELEMEKTARY GEOMETRT, PRACTICAL GEOMETRY, ELEMENTS OF SURVEYING, ELEMENTS OF DESCRIPTIVE AND ANALYTICAL GEOMETRY, ELEMENTS OF DIFFERENTIAL AND INTEGRAL CALCULUS, AND A TREATISE ON SHADES, SHAD- OWS, AND PERSPECTIVE. NEW YORK: PUBLISHED BY A. S. BARNES & CO. CINCINNATI:— H. W. DERBY & CO. 1850. DAVIE S' COURSE OF MATHEMATICS. DAVIES' FIRST LESSONS IN ARITHMETIC— For Beginners. I) A VIES' ARITHMETIC— Designed for the use of Academies and Schools. KEY TO DAVIES' ARITHMETIC. DAVIES' UNIVERSITY ARITIIMETIC—Embracing the Science of Num- bers and their numerous Applications. KEY TO DAVIES' UNIVERSITY ARITHMETIC. DAVIES' ELEMENTARY ALGEBRA— Being an introduction to the Sci- ence, and forming a counectiug link between Akithmetic and Algebra. KEY TO DAVIES' ELEMENTARY ALGEBRA. DAVIES' ELEMENTARY GEOMETRY.— This work embraces the ele- mentary principles of Geometiy. The reasoning is plain and concise, but at the same time strictly rigorous. DAVIES' ELEMENTS OF DRAWING AND 3IENSURATION — Ap- plied to the Mechanic Arts. DAVIES' BOURDON'S ALGEBRA— Inchiding Sturm's Theorem— Being an abridgment of the Work of M. Bourdon, with the addition of practical examples. DAVIES' LEGENDRE'S GEOMETRY and TRIGONOMETRY— Being an abridgment of the work of M. Legendre, with the addition of a Treatise on Men- suration OF Planes and Solids, and a Table of Logarithms and Logarithmic Sines. DAVIES' SURVEYING— With a description and plates of the Theodolite, Compass, Plane-Table, and Level; also, Maps of the Topographical Signs adopted by the Engineer Department — an explanation of the method of surveying the Public Lands, and an Elementary Treatise on Navigation. DAVIES' ANALYTICAL GEOMETRY — Embracing the Equations of the Point and Straight Line — of the Conic Sections — of the Line and Plank IN Space ; also, the discussion of the General Equation of the second degree, and of Surfaces of the second order. DAVIES' DESCRIPTIVE GE03IETRY— With its application to Spher- ical Projections. DAVIES' SHADOWS and LINEAR PERSPECTIVE. DAVIES' DIFFERENTIAL and INTEGRAL CALCULUS. Entered, according to Act of Congress, in the year 1844, by Charles Davies, in tlie Clerl s Office of tlie District Court of the United States, in and for the Soutliern District oi New York. Engineenng & Mathematical Sdences Jjbrary PREFACE The Treatise on Algebra, by M. Bourdon, is a work of singular excellence and merit. In France, it is one of the leading text books. Shortly after its first pubUcation, it passed through several editions, and has formed the basis of every subsequent work on the subject of Algebra. The original work is, however, a full and complete treatise on the subject of Algebra, the later editions containing about eight hundred pages octavo. The time which is given to the study of Algebra, in this country, even in those semin- aries where the course of mathematics is the fullest, is too short to accomplish so voluminous a work, and hence it has been found necessary either to modify it, or to abandon it al- together. The following work is abridged from a translation of M. Bourdon, made by Lieut. Ross, now the distinguished pro- fessor of mathematics in Kenyon College, Ohio. The Algebra of M. Bourdon, however, has been regarded only as a standard or model. The order of arrangement, in many parts, has been changed ; new rules and new methods have been introduced : and all the modifications which have 4 PREFACE. been suggested by teaching and a careful comparison with other standard works, have been freely made. It would, per- haps, not be just to regard M. Bourdon as responsible for the work in its present form. It has been the intention to unite in this work, the scien- tific discussions of the French, with the practical methods of the English school ; that theory and practice, science and art, may mutually aid and illustrate each other. CHARLES DAVIES. West Point, June, 1844 CONTENTS. CHAPTER I. PRELIMINARY DEFINITIONS AND REMARKS. ARTICLBa Algebra — Definitions — Explanation of the Algebraic Signs ^ . . 1 — 28 Similar Terms — Reduction of Similar Terms 28 — 30 Problems — Theorems — Definition of — Question 31 — 33 CHAPTER n. OF ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION. Addition— Rule 33—36 Subtraction — Rule — Remark 36 — 40 Multijdication — Rule for Monomials 40 — 42 Rule for Polynomials and Signs 42 — 45 Remarks — Theorems Proved 45 — 48 Of Factoring PoljTiomials 48 — 49 Division of Monomials — Rule 49 — 52 Signification of the Symbol ao 52 — 54 Division of Polynomials — Rule 54 — 56 Remarks 57 — 61 When m is entire, a"*—b"* is divisible by a—b 61 — 62 CHAPTER HI. ALGEBRAIC FRACTIONS. Definition — Entire Quantity — Mixed Quantity 62 — 65 Reduction of Fractions 65 — 69 To Reduce a Fraction to its Simplest Form 70 To Reduce a Mixed Quantity to a Fraction 71 6 CONTENTS. ABTICT.ES To Reduce a Fraction to an entire or Mixed Quantity 72 To Reduce Fractions to a Common Denominator 73 To Add Fractions 74 To Subtract Fractions 75 To Multiply Fractions 76 To Divide Fractions 77 Ri'sults from adding to both Terms of a Fraction 78 CHAPTER IV. EQUATIONS OF THE FIRST DEGREE. Definition of an Equation — DiiTerent Kinds — Properties of Equations 79 — 86 Principles in the Solution of Equations — Axioms 86 — 87 Transformation of Equations — First and Second 87 — 92 Resolution of Equations of the First De£;ree — Rule 92 — 94 Questions involving Equations of the First Degree 94 — 95 Equations with two or more Unknown Quantities 95 — 96 Elimination — By Addition — By Subtraction — By Comparison 96 — 103 Indeterminate Problems — Questions involving two or more unknown Quantities 103—104 Theory of Negative Quantities — Explanation of the Terms Nothmg and Infinity 104 — 114 Inequalities 114 — 116 CHAPTER V. EXTRACTION OF THE SQUARE ROOT OF NUMBERS. FORMATION OF THE SQUARE AND EXTRACTION OF THE SQUARE ROOT OF ALGE- BRAIC QUANTITIES. Extraction of the Square Root of Numbers 116—118 Of Incommensurable Numbers 118 Extraction of the Square Root of Fractions 118 — 124 Extraction of the Square Root of Algebraic Quantities 124 Of Monomials 124—127 Of Polynomials 127—130 Calculus of Radicals of the Second Degree 130 — 132 Addition and Subtraction— Of Radicals 132—133 Multiplication, Division, and Transformation 133 — 137 CHAPTER VI. EQUATIONS OF THE SECOND DEGREE. Equations of the Second Degree 137 — 139 Involving two Terms 139—110 Complete Equations of the Second Degree 14ii — 141 CONTENTS. 7 ARTICLES Discussion of Equations of the Second Decree 141 — 130 Problem of the Lights 150—151 Of Trinomial Equations 151 — 154 Extraction of the Square Root of the Binomial a ± VT 154 — 157 l&quations with two or more Unknown Quantities 157 — 159 CHAPTER VII. OF PROPORTIONS AND PROGRESSIONS. How Quantities may be compared together 159 Arithmetical Proportion Defined 159 Geometrical Proportion Defined 159 Arithmetical Proportion — Sum of Extremes 160 Arithmetical Progression — Increasing and Decreasing 161 Value of Last Term — How to find it 162 How to find last term in a Decreasing Series 163 Sum of two Terms equi-distant from Extremes 164 To find Sum of all the Terms 164 General Formulas 165 To find the first Term 165 To find the Common Difference 165 To find any Number of Means between two Numbers 166 The Whole makes a continued Series 167 GEOMETRICAL PROPORTION Ratio Defined 168 Proportion Defined 169 Antecedents and Consequents Defined 170 Mean Proportional Defined , 171 Proportion by Inversion — or Inversely 172 Proportion by Alternation 173 Proportion by Composition ... 174 Proportion by Division 175 Equi-multiples are Proportional 176 Reciprocal Proportion Defined 177 Product of Extremes equal to the Product of the Means 178 To make a Proportion from four Quantities 179 Square of middle Term equal to Product of Extremes 180 Four Proportionals are in Proportion by Alternation 181 Proportion by Equality of Ratios 182 Four Proportionals are Proportional Inversely 183 Four Proportionals are in Proportion by Composition or Division 1^4 Equi-multiples have the same Ratio as the Quantities ]S5 Proportion by augmenting Antecedent and Consequent 186 8 CONTENTS. ARTrC! ES Proportion by idding Antecedent and Consequent 187 The Powers of Proportionals are in Proportion 18S The Products of Proportionals are in Proportion 189 Geometrical Progression Defined 190 Value of the Last Term ■ 191' To find the Sum of the Series 192 To find the Sum of the Terms of a Decreasing Progression 193 When the Sum becomes ^ 194 To find any Number of mean Proportionals between two Num- bers 19f) Progressions having an Infinite Number of Terms 196 CHAPTER VITI. FORMATION OF POWERS, AND EXTRACTION OF ROOTS OF ANY DE- GREE WHATEVER. CALCULUS OF RADICALS. INDETERMINATE CO- EFFICIENTS. Formation of Powers 197 — 199 Theory of Permutations and Combinations 199 — 202 Binomial Theorem 202—206 Consequences of Binomial Theorem 206 — 209 Extraction of the Cube Roots of Numbers 209 — 213 To Extract the nth Root of a Whole Number 213—215 Extraction of Roots by Approximation 215 — 218 Cube Root of Decimal Fractions 218 Any Root of a Decimal Fraction 219 Formation of Powers and Extraction of Roots of Algebraic Quan- tities 220 Of Monomials— Of Polynomials 221—224 Calculus of Radicals — Transformation of Radicals 224 — 227 Addition and Subtraction of Radicals 227 Multiplication and Division 228 Formation of Powers and Extraction of Roots 229 Different Roots of Unity 230—232 Modifications of the Rules for Radicals 232 Theory of Exponents 233 Multiplication of Quantities with any Exponent 234 Division 235 Formation of Powers 23f Extraction of Roots 237 Method of Indeterminate Co-efficients 238—24? Recurring Series , 243 Binomial Theorem for any Exponent 244 — 24? Applications of the Binomial Theorem 245— 24t= CONTENTS. 9 ABTICLES Summation of Series 246 Summation of Infinite Series 247 CHAPTER IX. CONTINUED FRACTIONS. EXPONENTIAL QUANTITIES. LOGARirHiM-S FORMULAS FOR INTEREST. Continued Fractions 248 — 254 Expon-eiilial Quantities 255 Theory of Logarithms 256—258 Multiplication and Division 258 — 260 Formation of Powers and Extraction of Roots 260 — 262 General Properties 262—266 Logarithmic and Exponential Series — Modulus 266 — 270 Transformation of Series 270 — 272 Of Interpolation 272 Of Interest 273 CHAPTER X. GENERAL THEORY OF EQUATIONS. General Properties of Equations 274 — 285 Of the Greatest Common Divisor 285—294 Transformation of Equations 294 — 296 Remarks on Transformations 296 Derived Polynomials 297 — 300 Equal Roots 300—303 Elimination 303 By Means of Indeterminate Multipliers 304 By Means of the Common Divisor 305 — 307 Method of finding the Final Equation 307 — 309 CHAPTER XI. RESOLUTION OF NUMERICAL EQUATIONS. STURIWs' THEOREM. General Principles 309 First Principle 310 Second Principle 311 Third Prineii)le 313 Limits (if Heal Roots 314—317 Ordinary Limits of Positive Roots 317 10 CONTENTS. ARTICtES Smallest Limit in Entire Numbers 318 Superior Limit of Negative Roots — Inferior Limit of Positive and Negative Roots 319 Consequences 320— 32~ Descartes' Rule 327—330 Commensurable Roots of Numerical Equations 330 — 333 Sturms' Theorem 333—34 1 Young's Method of resolving Cubic Equations 342 — 345 Method of Resolving Higher Equations 3 J5 ELEMENTS ALGEBRA. CHAPTER I. PRELIMINARY DEFINITIONS AND REMARKS. 1. Quantity is a general tenii applied to everything which can be estimated or measured. 2. Mathematics is the science which treats of the properties and relations of quantities. 3. Algebra is that branch of mathematics in which the quanti- ties considered are represented by letters, and the operations to be performed upon them are indicated by signs. The letters and signs are called symbols. 4. The sign +, is called plus, and when placed between two quantities indicates that they are to be added together. Thus, 9 4- 5 is read, 9 plus 5, and indicates that the quantity repre- sented by 5 is to be added to the quantity represented by 9. In like manner, a + ^ is read, a plus h, and denotes that the quantity represented by b is to be added to the quantity repre- sented by a. 5. The sign — , is called minus, and indicates that one quantity is to he subtracted from another. Thus, 9 — 5 is read, 9 minus 5 or 9 diminished by 5. In like manner, a — 5 is read, a minus b, or a diminished by i 6. The sign X, is called the sign of multiplication. When placed between two quantities, it denotes that they are to be mul- tiplied together. Thus, 36 x 25, denotes that 36 is to be multi- plied by 25. The multiplication of two quantities may also be 12 ELEMENTS OF ALGEBRA. [CHAP. I. indicated by placing a point between them. Thus, 36.25 is the same as 36 X 25, and is read, 36 multiplied by 25, or the prod- uct of 36 by 25. 7. The multiplication of quantities, which are represented by letters, is generally indicated by simply writing the letters one after the other, without interposing any sign. Thus, ah is the same as a X b, or as a.b ; and ahc, the same as a X b x c, or as a.b.c. It is plain that the notation ah, or ahc, cannot be employed when the quantities are represented by figures. For, if it were re- quired to express the product of 5 by 6, we could not write 5 6, without confounding the product with the number 56. 8. In the product of several letters, as ahc, the single letters, a, h, and c, are called factors. Thus, in the product ah, there are two factors, a and b ; in the product acd, there are three, a, c, and d. 9. There are three signs used to denote division. Thus, a -^ b denotes that a is to be divided by b. -T- denotes that a is to be divided by b. b a\h denotes that a is to be divided by h. 10. The sign =, is called the sign of equality, and is read, is equal to. When placed between two quantities, it denotes that they are equal to each other. Thus, 9 — 5 = 4: that is, 9 mi- nus 5 is equal to 4 : Also, a -\- h =l c, indicates that the' sum of the quantities represented by a and h, is equal to the quantity de- noted by c. 11. The sign >, is called the sign of inequality, and is used to express that one quantity is greater or less than another. Thus, a > i is read, a greater than h ; and c < J is read, a less than h ; that is, the openmg of the sign is turned toward the greater quantity. 12. If a quantity is added to itself several times, as a-{-a-i-a-\-a-{-a, ii is generally written but once, and a figure is then placed before it, to show how many times it is taken. Thus, a-\-a-{-a-{-a-\-a^= 5a. CHAP. I.] DEFINITIONS AND REMARKS. 13 The number 5 is called the co-efficient of a, and denotes that a is taken 5 times. Hence, a co-efficietit is a number prefixed to a quantity, denoting the number of times which the quantity is taken. The co-efficient also indicates the number of times plus one, that the quantity is added to itself. When no co-efficient is written, the co-efficient 1 is alwa)fs understood. Thus, o = la. 13. If a quantity be multiplied continually by itself, as axaXaXaXa, the product is generally expressed by writing the letter once, and placing a number to the right of, and a little above it : thus, a X a X a X a X a = a^. The number 5 is called the exponent of a, and denotes the number of times which a enters into the product as a factor. Hence, the exponent of a quantity shows how many times the quantity is a factor. It also indicates the number of times, plus one, that the quantity is to be multiplied by itself. When no ex- ponent is written, the exponent 1 is always understood. 14. The product resulting from the multiplication of a quantity by itself any number of times, is called the power of that quantity : and the exponent denotes the degree of the power. For example, a^ =z a is the first power of a, a^ = a X a is the second power, or square of a, a^ z= a X a X a is the third power, or cube of a, a* = aXaXaXa is the fourth power of a, a^ = aXaXaXaXa is the fifth power of a, in which the exponents of the powers are, 1, 2, 3, 4, and 5; and the powers themselves, are the results of the multiplications. It ehoidd be observed that the exponent of a power is always greater by unity tha-n the number of multiplications. 15. As an example of the use of the exponent in algebra, lei it be required to express that a number a is to be multiplied three times by itself, that this product is then to be multiplied vhree times by b, and this new product twice by c ; we should write axaxaxaxbxhxbxcxc=z a*b^c^. If it were further required to repeat this result a certain num- 14 ELEMENTS OF ALGEBRA. [CHAP. I. ber of times, say seven times, that is, to add it to itself six timtSy we should simply Avrite This example shows the brevity of the algebraic language. 16. The root of a quantity, is a quantity which being multi- plied by itself a certain nmnber of times, will produce the given quantity. The sign y , is called the radical sign, and when prefixed to a quantity, indicates that its root is to be extracted. Thus, Y a or simply y a denotes the square root of a. V a denotes the cube root of a. a denotes the fourth root of a. The number placed over the radical sign is called the index of the root. Thus, 2 is the index of the square root, 3 of the cube root, 4 of the foiuth root, &c. 17. The reciprocal of a quantity, is unity divided by that quan- tity. Thus, — is the reciprocal of a; a and, r is the reciprocal of a + i. a -\- b 18. Every quantity written in algebraic language, that is, with the aid of letters and signs, is called an algebraic quantity, or the algebraic expression of a quantity. Thus, is the algebraic expression of three times the quantity denoted by a ; . 2 J is the algebraic expression of five times the ( square of a ; _ 3,2 ^ is the algebraic expression of seven times the product of the cube of a by the square of b ; is the algebraic expression of the difference between three times a and five times b ; is the algebraic expression of twice the square ., , o 1 , ^19 1 of a, diminished by three times the product 'Za^ — oab -\- Ab^^ r , i 1 ,. • 1 of a by 0, augmented by fom* times the square of b. la S 5a — 55 j CriAP. I.] DEFINITIONS AND REMARKS. 16 19. A single algebraic expression, not connected with any other by the sign of addition or subtraction, is called a monomial^ or simply, a term. Thus, 3a, Sa^, ld?lP-, are monomials, or single terms. 20. An algebraic expression composed of two or more terms separated by the sign + or — , is called a polynomial. For example, 3a — 5b, and 2a2 — 3cb + 4Z»2, are polynomials. A polynomial composed of two terms, is called a bino?mal ; and a polynomial of three terms is called a trinotnial. 21. The numerical value of an algebraic expression, is the num- ber obtained by giving a particular value to each letter which en- ters it, and performing the arithmetical operations indicated. This numerical value will depend on the particular values attributed to the letters, and will generally vary with them. For example, the numerical value of 2a^, will be 54 if we make a = 3 ; for, 3^ = 27, and 2 x 27 = 54. The numerical value of the same expression is 250 when we make a = 5; for, 5^ — 125, and 2 X 125 = 250. 22. It has been said, that the numerical value of an algebraic expression generally varies with the values of the letters which enter it : it does not, however, always do so. Thus, in the ex- pression a — b, so long as a and b are increased or diminished by the same number, the value of the expression will not be changed. * For example, make a == 7 and i = 4 : there results a — h = Z. Now, make a = 7 + 5 = 12, and /> = 4 + 5 = 9, and there results, as before, a — b =. \2 — 9 = 3. 23. Of the different terms which compose a polynomial, some are preceded by the sign +, and others by the sign — . The first are called additive terms, the others, suhtractive terms. When the first term of a polynomial is plus, the sign is gener- ally omitted ; and when no sign is written, it is always under- stood to be affected by the sign -)-. 24. The numerical value of a polynomial is not affected by- changing the order of its terms, provided the signs of all the terms remain unchanged. For example, the polynomial 4a3 _ 3a26 + bac"^ = 5ac2 — 'ia'^h + 4a3 = — Za^b f bac^ + 4cz* 16 ELEMENTS OF ALGEBRA, [CHAP. I. 25. Each of the literal factors which compose a term, is called a dimension of the term ; and the degree of a term is the number of these factors or dimensions. Thus, 3a is a term of one dimension, or of the first degree. bob is a term of two dimensions, or of the second degree. la^hc^ = laaahcc is of six dimensions, or of the sixth degree. In general, the degree, or the number of dimensions of a term, is determined by taking the sum of the exponents of the letters which enter this term. For example, the term 8a^bcd^ is of the seventh degree, since the sum of the exponents, 2+1+1 + 3 = 7. 26. A polynomial is said to be homogeneous, when all its terms are of the same degree. The polynomial 3a — 2J + c is of the first degree and homogeneous. — 4ab + b^ is of the second degree and homogeneous. 50^0 — 4c^ + 2c'^d is of the third degree and homogeneous. 8a^ — 4ab + c is not homogeneous. 27. A vinculum or bar , or a parenthesis ( ), is used to express that all the terms of a polynomial are to be considered together. Thus, a -{- b -^ c X b, or {a -\- b -\- c) x b denotes that the trinomial a -\- b -\- c is to be multiplied by b ; also, a + b-\- c X c + d-irf or {a + b + c) X {c + d +/) denotes that the trinomial a -^ b -^ c is to be multiplied by the trinomial c -\- d -\- f When the parenthesis is used, the sign of multiplication is usually omitted. Thus, {a + b -\- c) X b is the same as {a -{- b -\- c)h The bar is also sometimes placed vertically. Thus, -\- a X + i is the same as (a + i + c) a?, or, a + i + c X a;. + c 28. Those terms of a polynomial which are composed of the same letters, affected with the same exponents, are called similar terms. Thus, in the polynomial lab + Zab — AaW + Sa'i*, the terms lab and Zab, are similar, and so also are the terms — 4a'''Z»2 and ba^b"^, the letters and exponerts in each being the same. But in the binomial 8a26 + 7at2, CHAP. I.J DEFINITIONS AND RKMAUKS. 17 the. terms are not similar ; for, although they are composed of the same letters, yet the same letters are not ulTected with the same exponents. 29. When a polynomial contains several similar terms, it may often be reduced to a simpler form. Take the polynomial Aa^b — Sa^c + ^a^c — 2a'^b. It may be written (Art. 24) Aa^b — 2a'^b -f 7a^c — Sa'^c. But 4(2-6 — 2a'^b reduces to 2a-6, and 7a^c — 3a^c to Aa^c. Hence, 4a^ — 3a^c + la'^c — 2a^b = 2d^b + Aa^c. When we have a polynomial having similar terms, as + 2a^bc^ — Ad^bc^ + &a%c^ — 8a^c^ + lla^c^, anite the additive and subtractive terms separately : thus, Additive terms. Subtractive terms. + 2a3ic2 — 4a3ic2 + 6a^(y^ — 8a^c^ Hence, the given polynomial reduces to \9a^c^ — 12a3ic2 = 7a^c^. It may happen that the co-efllcient of the subtractive term, ob- tained as above, will exceed that of the additive term. In that case, subtract the positive co-efficient from the negative, prefix tria minus sign to the remainder, and then annex the litsral part. In the poljTiomial 2d^b + 2^2/, _ ^a^h _ ^a^b + Za% — 5d^b + 2a26 _ 3a^ + Hci'^b — 8aH H'lt, — 8aH = — ba^b — 2a^b : hence 5a^— 8a^ = 5aH — 5a^b — Sa^i z= — 3a^b. Hence, for the reduction of the similar terms of a poljmomial, we have the following RULE. I. Add together the co-efficients of all the additive terms, and an- nex to their sum the literal part: form a single subtractive term tn the same manner. II. Then, subtract the less co-efficient from the greater, and to the remainder prefix the sign of the greater co-efficient, and aimex the literal part. 2 18 ELEMENTS OF ALGKRRA. [CHAP. 1 EXAMPLES. 1. Reduce the pol^Tiomial \(P-h — 8«~/; — 9a^5 -f- Wcrh to its simplest form. Ans. — liP-h. 2. Reduce the polynomial lahc^ — nhc^ — lahc? — Sabc^ + ^ahc^ U) lis simplest form. Ans. — Sabc"^. 3. Reduce the polynomial 9iP — Bac"^ + ]5cP -\- Sea + Oi/c^ — 24c53 to its simplest form. Ans. ac^ -\- 8ca. 4. Reduce the polynomial 6nc~ — 5aP -\- lac''- — 3aP — 13ac^ 4- \8ab^ to its simplest form. Ans. lOah^. 5. Reduce the polynomial ahc^ — abc + ^ac^ — Qabc"^ + 6a^c — 8ac^ to its simplest form. A?is. — Sabc"^ -I- 5abc — Sac^. Remark. — It should be observed that the reduction affects only the co-efficients, and not the exponents. The reduction of similar terms is an operation peculiar to al- gebra. Such reductions are constantly made in Algebraic Addition^ Subtraction, Multiplication, and Division. 30. In the operations of algebra, there are two kinds of quan- tities which must be distinguished from each other, viz. 1st, Those whose values are known or given, and which are called known quantities ; and 2dly, Those whose values are unknovro, which are called un- knmon quantities. The known quantities are represented by numbers and the first letters of the alphabet, a. b, c, d, &c. ; and the unknown, by the final letters, x, y, z, &c. 31. A problem is a question proposed which requires a solution. It is said to be solved when the values of the quantities sought are discovered or found. \ theorem is a gen(>ral truth, which is proved by a course of reasoning called a demonstration. 32. The following questi;»n will tend to show the utility of the algebraic analysis, Quesllon. The siim of two numbers is 67, and their dijTerfnce 19 ; what art (hj- two numbers? CHAP. I.] DEFINITIONS AND REMARKS. 19 Solution. Let us first establish, by the aid of the algebraic symbols, the connexion which exists between the given and unknown num- bers of the question. If the least of the two numbers were knotvn, the greater could be found by adding to it the difference 19 ; or in other words, the less number, plus 19, is equal to the greater. If, then, we make x = the less number, X -f- 19 =r the greater, and 2x -{- 19 = the sum. But from the enunciation, this sum is to be equal to 67. There- fore we have 2a; -{- 19 = 67. Now, if 2x augmented by 19, gives 67, 2x alone is equal to 67 minus 19, or 2a; = 67 — 19, or performing the subtraction, 2x = 48. Hence, x is equal to half of 48, that is, X 1= — r= 24. The least number being 24, the greater is x+ 19 r=:24 -f 19 = 43. And, indeed, we have 43 + 24 = 67, and 43 — 24 = 19. Another Solution. Let X represent the greater number ; then, X — 19 will represent the less and 2x — 19 = 67 ; whence, 2x = 67 + 19 = 86 ; 86 therefore, x =:-—=: 43 = the greater, and consequently, x — 19 =: 43 — 19 = 24 = the less General Solution of this Problem. The sum of two numbers is a, and their difference is b. Whas are the two numbers ? Let X = the less number ; then will. x -j- 5 = the greater. 20 ELEMENTS OF ALGEBRA. [CHAP. I. Then, by the conditions of the question 2a: + ^ = a, the sum of the numbers ; therefore, 2a; = a — i and x — = . And by adding h to each side of the equality, we obtain the greater number, .7 a ^ . , a ^ ^ 2 2 ^ 2 2 Hence we have ^ . a b . , x-\-b=. — -\—-=i the greater number, and a? = = the less. 2 2 As the y^rm of these results is independent of any particular values attributed to the letters a and b, the expressions are called formulas, and may be regarded as comprehending the solution of all questions of the same nature, differing only in the numerical values of the given quantities. Hence, A formula is the algebraic enunciation of a general rule, or princi[)le. The principles enunciated by the formulas above, are these : Tlie greater of any two numbers is equal to half their suin in- creased by half their difference ; and the less, is equal to half their sum diminished by half their difference. To apply these formulas to the case in which the sum is 237 and difference 99, we have 237 99 237 + 99 336 ,^^ the greater number — — — ^- — = = -— = 1C8 ; 237 99 237—99 138 and the less = — — — = — = b9 ; and these are the true numbers ; for, 168 + 69 = 237 which is the given sum, and 168 — 69 = 99 which is the given difference. From the preceding explanations, we see that Algebra is a language composed of a series of symbols, by the aid of whicli, toe can abridge and generalize the operations required in the sul'c- tion of problems, and the reasonings pursued in the deinonstratwu af theorems CHAI (I.J ADDITION. 21 CHAPTER II. OF ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION. ADDITION. 33. Addition, in algebra, consists in finding the simplest equiv- alent expression for several algebraic quantities. Such equivalent expression is called their sum. 34. Let it be required to add together \ the monomials, j ( , 2c The result of the addition is - - - 3rt + 56 + 2c an expression which cannot be reduced to a more simple form. Again, add together the monomials - < 20^6' The result, after reducing (Art. 29), is - Vi'fih^ Let it be required to find the sura of rhe expressions, ") 2«' - 3ai + P 3a2 — 4ab Sab 2ab — 5^2 Their sum, after reducing (Art. 29), is - da^ — 5ab — 4/^^ 35. As a course of reasoning similar to the above would apply to all algebraic expressions, we deduce, for the addition of alge- braic quantities, the following general RULE. I. Write down the quantities to be added, with their respective signs, so that the similar terms shall fall under each other. I I . Reduce the similar terms, and annex to the results those terms which cannot be reduced, giving to each term its respective si!«:?. 27aic — 9ab + Gc^, 15. What is the sum of 8ahc -\- Pa — 2ca; — 6x1/ and 7cx — xy — \3Pa1 Ans. 8abc — 12b^a + 5cx —7xy. 16. What is the sum of 9a"c — 14a5y + ISa^fe^ ^nd — d'-c — 8aW\ A«.>\ 8a2c — 14aiy + laW. 17. What is the sum of 17a^Z»2 ^ 9^-/, _ 3^2^ _ 14^5^2 _|_ 7^^ — 9a3, — 15a36 + 7a^62 — c^, and 14a"6 - \9d^b ? Ans. . 18. What is the sum of 3aa;^ — 9ax^ — \laxy, — 9ax^ + 18ax^ •\- 34axy, and la^'b + Sax^ — Jax"^ + 46ca; ? Ans. . 19. Add together 3a2 + ^aWc"^ — 9a^r, la^ — 8a'^b\^ — lOa^a., 10a6 + IGa^^V + 19a^x. Ans. 10a- + ISaH^c^ + lOab. 20. Add together 7«-/y — 3uic — 8b^c — 9c^ + cd^, 8abc — 5aH -j_ 3c3 _ 4^2c _f_ c(l~, and 4a2J _ Sc^ + 952c — 3d^. Ans. 6d^b + 5a5c — 352c — \\c^ -\- 2cd? — 3d^. 21. Add together — 18a^ + 2ab^ + 60252 _ 8a5* + 7a^ — ba'^b^ — 5a^b 4- Gab* + lla252, ^„^, _ 15^35 ^ 12a252. 22. What is the sum of 3a352c — 16a*a; — 9ax^d, + 6a^^e — Gax^d + 17a*a:, + 16ax^d — a^x — 8a^^^c ? Ans. aH'^c + ax-^d. 23. WTiat is the sum of the following terms : viz., 8a^ — 10a*5 - 16a352 + 4^253 _ i2a4j ^ 15a352 _}_ 24a253 — 6ab* — 16a^^ 4- 20a253 + 32a5* — 85^ ? Ans. 8a5 — 22a*5 — 17^352 _{_ 48a2^,3 _|_ 26a5* — 85* 24 KLEMENTS OF ALGEBRA. [CHAP. J3. SUBTRACTION. 36. Subtraction, in Algebra, consists in finding the simplest expression for the difference between two algebraic quantities. Let it be required to subtract 46 from 5a. Here, as the quan- tities are not similar, their difference can only be indicated, and we write 5a — 4h. Again, let it be required to subtract 4a^5 from 7a^. These ('•rms being similar, one of them may be taken from the other, and their true difference is expressed !)y 7a^ — 4>Pb = 3a^. For another example, from - - - 4a take the binomial - - - - 26 — 3c. 4a — 26 + 3c True remainder. The subtraction may be indicated thus, - 4a — (26 — 3c) , that is, the quantity to be subtracted may be placed within a parenthesis, and written after the other quantity, with a minus sign. Now, in order to express this difference by a single polyno raial, let us see what the nature of the question requires. From 4a, we are to subtract the difference between 26 and 3e, and if 6 and c were given numerically, that difference would be known ; but since 3c cannot be taken from 26, 26 is first subtracted from 4a, which gives 4a — 26. Now, in subtracting the number of units contained in 26, the number taken away from 4a, is too great by the number of units contained in 3c, and the result 4a — 26 is therefore too small by 3c; this remainder must there- fore be corrected by adding 3c to it. Hence, there results from the proposed subtraction 4a — 26 + 3c. 37. Hence, for the subtraction of algebraic quantities, we have the following general RULE. I. Write the quantity to he subtracted vnder that from which it ts to he taken, placing the similar terms, if there are any, undn each other. II. Chiinrre the sirrns of all the terms of the pohpiotinnl to 6c subtracted, or conceive them to he chingrd, and then reduce the poly HOTnial result to its simplest form CHAP. II.] SUBTRACTION. EXAMPLES. 25 0)- 5 2-0 (1)- From Sac — 5ab -\- c^ s"=l 6ac — 5ab -\- c^ Take 3ac + Sab — 7c 2 = » ■ M 5 c- IS is 5 3 — Sac — Sab + 7c Remainder Sac — 8ab + c2 + 7c. Sac — Sab + c2 + 7c (2). (3). From 16a2 _ 56c + lac 19abc — IGax — 5axy Take 14«2 + 5bc + Sac 17 abc + 7 ax — Ibaxy Remainder 2a2 _ 106c — ac 2abc — 2Sax + lOaxr/ (4). (5). From 5a3 _ 4aH + Sb^c 4ab — cd -|- Sa^ Take - 2a3 + Sa^ — SPc 5ab — 4cd + 3a2 + 5b^ Remainder 7a3 — la'^b + ll/>2c - ah + Scd + - 562. 6. From 3a2x — 13a6c + 7a2, take 9a^x — ISabc. Ans. — da'^x -\- 7cP'. 7. From fiXa^b'^c — \Sabc — Wa'^y, take Ma^lfic — 27ahc — 14a2y. Ans. lOaWc + 9abc. 8. From 27a6c — 9ab -f 6c2, take 9a6c + 3c — 9<7,r. Ans. ISabc — 9ab + 6c2 — 3c + 9ax. 9. From 8a5c — 1253a + 5cx — 7xy, take 7cx — xy — ISb'^a, Ans. Sabc -j- b^a — 2ca; — 6xy. 10. From Sa^c — 14a6y + 7a262, take 9a2c — I4aby + 15a262. Ans. — a'^c — 8a262. 11. From 9a«x2 — 13 + 20a63a; — 4b^cx'^, take Sb^cx^ + QaV — G + Sab^x. Ans. 17 ab^x — 7bhx^ — 7. 12. From 5a* — 7aW — Sc^d!^ + 7d, take 3a* — 3a2 — 7c^J?- — 15a362. Ans. 2a* + Sa'^b'^ + 4c^d'^ + 7J + 3a2. 13. From 51a262 _ 48a36 + 10a*, take 10a* — Sa^* — 6a2i2. Ans. b7aW — 4Qa^b. 14. From 21x3y2 _|_ 25a;2y3 + GSxy* — 40y5, take 64^2^3 + 48a-y* — 40y5. Ans. 2idxy* — S9xhf + 21a;3y2. L5. From bSx^y"^ — Ibx^y"^ — ISxhj — bSx^ , take - l.^.r'y^ r 18a:3y2 _}. 24x*y. Ans. 35x3y2 _ \2x^y — SCx"* 26 ELEMENTS OF ALGEHRA. [CH^^P. II. 38. From what has preceded, we see that polynomials may l)e subjected to certain transformations. For example - - 60^ — 3ab + 2b'^ — 2hc, may be written - 6a^ — [Sab — 2b'^ + 2bc). In like manner - - 7a^ — Sa% — ib'^c + 6b\ may be written - 7a^ — {Sa?b + Ab^-c — W^) ; or, again, - - - la^ — Sa^b — [Ab'^c — 6b^). Also, - - - . 8a2 — 6a^^ + 5^253, becomes - - - Sa^ — [Ga^b^ — da^b^). Also, - - - - 9a2c3 — 8a* + ^»2 _ c, may be written - 90^6^ — (8a* — b'^ -\- c) ; or, it may be written Qa^c^ + b"^ — (8a* + c). These transformations consist in decomposing a polynomial into two parts, separated from each other by the sign — . It will be observed that the sign of each term which is placed within the parenthesis is changed. Hence, if we have one or more terms included within a parenthesis having the minus sign before it, if the parenthesis is omitted, the signs of all the terms must be changed. Thus, 4a — {Gab — 3c -^ 2b), is equal to 4a — Gab + 3c + 2b. xVlso Gab — ( — 4ac -\- 3d — 4ab), is equal to Gab -|- 4ac — 3d -{- 4ab. 39. Remark. — From what has been shown in addition and subtraction, we deduce the following principles. 1st. In Algebra, the words add and sum do not always, as in arithmetic, convey the idea of augmentation. For, a — b, which may result from the addition of — J to «, is properly speaking, the arithmetical difference between the number of units expressed by a, and the number of units expressed by b. Consequently, this result is numerically less than a. To distinguish this sum from an arithmetical sum, it is called the algebraic sum. Thus, the polynomial, 2a^ — 3a'^b + 3b'^c. is an algebraic sum, so long as it is considered as the resuU of the union of the monomials 2^3, — Sa^^, + 3b-c, with then respective signs ; but, in its proper acceptation, it is the arithmeli- CHAP. II.] MULTIPLICATION. 27 cal difference between the sum of the units contained in the ad- ditive terms, and the sum of the units contained in the subtractive terms. It follows from this, that an algebraic sum may, in the numeri- cal applications, be reduced to a negative number, or a number affected with the sign — . 2d. The words subtraction and dijference, do not always convey the idea of diminution. For, the difference between + a and — b being a — [— b) zzl a -\- b, is numerically greater than a. This result is an algebraic difference. MULTIPLICATION. 40. Algebraic raultiplication has the same object as arithmeti- cal, viz., to repeat the multiplicand as many times as there are units in the multiplier. The multiplicand and multiplier are called factors. It is proved in Arithmetic (see Davies' Arithmetic, § 26), that the value of a product is not affected by changing the order of its factors : that is, I2ab = ab X 12 = ba X 12 = a X 12 X b. For convenience, however, the letters in each term are generally arranged in alphabetical order, from the left to the right. Let it be required to multiply 7a^b^ by 4a'^b. By decomposing the multiplicand and multiplier into their fac- tors, we may write the product under the form 7a^b^ X 4a'^b = laaabh X Aaah ; and since we may change the order of the factors without affect- ing the value of the product, we have, laW X 4a^ = 7 X Aaaaaabbb = 2Sa^P ; a result which is obtained by multiplying the co-efficients to- gether for a new co-efficient, and adding the exponents of the same letter, for the new exponents. Again: multiply the monomial 12a%^c'^ by Qa%'^cP. We can place the product under the form, ■\2a^¥c'^ X ^aWd^ = 12 X Saaaaabbbbbbccdd = OGa^^c'^d^. By considering the manner in which these results are obtained, we see that any quantity, as a, must be found as many times 28 ELEMENTS OF ALGEBRA. [CHAP. 11. a factor in the product, as it is a factor in both the multiplicand and multiplier; which number will always be expressed by the sum of its exponents. 41. Hence, for the multiplication of monomials we have the following RULE. I. Multiply the co-efficients together for a new co-efficient. II. Write after this co-efficient all the letters which enter into the multiplicand and multiplier, affecting each with an exponent equal to the sum of its exponents in both factors. • EXAMPLES. (1) - - 8a2ic2 X labd"^ = 56a^^c^d^. (2) - - 2\a%Hc X 8a6r3 _ \QQa^b^cH. (3) (4) (5) (6) Multiply - - Sa^ft - - I'ia'^x - - 6xyz - - a^xy by - - 25a2 - - I2x'^i/ - - ay^z - - Ix.y^ Ga'^b^ I44a'^x^y 6ax7/^z'^ 2a'^x^y^. 7. Multiply Sa^b^c by 7a%^cd. Ans. 56a^Wc'^d. 8. Multiply 5abd^ by 12cd^. Ans. 60abcd^. 9. Multiply la^bd'^c'^ by abdc. Ans. la^Pd^c*. 42. We will now proceed to the multiplication of polynomials. In order to explain the most general case, we will suppose the multiplicand and multiplier each to contain additive and subtrac- tive terms. Let a represent the sum of all the additive terms of the multi- plicand, and b the sum of the subtractive terms ; c the sum of the additive terms of the multiplier, and d the sum of the sub- tractive terms. The multiplicand will then be represented by a — b and the multiplier, by c — d. We will now show how the multiplication expressed by {a — b) X (c — d) can be eflected. The required product is equal to a — 6 taken as many times as there are units in c — d. Let us first multiply by c ; that is, take a — b as many times as there are units in c. We begin by wri- ting ac, which is too great by b taken a -b c -d ac -be — ad -\- bd ac -bc- -ad + bd. CHAP. II.] MULTIPLICATION. 29 c times ; for it is only the difference between a and b, that is first to be multiplied by c. Hence, ac — be is the product of a — b by c. But the true product is a — b taken c — d times : hence, the last product is too great by a — b taken d times ; that is, by ad — bd, which must be subtracted. Changing the signs and subtracting this from the first product (Art. 37), we have (a — b) X (c — d) = ac — be — ad -{- bd : If we suppose a and c each equal to 0, the product will re- duce to + bd. 43. By considering the product of a — h by c — d, we may deduce the following rule for the signs, in the multiplication of two polynomials. When two terms of the multiplicand and multiplier are affected with the same sign, their product will be affected with the sign +, and when they are affected with contrary signs, their product will be affected with the sign — . Again, we say in algebraic language, that + multiplied by -\-, or — multiplied by — , gives -\- ; — multiplied by +> or + mul- tiplied by — , gives — . But since mere signs cannot be multi- plied together, this last enunciation does not, in itself, express a distinct idea, and should only be considered as an abbreviation of the preceding. This is not the only case in which algebraists, for the sake of brevity, employ expressions in a technical sense in order to se- cure the advantage of fixing the rules in the memory. 44. Hence, for the multiplication of polynomials we have the foUowidg RULE. Multiply all the terms of the multiplicand by each term of the tnultiplter in succession, affecting the product of any two terms with the sign plus, when their signs are alike, and with the sign minusy when their signs are unlike. Then reduce the polynomial result to its simplest form. 1. Mwhiply 3a2 + 4aJ + £2 by 2a -\- 5b Ca3 + 8aV} + 2ab^ The product after reducing, + \5aH + 20«Z»2 + 5P becomes 6m-< -f 23u^ + 22irb'^ + 5b^. 80 ELEMENTS OF ALGEBRa. [CHAP. II (2). (3). a' + y^ x^ -r xy^ + ^aa- ar — y ax -\- 5ax a^ + xy"^ ax^ 4- ax'^y^ -f- Va^a;^ — x'^y — y^ + 5ax^ + ^ax'^y^ + SSa^x' g ^ + a:?/^ — x^y—'^ 6ax^^+_6ax^^ j- 42a'^x ^. 4. Multiply x"^ + 2(7X + a^ by x -{- a. Ans. x^ + So-r^ + 3a^x + a^. 5. Multiply x^ -{- y^ by a: + y- -4^5^. x^ -f- xy^ 4- a"2y -\- y'. G. Multiply 3ai2 _^ e^^c^ by 3«62 _|_ Sa^c^. Ans. 9a^b* + 27aHh^ + 18a*c*. 7. Multiply 4a;2 — 2y by 2y. Ans. 8x^y — 4y2. 8. Multiply 2x + iy by 2t — 4y. Ans. 4x^' — 16y2. 9. Multiply a;^ + x^y + xy^ + y^ by x — y. Ans. -^ 10. Multiply x"^ -\- xy -\- y^ by x"^ — xy -{■ y^. Ans. X* + x'^y^ + y*. In order to bring together the similar terms, in the product of two polynomials, we arrange the terms of each polynomial with reference to a particular letter. 11. Multiply 4a3 — 5a'^b — 8ah^ -{- 2b^ by 2a2 — Sab — 4b^ 8a5 — lOa^b — IGa^b^ + 4a^b^ — 12a*b + l5aH^ + 24a2i3 _ Qab* — \(jaW + 2002^3 + 32ai4 — 86' 8a^ — 22a*b — 17 a'b^ + 48^2^,3 ^ 26«&^ — 865 . After ha-\ang arranged the polynomials, with reference to the letter a, multiply each term of the first, by the term 20^ of the second; this gives the polynomial 8a^ — lOa'^b ~ Ida^"^ -{- 4a'^b^, the signs of which are the same as those of the multiplicand. Passing then to the term — Sab of the multiplier, multiply each term of the multiplicand by it, and as it is affected with the gign — , affect each product with a sign contrary to that of the corresponding term in the multiplicand; this gives — 12a^i + I5a^^ + 24a^b^ - Gab* ^ CHAP. II.] MULTIPLICATION. 31 The same operation is also performed with the term — 4/>»', which is also subtractive ; this gives, — 1 6^352 _j_ 20^2^3 _|_ 32aJ4 _ 8J5, The product is then reduced, and we finally obtain, for the most simple expression of the product, 8a^ - 22a*i — l7aW + 48a^^ + 26ab* — 8b^. 12. Multiply 2a2 — 3ax + Ax^ by 5a2 _ Qax - 2x2. Ans. 10a* — 21a?x + ZAa^x"^ — ISax^ — Bx". 13. Multiply 3x2 _ 2yx + 5 by x'^-\-2xy — 3. Ans. 3x* + 4x3y — 4x2 — 4x2y2 + 16x3/ - 15. 14. Multiply 3x3 _|. 2x2y2 -f 3^2 by 2x3 _ 3^2y2 _j_ 5^3. Ans \ ^^^ ~ ^^^y^ ~ ^'^y* + ^^^y" "^ ^ ^^^'^ 15. Multiply Sax — Qab — c by 2i7x + a^ + c. .4«5. 16a2x2 — 4a26x — 6a2^2 _|_ Qacx — labc — c^. 16. Multiply 3a2 _ 5*2 + 3c2 by a^ — ^3. Ans. 3a* — 602^2 4. ^a'^c'^ — 3^263 + 5J5 _ ^b^c^- 17. Multiply 3a2_5irf+ c/ by — 5a2 + 4bd — 8cf Prod. red. — 15a* + 37a 2,^J — 29a\f^^0b^ (P + AAbcdf — 8c^ ^. 18. Multiply 4a352 — 5aH^c + 8a-bc^ — 3a2c3 _ 7abc^ by 2aJ2 _ 4^^^ _ 2/^c2 + c^. f 8a*b* — lOa^'c + 2Sa^'^c^ — 34a^b'^c^ Prod. red. < — 4a2i3c3 _ IGa*^^^ _^ Ua^c* + 7a2i2c4 C + ]4a25c 5 -}- I 4a ^2c5 _ 3a2c6 —7abc ^. 45. Resvlts deduced from the multiplication of polynomials. 1st. If the polynomials which are multiplied together are ho- mogeneous, Their product will also be homogeneous, and the degree of each term will be equal to the sum of the degrees of any two terrnn of the mvltiplicand and multiplier. Thus, in example 18th, each term of the multiplicand is of the 5th degree, and each term of the multiplier of the 3d de- gree : hence, each term of the product is of the 8lh degree. This remark serves to discover any errors in the addition of the exponents.* 82 ELEMENTS OF ALGEBRA. [CHAP. II 2(1. If no two of tlie partial products are similar, there will be no reduction among the terms of the entire product: hence. The total number of terms in the entire product will be equal to the number of terms m the multiplicand multiplied by the number of terms in the multiplier. •This is evident, since each term of the multiplier will produce as many terms as there are terms in the multiplicand. Thus, in example 16th, there are three terms in the multiplicand and two in the multiplier : hence, the number of terms in the product is equal to 3 X 2 =r 6. 3d. Among the difTerent terms of the product, there are always some which cannot be reduced with any others. For, let us consider the product with reference to any letter common to the multiplicand and multiplier. Then, the irreducible terms are, 1st. The term produced by the multiplication of the two terms of the multiplicand and multiplier which contain the highest ex- ponent of this letter ; and the term produced by the multiplica- tion of the two terms which contain the lowest exponent of this letter. For, these two partial products will contain this letter, affected with a higher and lower exponent than either of the other partial products, and consequently, they cannot be similar to any of them. This remark, the truth of which is deduced from the law of the exponents, will be very useful in division. Multiply - - 5«^^>2 + 3a^b — ab* — 2a&3 by - - - a'^b — ab"^ C 5a%^ + 3a^^ — a^b^ — 2a^^- Product, J - 5a^b* - 3a^^ + a~b^ + 2«^^>^ If we examine the multiplicand and multiplier, with reference to a, we see that the product of 5a'^b'^ by a'^b, must be irre- ducible ; also, the product of — 2ab^ by ab^. If we consider the letter b, we see that the product of — ab* by — ab"^, must be irreducible, also that of Ba'^b by a^b. 46. We will apply the rules for the multiplication of algebraic quantities in the demonstration of the following theorems. THEOREM I. The square of the sum of two quantities is equal to the square of the first, plus twice the product of the first by the second, plus the square of the second. * CHAP, II.] MULTIPLICATION'. 33 Let a denote one of the quantities and h the other : then a + ^'' = their sum. Now, Ave have from known principles, [a + hf = (a + J) X (a + i) = c2 + lah + V', which resuh is the enunciation of the theorem in the langiiag of Algebra. To apply this result to finding the square of the binomial we have (5a2 -f ^d^hf = 25a* + 80a*d + 64a*^»2. Also, (Go^i + 9aP) = 36^8^,2 ^ lOSa^M + Sla^js . also, (8a3 -f7ac6)2=. THEOREM II. The square of the difference between two quantities is equal to the square of the first, minus twice the product of the first hy the second, plus the square of the second. Let a represent one of the quantities and b the other : then a — 5 = their difference. Now, we have from known principles, [a — by = {a — b)x{a — h) = d^— 2ab + ^', which is the algebraic enunciation of the theorem. To apply this to an example, we have (7^2^,2 _ i2ab^) = 49a*b* — l68a^^ + lUa^b^. Also, (4a363 — 7c2(i^)2 = THEOREM III. The product of the sum of two quantities multiplied by thetr difference, is equal to the difference of their squares. Let the quantities be denoted by a and b. Then, a -\- b z=. their sum, and a — 6 = their difference We have, from known principles, {a-irb)x{a — b) = «2 _ ^2^ which is the algebraic enunciation of the theorem. To apply this principle to an example, we have (8a3 + 7ai2) x (Sa^ — lab"-) = 64^^ _ 49a2i*. Also, (Vc + lah^) X {9a^c — lab^) = 34 KLEMENTS OF ALGEBRA. [CHAP. II 47. By considering the last three results, it will be perceived that their composition, or the manner in which they are formed from the midtiplicand and multiplier, is entirely independent of any particular values that may be attributed to the letters a and b which enter the two factors. The manner in which an algebraic product is formed from its wo factors, is called the law of this product ; and this law re- mains always the same, whatever values may be attributed to the letters which enter into the two factors. Of factoring Pohpiomlals. 48. A given polynomial may often be resolved into two factors by mere inspection. This is generally done by selecting all the factors common to every term of the polynomial for one factor, and writing what remains of each term within a parenthesis foi the other factor. 1. Take, for example, the polynomial ab -j- ac ; in which, it is plain, that a is a factor of both terms : hence ah -\- ac ■= a [b -\- c). 2. Take, for a second example, the polynomial ab'^-c -}- 5oi^ "1- ab'^c^. It is plain that a and Jf- are factors of all the terms : hence ab-^c + 5aP + ab^"^ — ab"^ (c + 5i + c% 3. Take the polynomial 25a* — SOa^Z* + ISa^i^ . jt jg evident that 5 and d^ are factors of each of the terms. We may, there fore, put the polynomial under the form 5«2 (5a2 — 6«i + 3^2). 4. Find the factors of "^a^ -}- 9o^c + ISa^o-y. Ans. 3a2 (5 -f 3c + 6;ry). 5. Find the factors of Sa^ca: — \^acx^ + 2arc-^y — SOa'^c^x. Ans. 9.ac {Aax — 9x^ + c*y — ISaVa-). 6. Find the factors of 2ia%'^cx — ZOa%^c^y -\- 2(Sa^¥cd + (Jabc. Ans. 6abc {iabx - baU^c^y + 6a«^»V + 1). 7 Find the factors of a^ + 2ah -f b"^. Ans. {a + h) via + i) CHAP. II.] DIVISION. 35 8. Find the factors of a^ _ yi Ans. (a + 6) x (a — V). 9. Find the factors of a- — lah + h"^. Ans. (a — b) X (a — b). 10. Find the factors of the poljTiomial 6a^b -}- Sa'^b^ — I6ab'' - 2ab. 11. Find the factors of the polynomial ISaic^ — 3bc^ + Qa^iV - 12f///c2. 12. Find the factors of the polynomial 25a^bc^ — 30a^bc*d • 5ac* — 60ac^. 13. Find the factors of the polynomial 42a'^b'^ — 7 abed + labd Ans. lab (Joab — cd -{- d). DIVISION. 49. Division, in Algebra, explains the method of finding from iwo given quantities, a third quantity, which multiplied by the first shall produce the second. The first of the given quantities is called the divisor: the sec- ond, the dividend; and the third, or quantity sought, the quotient. Let U3 first consider the case of two monomials, and divide "ibaWc by lab. The dinsion may be indicated thus, — - = 5a^-^b'^-\^= baHc^. lab Now, since the quotient must be such a quantity as multiplied by the divisor will produce the dividend, the co-efficient of the quotient multiplied by 7 must give 35, the co-efficient of thv dividend ; hence, the new co-efficient 5 is found by dividing 35 by 7. Again, the exponent of any letter, as a, in the quotient, added to the exponent of the same letter in the divisor, must give the exponent of this letter in the dividend : hence, the ex- ponent in the quotient is found by subtracting the exponent in the divisor from that in the dividend. Thus, the exponent of a is 3 — 1 — 2, and of 6, 2 — 1 = 1, and since c is not found in the divisor, there is nothing to be subtracted from its exponent. bO. Hence, for the division of monomials, we have the following' 36 ELEMENTS OF ALGEBRA. [CHAP. 11. RULE. I. Divide the co-ejicient of the dividend by the co-ejicient of the divisor, for a new co-efficient. II. Write after this co-efficient, all the letters of the dividend, and affect each with an exponent equal to the excess of its exponent in the dividend over that in the divisor. From this rule we find, ASa^bh'^d \2ab-^c 1. Divide 16a:' 2. Divide = 4a^bcd ; l50a^Pcd^ 30a3'\ indicates that the letter b enters times as a factor in the quotient (Art. 13); or what is the same thing, that it does not enter it at all. Still, the notation shows that b was in the dividend and divisor with the same exponent, and has disappeared by division. 1 5(1 DC In like manner, = 5a%^c^ = 5&2. Sa^bc^ 53. We will now show that the power of any quantity whosy exponent is 0, is equal to unity. Let the quantity be represented jy a, and let 7/1 denote any exponent whatever 38 ELEMENTS OF ALGEBRA. [CHAP. 11 Then, — = a"'"'" = a", by the nile of division. But, — = 1, since the numerator and denominator are equal: hence, o° r= 1, since each is equal to—. ^ a"' We observe again, that the symbol a^ is only employed con- ventionally, to preserve in the calculation the trace of a letter which entered in the enunciation of a question, but which may disappear by division. Division of Polynomials. 54. The object of division, is to find a third polynomial called the quotient, w^hich, multiplied by the divisor, shall produce the dividend. Hence, the dividend is the assemblage, after reduction, of the partial products of each term of the divisor by each term of the quotient, and consequently, the signs of the terms in the quotient must be such as to give proper sign^ to the partial products. Since, in multiplication, the product of two terms having the same sign is affected with the sign +, and the product of two terms having contrary signs, with the sign — , we may conclude, 1st. That when the term of the dividend has the sign +» and that of the divisor the sign +, the term of the quotient must have the sign +. 2d. When the term of the dividend has the sign +, and that of the divisor the sign — , the term of the quotient must have the sign — ; because it is only the sign — , which, combined with the sign — , can produce the sign + of the dividend. 3d. When the term of the dividend has the sign — , and that of the divisor the sign +, the quotient must have the sign — . That is, when the two corresponding terms of the dividend and divisor have the same sign, their quotient will be affected with the sign +> and when they are affected with contrary signs, their quotient will be affected with the sign — ; again, for the sake of brevity, we say that + divided by +, and — divided by — , give -f ; — divided by +, and + divided by — , give — • CHAP. II.] 39 Dividend. Divisor. a2 g2 — 2ax ax + x' a;2 IL a — X a — X — ax Quotient. — ax + x2. FIRST EXAMPLK Divide a^ — 2ax + x^ by a — x It is found most convenient, in division in algebra, to place the divisor on the right of the dividend and the quotient di- rectly under the divisor. We first divide the term a"^ of the dividend by the term a of the divisor, the partial quotient is a, which we place under the divisor. We then multiply the divisor by a, and subtract the product a- — ax from the dividend, and to the remainder bring down x"^. We then divide the first term of the remainder, — ax, by a, the quotient is — x. We then multiply the divisor by — X, and, subtracting as before, we find nothing remains. Hence, a — X is the exact quotient. SECOND EXAMPLE. Let it be required to divide 26^2^2 ^ lOa* — 4Sa'% + 24uP by 4ab — 5a^ -f 3P. In order that we may follow the steps of the operation more easily, we will arrange the quantities with refer- ence to the letter a. Dividend. 10a* — 480^6 + 2Ga"^2 _|_ 24aP -f lOo* — 8a^ — &aW — 40a3r4-~32a2i2 _^ ^Aa^ — 40a^ + 32a2Z/'- -f 24aP. Divisor. 5a2 + 4ab + 3^2 — 2a2 -I- 8ab Quotient . It follows from the definition of division, and the rule for the multiplication of polynomials (Art. 44), that the dividend is the assemblage, after addition and reduction, of the partial products of each term of the divisor, by each term of the quotient sought. Hence, if we could discover a term in the dividend which was derived, without reduction, from the multiplication of a term of the divisor by a term of the quotient, then, by dividing this term of the dividend by that of the divisor, we should obtain a term of the required quotient. Now, from the third remark, of Art. 45, the term lOa"", affected with the highest exponent of the letter a, is derived, without re- duction from the two terms of the divisor and quotient, afl'ected 40 ELEMENTS OF ALGEBRA. [CHAP. 11. with the highest exponent of the same letter. Hence, by dividing the term 10a* by the term — 5a^, we shall have a term of the required quotient. Dividend. Divisor. 10a* — ASa^b + 2&aW + 2AaP + 10a* — Sa^b — 6aW 5a2 -(-. 4ab + 3b^ — 40a^ + 32a%^ + 2-iab^ — 40a^ + 'i2aW + 2\ab'^. — 2a2 + 8 Quotient. Since the terms 10a* and — 5a- are affected with contrary signs, their quotient will have the sign — ; hence, 10a*, divi- ded by — ott^, gives — 20^ for a term of the required quotient. After having written this term under the divisor, multiply each term of the divisor by it, and subtract the product, 10a* — 8a^ -h 6a^^, from the dividend, which is done by writing it below the divi- dend, conceiving the signs to be changed, and performing the re- duction. Thus, the remainder after the first partial division is — 40a^ + 32a2^>2 + 24ab\ This result is composed of the partial products of each term of the divisor, by all the terms of the quotient which remain to be determined. We may then consider it as a new dividend, and reason upon it as upon the proposed dividend. We will there- fore divide the term — iOa^^b, affected with the highest exponent of a, by the term — Sa^ of the divisor. Now, from the prece- ding principles, — 40a^, divided by — Sa^, gives -f 8a5 for a new term of the quotient, which is written on the right of the first. Multiplying each term of the divisor by this term of the quotient, and writing the products underneath the second divi- dend, and making the subtraction, we find that nothing remains. Hence. — 2a- 4- Sab or Sab — 2a^ IS the required quotient, and if the divisor be multiplied by it. f'le product will be the given dividend. Hy considering the preceding reasoning, we see that, in each partial operation, we dinde that term of the dividend which is CHAP, n.] DIVISION. 4l affected with the highest exponent of one of the letters, by that term of the divisor affected with the highest exponent of the same letter. Now, we avoid the trouble of looking out these terms by writing, in the first place, the terms of the dividend and divisor in such a manner that the exponents of the same letter shall go on diminishing from left to right. This is Avhat is called arranging the dividend and divisor with reference to a certain letter. By this preparation, the first term on the left of the dividend, and the first on the left of the divisor, are always the two which must be divided by each other in or- der to obtain a term of the quotient. 55. Hence, for the division of polynomials we have the fol- lowing RULE. I. Arranore the dividend and divisor with reference to a certain letter, and then divide the first term on the hft of the dividend by the first term on the left of the divisor, for the first term of the quotient ; muliply the divisor by this term and subtract the prod- uct from the dividend. II. Then divide the first term of the remainder by the first term of the divisor, for the second term of the quotient ; multiply the divisor by this second term, and subtract the product from the re- sult of the first operation. Continue the same process, and if the remainder is 0, tlie division is said to be exact. THIRD EXAMPLE. Divide 21a;3y2 + 25xh/ + 68^3/* — 40y5 — bQx^ — IQx^y by 5y^ — S-r^ — Qxy. — 40y5 -f 68a:y* + 25a;2y3 + 2lx^y'^ — IQx^y — 56*5 llSy^ — Qxy — 8x^ — 40y' + 48.ry-t -f 64x^^3 _ g^g ^ ^^^^ _ 3^,2^ 4. 7^^ 1st rem 20xy* — SOx^y^ + 2 1 xhf 20.Ty* — 24a:2y3 _ 32a:^y2 2d rem. ~^^ — 1 5x2y3 + 53^^ — 18a-*y — 15x2y3 ^ iSxy- + 24jiy 3d rem. ... - '3oj-^y^ — 42x*y — 56x^ 35a:3y2 — 42x*y — 56a^ Final remainder - - . - 42 KLKMENTS OK ALOKBRA. [CHAP. 11. 56. Remark. — In performing the division, it is not necessar}- to bring down all the terms of the dividend to Ibrin the first re- mainder, but they may be brought down in succession, as in tlie example. As it is important that begiimers should render themselves familiar with the algebraic operation, and acquire the habit of calculating promptly, we will treat this last example in a different manner, at the same time indicating the simplifications which should be introduced. These, consist in subtracting eacli jiartial product from the dividend as soon as this product is formed. — 40 If + 68xf + •35a:-y3 -|- 2 1 x^f — 1 8x^y — 56af ^ 1 1 5f — 6xy — So-^ 1st rem. 20.ry^ — 39jY + 2\xY — 8f + Axf — 3x^y -f Ta^ 2d rem. - — Ibx'^f + 53a:^y- — ISar^y 3d rem. - - - SSjr^y^ — A2x^y — 56a;^ Final remainder - - - - - 0. First, by dividing — 40y^ by Sy^, we obtain — Sy^ for the quo- tient. Multiplying Sy^ by — 8y?, we have — AOy^, or by chan- ging the sign, + 40y^, wliich destroys the first term of the divi- dend. In like maimer, — Qxy X — Sy-' gives + 48ary*, and for the subtraction — 4S.ry'', which reduced with + QQxy^, gives 20a:y* for a remainder. Again, — 8x!^ x — Sy^ gives +, and changing the sign, — QAx'^y'^, which reduced with 2ox'^y^, gives — 39x^y^ Hence, the result of the first operation is 20a;y* — 39a;^y^, fol- lowed by those terms of the dividend which have not been re- duced with the partial products already obtained. For the sec- ond part of the operation, it is only necessary to bring down the next term of the dividend, to separate this new dividend from the primitive by a line, and to operate upon this new dividend in the same manner as we operated upon the primitive, and so on. FOURTH EXAMPLE. Divide - - 95a — 73«2 -f- 560* — 25 — 59a3 by — 3u^ -1-5 — 11a — 7a3. 56a^ — 59a3 — 73a2 + 95a — 25 ist rem. — 35a3 -f 15a2 + 55a — 25 7«3_ 3a2 _ Ua _^ 5 8a — 5 2d remainder - - 0. CHAP. 11.] DIVISION. 43 GENERAL EXAMPLES. 1. Divide lOab + loac b}' 5a. Ans. 2b + 3c. 2. Divide 30ax — 54x by 6x. Ans. 5a — 9. 3. Divide lOx'^y — \5y^ — 5y by 5y. Ans. 2x^ — 2y — 1. 4. Divide 12a + 'iax — ISax^ by 3a. Ans. 4 + x — &x^. 5. Divide Qax^ -\- 9a"x + a^^^ by ax. Ans. 6x + ''^a -j- aoc. G. Divide a- + 2aj? -4- x^ by a + a:. A/i^. a -}- a-. 7. Divide a^ — 3a~y + Say^ — y^ by a — y. Ans. c^ — 2ay 4" v^ 8. Divide 24a2i — VlaHlP- — 6ab by — 6ab. Ans. - Aa +2aHb + 1. 9. Divide 6x<— 96 by 3x— 6. Ans. 2x^ -\- Ax- + Qx + \Q. 10. Divide - - a^ — 5a^x + lOu^x"^ — lOa^x^ + Sox* — x^ by a^ — 2aa: + x"^. Ans. a^ — Sa'^x + Saj;^ — x^ 11. Divide 48x3 _ 75^:^2 — 64a2x + lOSa^ by 2x — 3a. Ans. 24x2 — 2ax — 350^. 12. Divide y^ — Sy^x^ -f Sy^x* — x^ by y^ — 3y'^x -f- Syx^ — x^, Ans. y^ -f Sy^x + Syx^ -j- x^. 13. Dinde %\a^b^ —25a'^h^ by ^aW ^ 5abK Ans. 8a^^ — 5ab\ 14. Divide 6a^ + 23a2^ + 22ai2 4. 5P by 3a2 + 4a^' + b^. Ans. 2a -j- 5b. 15. DiA'ide 6ax^ + 6ax^y^ -{- A2a'^x'^ by ox + 5ax. Ans. x^ + xy*^ + 7ax. 16. Divide — ISa" + 37a2iJ — 29a''-cf — 20b^

the new numerators. e X b X d = ebd ) And - b X d X f = bdf the common denominator. -_ adf cbf ebd adf -\- cbf-\- ebd 2. To a- ^ add b + ^. c . . , . 2abx — Scjr' Ans. a -{- A ; . be ^ Ji -4^ -^ ■ r -r::,, '•-\ CHAP. III.] ALGEBRAIC FRACTIONS. 53 3. Add — , — , and — together. Ans. a; + — -. 2 3 4 12 ..,,« — 2 4a; 19.r — 14 4. Add — - — and — - together. Ans. . 3 7" 21 .5. Add X H to 3a: -\ . 3 4 Ans. 4a: -\ . 12 03!? ST I // 6. It IS required to add 4a?, , and toirether. . ^ , bx^ -{- ax -\- o^ Ans. 4x -i . 2fla; 2* 7a! 2a; 4- 1 7. It is required to add — ^, — , and together. 19.r + 12 Ans. 2x 4 . ^ 60 8. It is required to add 4ar, — , and 2 -\ togeiiier. y o . ^ , 4-la- + 90 Ans. 4x A . ^ 45 2x 8a; 9. It is required to add 3a; H and x — together. 23r Ans. 3x -\ . 45 10. What is the sum of -, -, and Ans. a — b[ a -\- b' a -^ x' cfi — ax^ + cP-h — hx^ + «^<^ + the numerators. (2a — 4a;) X 2b = Aab — 8bx ) And, 2b x 3c = 6bc the common denominator. 3cx — 3ac 4ab — 8bx 3cx — 3ac — 4ab + 8bx llence, 6bc 6bc 6bc 12a; 3a; 30x 2. Required the difference of — - and — . Ans. -^-. 3. Required the difference of 5y and -—. Ans. — --. 8 o 3x 2x 1 3x 4. Required the difference of -— and -— . Ans. — -— . ^ 7 9 63 5. Required the difference of — r — and -r-. a dx -\- ad — be ^"^- Vd — 3x A- a 2a; + 7 6. Required the difference of — -r — and — - — . 24a; -\- Sa — lObx — 3bb Ans. ; . 406 7. Required the difference of 3a; H — ~ and x • b c ex -^ bx — uh be Ans. 2x + CASE vn. 76. To multiply fractional quantities together. RULE. If the quantities to be midtipJied are mixed, reduce them fo a fractional form ; then multiply the numerators together for a nu- merator and the denominators together for a denominator. CHAP. III.] ALGEBRAIC FRACTIONS. EXAMPLES. 55 bx c i. Multiply a -\ by — . bx a^ + hx First, - - - a -\ = ; a a d^ -\-hx c a^c + hex Hence, - X -j- = 9 5 a a ad r 3x , 3a . 9ffT 2. Required the product of — and — . Ans. —j-. 2x Sx"^ 3a;' 3. Required the product of ^ and — -. Avs. — — . 2a!; Sab , 3r/r 4. Find the continued product ot — , , and -— -. * a c ■lo Ans. dux. hx ft 5. It is required to find the product of b -\ and — . oh 4- bx Ans. . ,r a-2 _ ^2 a:2 4- b"^ 6. Required the product of — ; and — — ■ . ^ ^ be 6 + c x^ — h* Ans. b'^c + hc^ a; -f- 1 X — 1 7. Required the product of x -\ , and — T~T' ax"^ — ax + a'2 — 1 Ans. + ah . ax a'' — X '^ 8. Required the product of a -\ by — — — ^. a^{a + x) Ans. — r. x (1 -+- x] CASE VIII. 77. To divide one fractional quantity by another. RULE. Reduce the mixed quantities, if there are any, to a fractiounl firm : theji invert the terms of the divisor and multiphj the frac- tions together as in the last case. 56 ELEMENTS OF ALGEBRA. [CHAP III EXAMPLES. X.. ., b f 1. Divide - - - a by ■^. b 2ac — b a — 2c 2c Hence, g- ^ ^ L ^''j^ yS^'^acg -bg 2c g 2c f 2cf -. T 7a; ,,..,,, 12 , 9\x 2. Let — be divided by — . Ans. — . 5 -^ 13 60 Ax^ Ax 3. Let be divided by 5x. Ans. — '■. 1 ^ 35 ^- a+l,,. .,-, 2x ^ a?+l 4. Let — - — be divided by — . Ans. 6 -^3 4,r 5. Let be divided by — . Ans. X -\ '2 X — \ 6. Let -— be divided by — -. Ans. . 3 -^ 36 2a m T X — b l-.ji, ^cx ^ x — h 7. Let -r— T- be divided by — -. Ans. — - — . Scd ^ Ad 6c2a: „ - a;* — 6* , ,..,,, x'^-\-bx 8. Let — — — — — be divided by r-. x^ — 2bx-\-b^ ^ x — b 62 Ans. X -\ . X „ ^. ., ax — 1 , a , aa;(l + a;) — ac— 1 9. Divide — by -. Ans. -^ — - — '- . 1— a;l — x^ a 10. Divide by -. Ans. — (1 -j- o). a+ 1 , • 1 4- a If we have a fraction of the form a T^'^' a - c and = c ; that is. — -b~ we may observe that — T = — c, also The sign of the quotient will be changed by changing the sign Cither of the numerator or denominator, but will not be affected by changing the signs of both the terms. CHAP. III.] ALGEBRAIC FRACTIONS. 57 78. We will add but two propositions more on the subject of fractions. If the same number be added to each of the terms of a proper fraction, the new fraction resulting from this addition will be greater than the first ; but if it be added to the terms of an improper fraction, the resulting fraction will be less than the first. a Let the fraction be expressed by — , and suppose a <^b. Let m represent the number to be added to each term : then the new fraction becomes . b -{- m In order to compare the two fractions, they must be reduced to the same denominator, which gives for the first fraction, and for the new fraction, a ab -{- am b b'^ -\- bm a -\- m ab -\- bm b -\~ m b"^ -\- bm Now, the denominators being the same, that fraction will be the greatest which has the greater numerator. But the two nu- merators have a common part ab, and the part bm of the sec- ond is greater than the part am of the first, since b ^ a: hence ab -\r bm "^ ab -{• am ; that is, the second fraction is greater than the first. If the given fraction is improper, that is, if a > J, it is plain •hat the numerator of the second fraction will be less than that of the first, since bm would then be less than am. If the samjB number be subtracted from each term of a proper fraction, the value of the fraction will be diminished; but if it be subtracted from the terms of an improper fraction, the value of the fraction will be increased. Let the fraction be expressed by — , and denote the number to be subtracted by m. Inen, — z= the new fraction — m 58 ELEMENTS OF ALGEBRA. By reducing to ilie same denominator, we have, a ah — am and b IP' — bm a — m ah — hm b — m b"^ — bm Now, if we suppose a <^b, then am < hm; and if am bm, and ah — am <^ ab — hm, that is, the new fraction will be greater than the first. GENERAL EXAMPLES. 1. Add to Ans. -\ f. 1 — a:^ 1+0.-2 I — X* . , , 1 1 . *-^ 2. Add to . Ans. I + X \ ~ X 1 — x2 a -\- b . a — b 4ab 3. From 7 take r. Ans. a — b a -{- b cP- — h"^ 4. From take -. Ans. \ — X- 1 -f .1-2 1 — x^ ,. , . , 0-2 _ 9a: + 20 , x2 - ] 3.r + 42 ^- ^^^"^^^P^>^ .2_6, by ^,_,^ ■ x"^ — WxA 28 Ans. x" y^ ^4 Xp' -\- JlX 6. Multiply ^ , ^ . — —Tz by —• ^"•^- *^ + ^'^^■ x^ -\- 2bx -{- h^ X — b « T^- -J a + X a — X a -\- X a — x 7. Divide 1 — by — . a — a: a -\- x a — x a -\- x o2 _|_ r2 Ans. "Zax ^. ., , " — 1. , n — \ 8. Divide 1 H -— - by 1 -— -. Ans. n. n -\- 1 n -f 1 CHAP. IV.J EQUATIOXS OF THE FIRST DEGRFE. 59 CHAPTER IV. OF EtiUATlONS OF THE FIRST DEGREE. 79. Ax Equation is the algebraic expression of two equai quan- ties with the sign of equality placed between them. Thus, is an equation, in which x is equal to the sum of a and b. 80. By the definition, every equation is composed of two parts, separated from each other by the sign =. The part on the left of the sign, is called the first member, and the part on the right, is called the second member; and each member may be composed of one or more terms. 81. Every equation may be regarded as the enunciation, in al- gebraic language, of a particular problem. Thus, the equation X -\- X z^ 30, is the algebraic enunciation of the following problem : To find a number which, being added to itself, shall give a sum equal to 30. Were it required to solve this problem, we should first express it in algebraic language, which would give the equation X -\- X =^ 30, by adding x to itself, - - 2x = 30, and dividing by 2, - - a: == 15. Hence we see that the solution of a problem by algebra, con- sists of two distinct parts : viz., the statement, and the solution oS ail equation GO ELEMENTS OF ALGEBRA. ' fCHAP. IV. The STATEMENT consists in finding an equation which shall ex press the relation between the known and unknown quantities of the problem. The SOLUTION of the equation consists in finding such a value for the unknown quantity as being substituted for it in the equa- tion null satisfy it ; that is, make the first member equal to the second. 82. An equation is said to be verified, when such a value is substituted for the unknown quantity as will prove the two mem- bers of the equation to be equal to each other. 83. Equations are divided into classes, with reference to the highest exponent with which the unknown quantity is affected. An equation which contains only the first power of the un- known quantity, is called an equation of the first degree: and generally, the degree of an equation is determined by the greatest of the exponents with which the imknown quantity is affected, without reference to other terms which may contain the unknown quantity raised to a less power. Thus, ax -\- b =: ex -\- d is an equation of the 1st degree. 2x'^ — 3a; = 5 — 2x'^ is an equation of the 2d degree. 4x^ — 5x2 =: 3x + 11 is an equation of the 3d degree. It more than one unknown quantity enters into an equation, its degree is determined by the greatest sum of the exponents with which the unknown quantities are affected in any of its terms. Thus. xy + bcx = d* is of the second degree. xyz"^ 4- cx^ = a^ is of the fourth degree. 84. Equations are also distinguished as numerical equations and literal equations. The first are those which contain numbers only, with the exception of the unknown quantity, which is always de- noted by a letter. Thus, 4« — 3 = 2a; + 5, 3a;2 — a; = 8, are numerical equations. They are the algebraical translation of problems, in which the known quantities are particular numbers. A literal equation is one in which a part, or all of the known quantities, are represented by letters. Thus, bx^ -\- ax — 3x = 5, and ex + dx^ = c -{-f are literal equations. CHAP. IV.] EQUATIONS OF THE FIRST DEGREE. 61 85. It frequently occurs in Algebra, that the algebraic sign + or — , which is written, is not the true sign of the term before which it is placed. Thus, if it were required to subtract — b from a, we should write a — ( — h) := a -\- h. Here the true sign of the second term of the binomial is plus although its algebraic sign, which is written in the first member of the equation, is — . This minus sign, operating upon the sign of b, which is also negative, produces a plus sign for b in the result. The sign which results, after combining the algebraic sign with the sign of the quantity, is called the essential sign of the term, and is often different from the algebraic sign. By considering the nature of an equation, we perceive that it must possess the three following properties : 1st. The two members are composed of quantities of the same kind. 2d. The two members are equal to each other. 3d. The essential sign of the two members must be the same. 86. An axiom is a self-evident proposition. We may here stale the following : 1. If equal quantities be added to both members of an equa- tion, the equality of the members will not be destroyed. 2 If equal quantities be subtracted from both members of an equation, the equality will not be destroyed. 3. If both members of an equation be multiplied by the same number, the equality will not be destroyed. 4. If both members of an equation be divided by the same number, the equality will not be destroyed. Solution of Equations of the First Degree. 87. The transformation of an equation is any operation by which we change the form of the equation without affecting the equality of its members First Transformation. 88. When some of the terms of an equation are fractional, to reduce the equation to one in which the terms shall be entire. 62^ ELEMENTS OF ALGEBRA. [CHAP. IV. Take the equation, 2x 3 X First, reduce all the fractions to the same denominator, by the Known rule ; the equation then becomes 4aT 5ix 12a; "72"~ W^ 72"" ■^^' If now, both members of this equation be multiplied by 72, the equality of the members will be preserved, and the common de- nominator will disappear ; and we shall have 48x — 5ix + 12a; z= 792 ; or dividing by 6, 8x — 9x -\- 2x = 132. .89. The last equation could have been found in another man- ner by employing the least common multiple of the denominators. The common multiple of two or more numbers is any number which each will divide without a remainder ; and the least com- mon multiple, is the least number which can be so divided. The least common multiple can generally be found by inspec- tion. Thus, 24 is the least common multiple of 4, 6, and 8 ; and 12 is the least common multiple of 3, 4, and 6. Take the last equation, 2x 3 X a;-|----= 11. 3 4 6 We see that 12 is the least common multiple of the denomina- tors, and if we multiply each term of the equation by 12, divi- ding at the same time by the denominators, we obtain Qx — ^x + 2x = 132, the same equation as before found. 90. Hence, to transform an equation invohang fractional terms to one involving only entire terms, we have the following RULE. Form the least common mnUiplc of all the denominators, and then nnihiphj every term of the equation hy it, reducing at the same tunc the fractional to entire terms. CHAP. [V.] EQUATIO^f« OF THK FIRST DEGREE. 63 EXAMPLES. 1. Reduce 1 3 = 20, to an equation involving entire 5 4 terms. We see, at once, that the least common multiple is 20, by uliich each term of the equation is to be multiplied. ^ r. 20 ^ Now, — X 20 = a; X -^ = 4a;, 5 o ^ ^r. 20 ^ and , — X20=:a;x— - = 5x; 4 4 that is, we reduce the fractional to entirfj terms, by multiplying the numerator by the quotient of the common multiple divided by the denominator, and omitting the denominators. Hence, the transformed equation is 4a; -f 5.r - 60 = 400. 2. Reduce \- — 4 — 3 to an equation involving only entire terms. Ans. 7a; + 5a; — 140 = 105. a c 3. Reduce -— r'\~f=g to an equation involving only en- d tire terms. Ans. ad — be -{- hdf ■— bdg. 4. Reduce the equation ax 2c'^x 4bc^x 5a^ 2c^ „, 7- + 4a = — —-{ 36 ab a-^ 0'' a to one involving only entire terms. Ans. a^bx — 2a%c'^x + Aa^b'^ = ib^-'^x — 5a^ + 202^^ _ 2a%^. Second Transformation. 91. When the two members of an equation are entire polyno- mials to transpose certain terms from one member to the other. Take for example the equation 5x — 6 = 8 + 2.t. If, in the first place we subtract ") 2a; from both members, the equality V 5x — 6 — 2a: =: 8 -f- 20" — 2.>-.- will not be destroyed, and we have or, by reducing the terms in the second member. > 5a; — 6 — 2a; = 8. 64 ELEMENTS OF ALGEBRA [CHAP. IV. Whence we see that the term 2x, which was additive in the second member, becomes subtractive in the first. In the second place, if we add 6 '\ to both members, the equality will > 5a; — 6 — 2a; +6 = 8 + 6; still exist, and we have ) or, since — 6 and + 6 destroy each other 5x — 2a? = 8 + 6. Hence, the term which was subtractive in the first member, passes into the second member with the sign plus. For a second example, take the equation aa? + & =3 fZ — ex. ' If we add ex to both ^ members and subtract b, > ax + b + cx — b = d — cx-\-cx — b: the equation becomes ) or reducing - - - qx -\- ex = d — b. Hence, we have the following principle : Any term of an equation may be transposed from one member to the other by changing its sign. 92. We will now apply the preceding principles to the resolu' tion of equations. 1. Take the equation 4a; — 3 = 2a; + 5. By transposing the terms — 3 and 2a;, it becomes 4a; — 2x = 5 + 3 ; and by reducing 2a; = 8 : 8 di^ading by 2 a? = - = 4. Now, if 4 be substituted in the place of x in the given equa- tion, it becomes 4x4 — 3=2x4 + 5, that is, 13 = 13. Hence, 4 is the true value of x ; for, being substituted for x in the given equation, that equation is verified. 2. For a second example, take the equation 5x Ax 7 13a; 12 "Y" ~ Y 6~' CHAP. IV.] EQUATIONS OF THE FIRST DEGREE. 6i> By making the denominators disappear, we have 10^ — 32a: — 312 = 21 — 52a? b)' transposing 10a; — 32a; + 52a: = 21 + 312 by reducing 30a; = 333 333 111 dividing by 30 x = — — = — — = 11.1 ; * -^ 30 10 ' a result which, being substituted for x, will verify the givct equation. 3. For a third example let us take the equation (3a — x) [a — 6) + 2aa; ■=: 4b (x -\- a). It is first necessary to perform thd multiplications indicated, in order to reduce the two members to polynomials, and thus be able to disengage the unknown quantity x from the known quan- tities. Having performed the multiplications, the equation be- comes, 3a^ — ax — Sab -{- bx -\- 2ax = Abx + 4a5 ; by transposing — ax -\- bx -{■ 2ax — 4bx =. 4ab + Sab — Sa^, by reducing ax — 35a; = 7ab — Sa^ ; or, (Art. 48), (a — Sb)x = lab — Sa^. Dividing both members by a — Sb, we find _ lab — 3a2 *■" a — Sb' 93. Hence, in order to resolve any equation of the first de- ^ee, we have the following general RULE. I. If the equation contains fractional terms, reduce it to one tn which all the terms shall be entire, and then transpose all the terma affected with the unknown quantity into the first member, and all the known terms into the second. H. Reduce to a single term all the terms involving the unKnowi> quantity : this term will be composed of two factors, one of whicJi will be the unknown quantity, and the other all its co-efficients con- nected by their respective signs. 111. Then divide both members of the equation by the multiplier of the unknoion quantity. 65 ELEMENTS OF ALGEBRA. [CHAP. IV. EXAMPLES. 1. Given 3a; — 2 + 24 = 31 to find x. Ans. x = 3. 2. Given a;+18 = 3a? — 5 to find x. Ans. x = 11|. 3. Given 6 - 2a; + 10 = 20 — 3x — 2 to find x. Ans. X = 2. 4. Given x-{-— x-{----x=:ll to find x. Ans. x = 6. 1 fi 5. Given 2x — x + 1 = 5x — 2 to find a;. Ans. x = — . 2 7 6. Given 3ax -\ 3 = bx — a to find x. e —3a Ans. X =: -. 6a — 2b -. /-• X -~ 3 , X ^„ a? — 19 ^ , 7. Given — f- — = 20 to find x. .i o 2 „^. a; + 3,a; ^ x — 5 ^, 8. Given — 1- — = 4 — to find x Ans. X = 23^. t. Ans. 0? = 3y3. „ _,. ax — b a bx bx — a „ . 9. Given — 1- — = — — to find x. 4 o ^ o Ans. X = 3a ~2b' 10. Given ; 4 =r f. to find x. c d *^ cdf + Acd Ans. X ■=. -^ — . 3aa — 2bc . , ^. Sax — b 3b — c ^ , ^ . 1 1 . Given — — = 4 —b, to find x. 56 -r 9i - 7c Ans. X = ir. n- ^ X — 2 _ X 13 - - 12. Given — f- _ = _, to find x. O o id o 16a Ans. X =. 10. (3. Given -\ = f, to find a:. a b c a -^ , abcdf Ans. X — -^ ocd — acd -(- adb — abc CHAP TV.] EQUATIONS OF THE FIRST DEGREE. 67 3^ 5 4^ 2 14. Given a — 1 — — = x -\- 1, to find a;. Ans. X = 6. 3C Qx cc ^— 3 15. Given — — — =: — 12^, to find re. 7 9 5 *^ ^ ^. 4x — 2 3x — 1 16. Given 2x r = — ^ — , to find x. Ans. « = ] 4 Ans. a; = 3- 17. Given Sx -\ — — x -\- a, to find x. Za + d 18. Find the value of x in the equation {a -{-b){x-b) 4ab — 62 a^ — bx 3a — 1 — 7 2x -\ - — . a — a -\- a* + 3a^ + 4a^^ — 6ab^ + 26* Ans. X = 2b (2a2 + ai — b"^) Questions producing Equations of the First Degree, i?ivolving hut one Unknown Quantity. 94. It has already been observed (Art. 81), that the solution of a problem by Algebra, consists of two distinct parts 1st. The statement; and 2d. The solution of the equation. We have already explained the methods of solving the equa- tion ; and it only remains to point out the best manner of making the statement. Tliis part cannot, like the second, be subjected to any w^ell- defined rule. Sometimes the enunciation of the problem furnishes the equation immediately ; and sometimes it is necessary to dis- cover, from the enunciation, new conditions from which an equa- tion may be formed. The conditions enunciated are called explicit conditions, and those which are deduced from them, implicit conditions. In almost all cases, however, we are enabled to discover the equation by applying the following ELEMENTS OF ALGEBRA. [CHAP. IV. RULE. Represent the unknown quantity by one of the final letters of the alphabet, and then indicate, by means of the algebraic signs, the same operations on the known and unknown quantities, as would verify the value of the unknown quantity, were such value known. QUESTIONS. 1. Find a number such, that the sum of one half, one third and one fourth of it, augmented by 45, shall be equal to 448. Let the required number be denoted by - - - - at . Then, one half of it will be denoted by - one third of it - - - - by one fourth of it - - - - by - And by the conditions, — - + — + — - + 45= 448, Now, by subtracting 45 from both members, ^ + ^ + ^^403. 2 ^ 3 ^ 4 By making the terms of the equation entire, we obtain 6a; + 4x + 3a; = 4836 ; or - - 13x=4836. 4836 Hence - a; = -— - = 372. Let us see if this value will verify the equation of the prob- lem. We have ?!^ + ^ + !Z! + 45 = 186 -f- 124 + 93 + 45 = 448. 2^3^4^ 2. What number is that whose third part exceeds its fourth. by 16. Let the required number be represented by x. Then — X =z the third part. 3 ^ — X =z the fourth part. 4 CHAP. IV.] EQUATIONS OF THE FIRST DEGREE. 69 And by the question -— x x = 16. or, - - - - 4a; — 3a; = 192. a; = 192. Veri/ication. 1|? _!!_-= 64 -48 =16. 3. Out of a cask of wine which had leaked away a third part, 21 gallons were afterward drawn, and the cask being then gauged, appeared to be half full : how much did it hold ? Suppose the cask to have held x gallons. X Then, — = what leaked away. And "5" + 21 = what leaked out, and what was drawn X 1 Hence, -^-f 21=— a; by the question. or 2x + 126 =: 3a:. or — X = — 126. or a; = 126, by changing the signs of both members, which does not destroy their equality. Verification. 126 126 ^- + 21 = 42 + 21 = = 63. 3 2 4. A fish was caught whose tail weighed 9?i. ; his head weighed as much as his tail and half his body, and his body weighed as much as his head and tail together : what was the weight of the fish? Let - - 2a; = the weight of the body. Then - - 9 + a; = weight of the head. And since the body weighed as much as both head and tail 2a? = 9 + 9 + a; or- - - 2a; — a; = 18 and - - - a; = 18. 70 ELEMENTS OF ALGEBRA. [CHAP. IV Verification. 2x =z 36 lb =: weight of the body. 9 -\- X z= 27 lb =z weight of the head. 9/5 = weight of the tail. Hence, 72/6 = weight of the fish. 5. A person engaged a workman for 48 days. For each day that he labored he received 24 cents, and for each day that he was idle, he paid 12 cents for his board. At the end of the 48 days, the account was settled, when the laborer received 504 cents. Required the iiumber of working days, and the number of days he was idle. If these two numbers were known, by multiplying them respec- tively by 24 and 12, then subtracting the last product from the first, the result would be 504. Let us indicate these operations by means of algebraic signs. Let - - a; = the number of working days. Then 48 — a; = the number of idle days. 24 X a: = the amount earned, and 12(48 — a?) = the amount paid for his board. Then 24a? — 12 (48 — x) = 504 what he received. or 24a; — 576 + 12a; = 504. or 36a; = 504 -|- 576 = 1080 1080 , , . , and X = — ^TT- = 30 the working days. 3b whence, 48 — 30 = 18 the idle days. Verification. Thirty day's labor, at 24 cents a day, amounts to 30x24 = 720 cts. And 18 days' board, at 12 cents a day, amounts to 18x12 = 216 cts. And the amount received is their difference 504. CHAP. IV.] EQUATIONS OF THE FIRST DEGREE. 71 General Solution. the whole number of working and idle days, the amount received for each day he worked, the amount paid for his board, for each idle day. the balance due, or the result of the account, the number of working days, the number of idle days, what he earned ; the amount deducted for board. The equation of the problem will then be, ax — i (« — x) =z c whence ax — bn -\- bx z=. c [a -\- b) X =: c -\- bn c -\- bn ^ , ^.. an -\- bn — c — bn and consequently, n — a; = n — L,e1 t, n = a =z b = c = X = n — X =: Then > ax z=: and, b{n- -X) = or a + b c + bn a + b an \ — c + b u-\-b 6. A fox, pursued by a greyhound, has a start of 60 leaps. He makes 9 leaps while the greyhound makes but 6 ; but '6 leaps of the greyhound are equivalent to 7 of the fox. How many leaps must the greyhound make to overtake the fox ? From the enunciation, it is evident that the distance to be passed over by the greyhound, is equal to the 60 leaps of the fox, plus the distance which the fox runs after the gi'eyhound starts in pursuit. Let X = the number of leaps made by the greyhound from the time of starting till he overtakes the fox. Now, since the fox makes 9 leaps while the greyhound makes 3 6, the fox will make 1 ^, or — leaps while the greyhound makes 1 ; and, therefore, while the greyhound makes x leaps, the 3 {oji will make — x leaps. Hence, 3 60 + ^* = 72 ELEMENTS OF ALGEBRA. [CHAP. IV the number of leaps made by the fox, in passing over the entire distance. It might, at first, be supposed that the equation of the problem would be obtained by placing this number equal to x; but in doing so, a manifest error would be committed ; for the leaps of lie greyhound arps greater than those of the fox, and we should ihus equate nmnhers referred to different units. Hence, it is ne- cessary to express the leaps of the fox by means of those of the greyhoimd, or reciprocally. Now, according to the enunciation, 3 leaps of the greyhound are eq-iivalent to 7 leaps of the fox ; and hence, 1 leap of the 7 greyhound s equivalent to — leaps of the fox ; consequently, 7x X leaps of the gre) ^'•ound are equivalent to — of the fox : that is, had the leaps of the grt\^"nund been no longer than fhose of the fox, he would have made — le^.^.? instead of x leaps. 7x 3 Hence the true equation is, — ,=z 60 -| x ; or, by making the terms entire 14a; = 360 + 9a;, whence - ' - - - - 5a; =: 360 and x = 72. Therefore, the greyhound will make 72 leaps to overtake the fox, 3 and during this time the fox will make 72 x — =108. Verification. The 72 leaps of the greyhound are equivalent to 72 X 7 ■ — = 168 leaps of the fox = the whole distance. And 60 + 108 = 168, the leaps which the fox made from the beginning. 7. A can do a piece of work alone in 10 days, and B in ^3 days : in what time can they do it if they work together ? Denote the time by x, and the work to be done by 1. Then m I day A could do — of the work, and B could do — of ';HAP. IV.] EQUATIONS OF THE FIRST DEGREE. 73 X it : and in x days A could do — of the work, and B •^ 10 13 hence, by the conditions of the question, - + --1 10^ 13 -' wliich gives 13a; + lOx = 130: 130 bence, 23a; = 130, x = — ;— = 5^| days. 8. Divide $1000 between A, B, and C, so that A shall have $72 more than B, and C $100 more than A. Ans. A's share = $324, B's = $252, C's = $424. 9. A and B play together at cards. A sits down with $84 and B with $48. Each loses and wins in turn, when it ap- pears that A has five times as much as B. How much did A win? Ans. $26. 10. A person dying leaves half of his property to his wife, one sixth to each of two daughters, one twelfth to a servant, and the remaining $600 to the poor : what was the amount of his prop- erty ? Ans. $7200-. 11. A father leaves his property, amounting to $2520, to four sons, A, B, C, and D. C is to have $360, B as much as C and D together, and A twice as much as B less $1000: how much does A, B, and D, receive ? Ans. A $760, B $880, D $520. 12. An estate of $7500 is to be divided between a widow, two sons, and three daughters, so that each son shall receive twice as much as each daughter, and the widow herself $500 more than all the children : what was her share, and what the shai-e of each child? r Widow's share $4000. Ans. ) Each son $1000. ( Each daughter $500. 13. A company of 180 persons consists of men, women, and children. The men are 8 more in number than the women, and the children 20 more than the men and women together : how many oi each sort in the company ? Ans. 44 men, 36 women, 100 children. 74 ELEMENTS OF ALGEBRA. [CHAP. IV 14. A father divides $2000 among five sons, so that each elder should receive $40 more than his next younger brother : what is the share of the youngest ? Ans. $320. 15. A purse of $2850 is to be divided among three persons, A, B, and C; A's share is to be to B's as 6 to 11, and C is to have $300 more than A and B together : what is each oiie'.s share? Ans. A's $450, B's $825, C's $1575. 16. Two pedestrians start from the same point; the first steps twice as far as the second, but the second makes 5 steps while the first makes but one. At the end of a certain time they are 300 feet apart. Now, allowing each of the longer paces to be 3 feet, how far will each have travelled ? Ans. 1st, 200 feet; 2d, 500. 17. Two carpenters, 24 journeymen, and 8 apprentices, re- ceived at the end of a certain time $144. The carpenters received $1 per day, each jour.ieyman half a dollar, and each apprentice 25 cents : how many days were they employed ? Ans. 9 days. 18. A capitalist receives a yearly income of $2940: four fifths of his money bears an interest o*" 4 per cent., and the remainder of 5 per cent.: how much has he at interest? Ans. $70000. 19. A cistern containing 60 gallons of water has three unequal cocks for discharging i' , the largest will empty it in one hour, the second in two hoiirs, and the third in three : in what time will the cistern be emptied }f they all run together ? Ans. 32^ min. 20. In a fi^rtam orchard ^ are apple-trees, \ peach-trees, \ plum-treos, . 20 cherry-trees, and 80 pear-trees : how many trees in the orchard^ A as. 2400. 21. A farmer being an kid how many sheep he had, answen;d that he had them in five fields; in the 1st he had \, in the 2d 1, in the 3d i, in the 4th ^^, and in the 5th 450: how many had he ? Ans. 1200. 22. My horse and saddle together are wor'h $132, and tlie horse is worth ten times as much as the saddle : what is the value of the horse? Ans. $120. CHAP. IV.] EQUATIONS OF THE FIRST DEGREE. 75 23. The rent of an estate is this year 8 per cent, greater than it was last. This year it is $1890: what was it last year? Ans. $1750. 24. What number is that from which, if 5 be subtracted, f of the remainder will be 40 ? Ans. 65. 25. A post is 1 in the mud, ^ in the water, and ten feet above the water : what is the whole length of the post 1 Ans. 24 feet. 26. After paying I and i of my money, I had 66 guineas left in my purse : how many guineas were in it at first ? Ans. 120. 27. A person was desirous of giving 3 pence apiece to some beggars, but found he had not money enough in his pocket by 8 pence ; he therefore gave them each two pence and had 3 pence remaining: required the number of beggars. Ans. 11. 28. A person in play lost A of his money, and then won 3 shillings ; after which he lost i of what he then had ; and this done, found that he had but 12 shillings remaining : what had he at first ? Ans. 20s. 29. Two persons, A and B, lay out equal sums of money in trade; A gains $126, and B loses $87, and A's money is now double of B's : what did each lay out ? Ans... $300. 30. A farmer bought a basket of eggs, and offered them at 7 cents a dozen. But before he sold any, 5 dozen were broken by a careless boy, for which he was paid. He then sold the re- mainder at 8 cents a dozen, and received as much as he would have got for the whole at the first price. How many eggs had he in his basket ? Ans. 40 dozen. 31. A person goes to a tavern with a certain sum of money in his pocket, where he spends 2 shillings ; he then borrows as much money as he had left and going to another tavern, he there spends 2 shillings also ; then borrowing again as much money as was left, he went to a third tavern, where likewise he spent 2 shillings and borrowed as much as he had left ; and again spending 2 shillings at a fourth tavern, he then had nothing remaining. What had he at first ? Ans. 3s. 9d 76 ELEMENTS OF ALGEBRA. [CHAP. IV. Of Equations of the First Degree, involving two or more Unknown Quantities. 95. Although several of the previous questions contained in their enunciation more than one unknown quantity, we have neverthe- less resolved them all by employing but one symbol. The rea- son of this is, that we have been able, from the conditions of the enunciation, to represent the other unknown quantities by means of this symbol and known quantities ; but this cannot be done in all problems containing more than one unknown quantity. To explain the methods of resolving problems of this kind, let us take some of those which have been resolved by means of one unknown quantity. 1. Given the sum of two numbers equal to a, and their differ ence equal to h ; it is required to find the numbers. Let X = the greater, and y the less number Then by the conditions x -\- y = a ; and a; — y =: b. By adding (Art. 86, Ax. 1), 2x — a -{- b. By subtracting ( Irt. 86, Ax. 2), 2y = a — b. Each of these equations contains but one unknown quantity From the first we obtain x And from the second y 2 a-h Verification, a -\- b a — b 2a a -\- b a — b 2b ~~2 2~ ~ T ~ ^ ' ^^ ~2 2~ ~ '2~ ' 2. A person engaged a workman a number of days, denoted by n. For each day that he labored he was to receive a cents, and for each day that he was idle he was to pay b cents for his board. At the end of the n days, the account was settled, when the laborer received c cents. Required the number of working days and the number of days he was idle. Let X = the number of working days. y = the number of idle days. Then, ax = what he earned, and by — what h-e paid for his board ; CHAP. IV.] EQUATIONS OF THE FIRST DEGREE. 77 and by the question, we have < X -\- y = n ax — by = c. It has already been shown that the two members of an equa- tion can he multiplied by the same number, without destroying the equality ; therefore, multiply both members of the first equa- tion by b, the co-efiicient of y in the second, and we have the equation - - - - - bx -j- by = bn, which, added to the second - ax — by = e, gives - - - - - - ax -\- bx = bn -{- c. Whence ------ x = ■ — - a -{- In like manner, multiplying the two members of the first equa- tion by a, the co-efficient of x in the second, it becomes ax -{- ay = an; from which, subtract the second equation, ax — by = c, and we obtain - - - - - - - ay -\- by =^ an — c. Whence -------- « = -. ^ a + 6 By introducing a symbol to represent each of the unknown quantities of the problem, the above solution has the advantage of making known the two required numbers, independently of each other. What will be the numerical values of x and y, if we suppose n = 48, a = 24, b = 12, and c = 504. Elimination. 96. The method which has just been explained, of combining two equations, involving two unknown quantities, and deducing therefrom a single equation involving but one, may be extended to three, four, or any number of equations, and is called Elimtnc^ tion. There are three principal methods of elimination : 1st. By addition and subtraction. 2d. By substitution. 3d. By comparison. We shall discuss these methods separately. 78 ELEMENTS OF ALGEBRA. [CHAP. IV Elimination hy Addition and Subtraction. 97. Before considering the case of Elimination, we will ex plain a new notation which is about to be used. It often happens, in Algebra, that some of the known quantities of an equation or problem, though entirely independent of each other in regard to their values, have, nevertheless, certain rela- tions which it is desirable to preserve in the discussion. In such case, the second quantity is represented by the same letter, with a small mark over it. Thus, if the first quantity was denoted by a, the second would be denoted by a', and is read, a prime If there were a third, it would be denoted by a'^, and read, a second, &c. Let us now take the two equations, ax -\- by = c a'x -\- h'y = \ ab — ab or, if we wish the value for y to have the same denominator with that for x, we change the signs of the numerator and de- nominator, and write ac' — a'c y^ ab' - a'b ^ The method of elimination just explained, is called the methoa by addition and subtraction, because the unknown quantities dis- appear by additions and subtractions, after having prepared the equations in such a manner that the same unknown quantity shall have the same co-efficient in both equations. b'c — be' ac' — afc The formulas x = — — ■, y = ab'-a'V " ab'-a'b deduced from the equations ax -\- by = c a'x + b'y = d will enable us to write the values of x and y immediately, with- out the trouble of elimination. They contain the germe of a gen- eral rule, not before given, for the solution of all similar equations. RULE. I. The first term in the numerator for the value of x, is foimd by beginning at h' and crossing up to c — giving b'c ; the second term is found by crossing from b to c' — giving he'. II. For the first term in the numerator of the value for y, begin at a and cross down to c' — giving ac' ; and for the second term, cross from df to c — giving a'c. III. The first term of the common denominator is found by cross- ing from a to h' — giving ab' ; and the second, by crossing from a' to b — giving a'b. The manner of obtaining these formulas will be easily remem- bered, and their applications will be found very simple. 1. What are the values of x and y in the equations, 5x + 7y = 43 Ua; -f % = 69. 80 ELEMENTS OF ALGEBRA. [CHAP. IV = 3 We write immediately, _ 9 X 43 — 7 X 69 _ 387 — 483 _ — 96 * ~ 5 X 9- 11 X 7 ^ 45-77 ~ - 32 _ 5 X 69 - 11 X 43 _ 345 - 473 _ - 128 _ y ~ — 32 ~ — 32 "~ — 32 "" 2. What are the vakies of x and y in the equations, 3a; - 4- = 14 4 X — 4y = — 11. We write _ -4 X 14-(-l X -11) _ -56-V _ ^ *~ 3x-4-lX-i ~-12+i~ 3 X — 11 —1 X 14 —33-14 ^ y =. r= =: 4. ^ -12 + i -12+^ Elimination by Substitution. 98. Let us take the two equations 5a;+7y = 43 and lla;+9y=69. Find the value of a; in the first equation, which gives 43 - 7y X = 5 Substitute this value of x in the second equation, and we have 43 7y 11 X -— ^ + 9y = 69. 5 or 473 — 77y + 45y = 345 : or — 32y = — 128. Hence y = 4. 43 - 28 „ And X = = 3. 5 This method, called the method by substitution, consists in find- ing in one equation the value of one of the unknown quantities, as if the others were already determined, and then substituting this value in the other equations. In this way, new equations are formed from which one of the unlmown quantities has been eliminated. We then operate in a similar manner, on the new equations. CHAP. IV.] EQUATIONfS OF THE FIItST DF.GREE. SI Elimination htj Comjiarison. 99. Let us take the two equations, 5a; + 7y = 43 and 11a: + 9y = 69. Finding the value of x in the first equation, we have 43 — 7v/ X — 5 And finding; the value of x in the second, we obtain 69 — 9y * = -n-- Let these two values of a; be placed equal to each other, and 43 — 7y 69 — 9y we have, = r^ ; o 11 or, 473 — 11 y !^ 345 — 45y ; or, - 32y = - 128. Hence, y = 4 69 — 36 and, X - 11 This method of elimination is called the method by compari- son, and consists in finding the value of the same unlinown quantity in all the equations, and then placing those values equal to each other, two and two. This will give rise to a new set of equations containing one less unknown quantity, and upon whir'i we operate as on the given equations. The new equations v/hich arise, in the last two methods of elimination, contain fractional terms. This inconvenience is avoid- ed in the first method. The metJtod hy substitution is, however, advantageously employed whenever the co-eftieient of either of the unknown quantities in one of the equations is equal to unity, because then the inconvenience of which we have just spoken does not occur. We shall sometimes have occasion to employ this method, but generally the method by addition and subtraction is preferable. When the co-efficients are nut too great, we c- find .x = Substituting for x its value in equation (2), it becomes, 4m — 6 ^ 30, whence - - m = And substituting for y its value in equation (3), there results ------ z ■=. Of 'indeterminate Problems. 103. In all the preceding reasoning, we have supposed the number of equations equal to the number of unknown quantities This must be the case in every problem, in order that it may be determinate ; that is, in order that it may admit of a finite num- ber of solutions. Let it be required, for example, to find two quantities such, that five times one of them, diminished by three times the other, shall be equal to 12. 84 ELEMENTS OF ALOEBRA. [CHAP. IV. If we denote the quantities sought by x and y, we shall have the equation 5a; — 3y = 12, V . 12 + 3y whence, x = — -. 5 Now, by making successively, y =: 1, 2, 3, 4, 5, 6, &c., o 18 21 24 27 ^ „ there results, rr = 3, — , — , — , — 6, &c., ' ' 5 5' 5 5 and any two corresponding values of x, y, being substituted in the given equation, 5a; — 3y = 12 will satisfy it equally well : hence, there are an infinite number of values for x and y which will satisfy the equation, and conse- quently, the problem is indr terminate ; that is, it admits of an in- finite number of solutions. If, however, we impose a second condition, as for example, that the sum of the two (juaiitities sliall be equal to 4, we shall have a second equation, a; 4- y = 4 ; and this, combined with the equation already considered, will give determinate values for x and y. If we have two equations, involving three unknown quantities, we can eliminate one of the unknown quantities, and thus ob- tain an equation containing two unknown quantities. This equa- tion, like the preceding, woidd be satisfied by an infinite num- ber of values, attributed in succession, to the unknown quanti- ties. Since each equation expresses one condition of a problem, therefore, in order that a prohlcm may be detenninate, its enun- nation must contain at least as many different conditions as there are iinknotim quantities , and these conditions must be such, that each of them may be expressed by an independent equation ; that ts, art equation not produced by any combination of the others of the system. If, on the contrary, the number of independent equations ex- ceeds the number of utiknown quantities involved in them, the conditions which they express cannot be fulfilled. CHAl'. IV. J EQUATIONS OF THE FIRST DEGREE. 86 For example, let it be required to find two numbers such thai their sum shall be 100, their difference 80, and their product 700. The equations expressing these conditions are, a: + y = 100 X — y = 80 and X X y = 700. Now, the first two equations determine the values of x and y, viz., X = Q0 and y =z 10. The product of the two numbers is therefore known, and equal to 900. Hence, the third condition cannot be fulfilled. Had the product been placed equal to 900, all the conditions would have been satisfied, in which case, however, the third would not have been an independent equation, since the condition ex- pressed by it, is implied in the other two. EXAMPLES. 1. Given 2jr + 3^=16, and 3x— 2y=:ll to find the values of X and y. Ans. a; = 5, y = 2. 2x 37/ 9 , Sx 2y 61 ^ , , 2. Given — ^ = — , and — ~ z= — - to find the 5 4 20 4 5 120 values of x and y. A 1 1 Ans. X =. — , y =^ — . 2' -^ 3 3. Given -— + 7y = 99, and -^ + 7x = 51 to find the values of T and y. A71S. X = 7, y 14. ^. X V , X -{- y X 2// — X 4. Given 12=^1- + 8, and -— -i H 8 = -^ h 27 2 4 5 3 4 to find the values of x and y. Ans. x = 60, y = 40. X -f y + z = 29 X -{- 2y + 3^ =r 62 ft. Given < 1 1 1 > to find X, y, and 2 Ans. a; = 8, y =: 9, ^ = 12 m ELEMENTS OF ALGEBRA. [CHAP. IV, 2a; + 4y — 3z = 22 1 6 Given -^ 4a; — 2y + 5z — 18 ^ to find x, y, and z _ 6a; + 7y — ;? :^ 63 J Aw5. a; = 3, y = 7, ^ = 4. 7. Given -: 111,. -3-^ + T^ + T" = '' 1 1 1 =■ to find X, y, and 2. 8. Given -! Ans. a; = 12, y = 20, 2 = 30 7a; — 2;^ + 3m = 17 4y — 20 + t= 11 5y — 3a; — 2ii ■■= 8 4y — 3i« + 2t — 9 3z + 8m = 33 Ans. a; = 2, y = 4, ^ = 3, m = 3, t=-l. > to find a;, y, z, «, and i. QUESTIONS. 1. What fraction is that, to the numerator of which, if 1 be added, its value will be one third, but if one be added to its de nominator, its value will be one fourth. Let the fraction be represented by — . y a; + 1 1 X 1 Then, by the question = -— and — -— = — -. y 3 y + 1 4 Whence 3a; + 3 = y, and 4a; = y + 1 . Therefore, by subtracting, a? — 3=1 or a;=:4; and 3 X 4 + 3 =- 15 = y. 2. A market woman bought a certain number of eggs at 2 for a penny, and as many more, at 3 for a penny, and having sold them again altogether, at the rate of 5 for 2d., found that she had lost 4J. : how many eggs had she ? CHAP. IV.] EQUATIONS OF THE FIRST DEGREE. 87 Let 2x = the whole number of eggs ; then X = the number of eggs of each sort ; and - X := the cost of the first sort ; 2 and - a; = the cost of the second sort ; But 5 : 2a; : : 2 : ^ ; 5 4x hence, — the amount for which the eggs were sold. 5 Hence, by the question, 1 1 4x ■* —x-\ X = 4: 2^3 5 therefore 15a; + lOx — 24x = 120. Or, X = 120 the number of eggs of each sort. 3. A person possessed a capital of 30,000 dollars, for which 'la: drew a certain interest per annum ; but he owed the sum o! 20,000 dollars, for which he paid a certain interest. The inter- est that he received exceeded that which he paid by 800 dollars. Another person possessed $35,000, for which he received interest at the second of the above rates ; but he owed 24,000 dollars, for which he paid interest at the first of the above rates. Tlu' interest that he received exceeded that which he paid by 310 dollars. Required the two rates of interest. Let X and y denote the two rates of interest: that is, the in- terest of SI 00 for one year. To obtain the interest of $30,000 at the first rate, denoted by x, we form the proportion 30,000:r 100 : a; : : 30,000 : : — ——- or 300a;. And for the interest $20,000, the rate being y, 20,000y 100 : y : : 20,000 : : — ——^ or 200y. •^ 100 But from the enunciation, the difference between these two in lerests is equal to 800 dollars. We have, then, for the first equation of the problem. 300a: — 200i/ = 800. SS ELEMENTS OF ALGEBRA. [CHAP. IV. By expressing the second condition of the problem algebraically, we obtain the other equation, 350y — 240,x' = 310. Both members of the first e at the end of the re- and b -\- X = the age of the son ) quired time. Hence, by the question a -{- X a — 4h — : — = b -\- x: whence, x = . 4 ' 3 o =.^ 1 I n .1, 54 - 36 18 buppose a — 54, and 6 = 9; then x = = — = (J. " 3 3 06 ELEMENTS OF ALGEBRA. [CHAP. IV. The father being 54 years old, and the son 9, in 6 years the father will be 60 years old, and his son 15; now 15 is the tDurili of 60 ; hence, a; = 6 satisfies the enunciation. Let us now suppose a = 45, and i =: 15 ; 45 — 60 then, X = = — 5. If we substitute this value of x in the equation of condition, b -\- X, 4 -'^ ' we obtain, and '^^ - ' - 15 4 -^^ 10 = 10, Hence, — 5 substituted for x, verifies the equation, and therefore js a true answer. Now, the positive result which was obtained, shows that the age of the father will be four times that of the son at the ex- pirn,tioii of 6 years from the time when their ages were con- sidered ; while the negative result, indicates that the age of the father was four times that of his son, 5 years previous to the time when their ages were compared. The question, taken in its general, or algebraic sense, demands the time, at which the age of the father was four times that of the son. In stating it, we supposed that the age of the father was to be augmented ; and so it was, by the first supposition. But the conditions imposed by the second supposition, required the age of the father to be diminished, and the algebraic result con- formed to this condition, by appearing with a negative sign. If we wished the result, under the second supposition, to have a positive sign, we might alter the enunciation by demanding, how many years since the age of the father was four times that of the son. U X = the number of years, we shall have a — X , . 4b — a =: 6 — X : hence, ,t = — . 4 3 If a = 45 and h — 15, x will be equal to 5. Reasoning from analogy, we establish the following general principles, CHAP. IV.] EQUATIONS OF THE FIRST DEGREE. 97 Ist. Every negative value found for the unknown qua7itity in a problem of the first degree, will, when taken with its proper sign, verify the equation from which it was derived. 2d. That this negative value, taken with its proper sig7i, will also satisfy the enunciation of the problem, understood in its algebraic sense. 3(1. The negative result shows that the enunciation is impossible legarded in its arithmetical sense. The language of Algebra de- tects the error of the arithmetical enunciation, and hidicates the gen- eral relation of the quantities. 4th. The negative result, considered without reference to its sign, may be regarded as the answer to a problem of which the enuncia- tion only differs from the one proposed in this : that certain quan- tities which were additive have become subtractive, and reciprocally 106. As a further ilhistration of the change which an alge- braic sign may produce in the enunciation of a problem, let us resume that of the laborer (page 76). Under the supposition that the laborer receives a sum c, we have the equations X -\- y :=z n^ , hn -{- c an — c ^ \ whence, x= — -, y~ -— . ax — by ^ c ) a -\- b a -\- b If at the end of the time, the laborer, instead of receiving a sum c, owed for his board a sum equal to c, then, by would be greater than ax, and under this supposition, we should have the equations a? + y == ra and ax — by = — c. Now, it is plain that we can obtain immediately the values of X and y, in the last equations, by merely changing the sign of c in each of the values found from the equations above ; this gives In — c an -\- e The results for both enunciations, may be compreiiended in the same formulas, by writing bn ± c an ^ c X = r ; y =: — —r- a + b -^ a + b The double sign ±, is read plus or minus, and q:, is read, rni nus or plus. The upper signs correspond to the case in wliich 98 ELEMENTS OF ALGEBRA. [CHAP. IV the laboier received, and the lower signs, to the case in which he owed a sum c. These formulas also comprehend the case in which, in a settlement between the laborer and his employer, their accounts balance. This supposes c = 0, which gives bn an ^ = 7+T' ^ = 7+1' Discussion of Problems. Explmiat'ion of the terms Nothing a7id lujinity. 107. When a problem has been resolved generally, that is, by means of letters and signs, it is often required to determine what the values of the unknown quantities become, when particular sup- positions are made upon the quantities which are given. The determination of these values, and the interpretation of the pe- culiar results obtained, form what is called the discussion of the problem. The discussion of the following question presents nearly all the circumstances which are met with in problems of the first degree. 108. Two couriers are travelling along the same right line and in the same direction from R^ toward R. The number of miles travelled by one of them per hour is expressed by m, and the number of miles travelled by the other per hour, is expressed by n. Now, at a given time, say 12 o'clock, the distance be- tween them is equal to a number of miles expressed by a : re- quired the time when they will be together. R^ A B R. At 12 o'clock suppose the forward courier to be at B, the other at A, and R to be the point at which they will be together. Then, AB = a, their distance apart at 12 o'clock. Let t = the number of hours which must elapse, be- fore they come together ; and X = the distance BR, which is to be passed over by the forward courier. Then, since the rate per hour, multiplied by the number of hours will give the distance passed over by each, we have, t X m =z a -\- X ~ AR t X n zz: ic — BR ■JHAP. IV.] EQUATIONS OF THE FIRST DEGREE. 99 Hence by subtracting, t (m — n) =z a, and hence, t = . m — n Now, so long as m '^ n, t will be positive, and the problem will be solved in the arithmetical sense of the enunciation. For if 771 > 71, the courier from A will travel faster than the courier from B, and will therefore be continually gaining on him : the interval which separates them will diminish more and more, un- til it becomes 0, and then the couriers will be found upon the same point of the line. In this case, the time t, which elapses, must be added to 12 o'clock, to obtain the time when they are together. But, if we suppose m 20 and 18 > 10, or 6<8 and 7 < 9, are inequalities which subsist in the same sense ; and the in- equalities 15 > 13 and 12 < 14, subsist in a contrary sense. \. If we add the same quantity to both members of un inequality, or sxibtract the same quantity from both members, the resulting in- equality will subsist in the same sense. Thus, take 8 > 6 ; by adding 5, we still have 8 + 5>6+5; and subtracting 5, we have 8 — 5 > 6 — 5. When the two members of an inequality are both negative, that one is the least, algebraically considered, which contains the great- est number of units. Thus, — 25 < — 20 ; and if 30 be added to both members, we have 5 <| 10. This must be understood entirely in an algebraic sense, and arises from the convention be- fore established, to consider all quantities preceded by the minus sign, as subtractive. The principle first enunciated, serves to transpose certain terms from one member of the inequality to the other. Take, for ex- ample, the inequality a2 + Z,3 > 3b^^ — 2a2 ; there will result, by transposing, a3 + 2a2 > 3/;2 _ i2^ or 3a2 > 2J2. 2. Jf two inequalities subsist in the same sense, and we add them member to member, the resulting inequality will also subsist in the same sense. Thus, add o !> ^, c > J, e y> f: and there results a-\-c-{-e^b-{-d-\-f. But this is not always the case, when we subtract, member from member, two inequalities established in the same sense. Let there be the two inequalities 4 < 7 and 2 < 3, we have 4 — 2 or 2 < 7 — 3 or 4. 106 ELEMENTS OF ALGEBRA. [CHAP. IV. But if we have the inequalities 9 < 10 and 6 < 8, by sub- tracting we have 9 — 6 or 3 > 10 — 8 or 2. We should then avoid this transformation as much as possible, or if we employ it, determine in which sense the resulting in- equality exists. 3. If the two members of an inequality he multiplied hy a positive number, the resulting inequality will exist in the same sense. Thus, a < ^, will give 3a < 3J ; and, — a < — i, — 3a < — 35. This principle serves to make the denominators disappear, „ , . ,. a"^ — h"^ c^ — d^ T , , , From the mequality — > - — , we deduce, by mul- 2a 6a tiplying by 6a^, 3a (a2 - i2) > 2d (c^ - J2), and the same principle is true for division. But, when the two members of an inequality are multiplied oi divided hy a negative number, the inequality will subsist in a con trary sense. Take, for example, 8 > 7 ; multiplying by — 3, we have — 24 < — 21. Q R 7 In like manner, 8 > 7 gives — — , or — < — . Therefore, when the two members of an inequality are multi- plied or divided by a number expressed algebraically, it is ne- cessary to ascertain whether the multiplier or divisor is negative ; for, in that case, the inequality will exist in a contrary sense. 4. It is not permitted to change the signs of the two members of an inequality, unless we establish the resulting inequality in a con- trary sense ; for this transformation is evidently the same as mul- tiplying the two members by — 1. 5. Both members of an inequality between positive numbers can be squared, and the inequality will exist in the same sense. Thus, from 5 > 3, we deduce, 25 > 9 ; from a + i > c, Ave find (a + bf > c2. CHAP. IV.] OF INEQUALITIES 107 6. When the signs of both members of the irieqiiality are not hiown, we cannot tell hfore the operation is performed, in which sense the resulting inequalitij will exist. For example, — 2<3 gives (—2)2 or 4<9; but 3>— 5 gives, on the contrary, (3)^ or 9 < (— 5)^ or 25. We must, then, before squaring, ascertain the signs of the tv.o members. EXAMPLES. 1. Find the limit of the value of x in the expression 5a; — 6 > 19. Ans. a; > 5 2. Find the limit of the value of x in the expression 14 3a; H a; — 30 > 10. Ans. a? > 4. 3. Find the limit of the value of x in the expression 1 1 a; 13 17 , -— X X -\ f > — . Ans. a; > o. 6 322'^2 '^ 4. Find the limit of the value of x in the inequalities era; d^ y — oa; + «6 < y. 5. The double of a number diminished by 5 is greater than 25, and triple the number diminished by 7, is less than double the number increased by 13. Required a number which shall satisfy the conditions. By the question, we have 2a; — 5 > 25. 3a; — 7 < 2a; + 13. Resolving these inequalities, we have a; > 15 and x < 20. Any number, therefore, either entire or fractional, comprised be- tween 15 and 20, will satisfy the conditions. JOS ELEMENTS OF ALGEBRA. [CHAP. * CHAPTER V. EXTRACTION OF THE SQUARE ROOT OF NUMBERS. FORMATION OF THE SQUARE AND EXTRACTION OF THE SQUARE ROOT OF ALGE- BRAIC QUANTITIES. CALCULUS OF RADICALS OF THE SECOND DEGREE. 116. The square or second power of a number, is the product which arises from muhiplying that number by itself once : for example, 49 is the square of 7, and 144 is the square of 12. The square root of a number, is that number which multiplied by itself once will produce the given number. Thus, 7 is the square root of 49, and 12 the square root of 144 : for, 7x7 = 49, and 12 X 12 = 144. The square of a number, either entire or fractional, is easily found, being always obtained by multiplying the number by itself once. The extraction of the square root is, however, attended with some difficulty, and requires particular explanation. The first ten numbers are, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and their squares, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100: 'and reciprocally, the numbers of the first line are the square roots of the corresponding numbers of the second. We may also re- mark that, the square of a number expressed by a single figure, will contain no figure of a higher denomination than tens. The numbers of the last line, 1, 4, 9, 16, &c., and all other numbers which can be produced by the multiplication of a num- ber by itself, are called perfect squares. It is obvious, that there are but nine perfect squares among all the numbers which can be expressed by one or two figures : the square roots of all other numbers expressed by one or two figures CHAP, v.] EXTRACTION OF THE SQUARE ROOT OF NUMBERS. 109 will be found betv/een two whole numbers differing from each other by unity. Thus, the square root of 55, comprised between the perfect squares 49 and 64, is greater than 7 and less than 8. Also, the square root of 91, comprised between the perfect squares 81 and 100, is greater than 9 and less than 10. Every number may be regarded as made up of a certain num- ber of tens and a certain number of units. Thus 64 is made up of 6 tens and 4 units, and may be expressed under the form 60 + 4 = 64. Now, if we represent the tens by a and the units by 6, we shall have a-\-h =60 + 4, and (a + bf = (60 + 4)2, and consequently, a2 + 2ab + b^ = (60)2 _^ 2 X 60 X 4 + (4)2 = 4096. Hence, the square of a number composed of tens and units con- tains, the square of the tens, plus twice the product of the tens by the units, plus the square of the units. 117. If now, we make the units 1, 2, 3, 4, &c., tens, by an- nexing to each a cipher, we shall have, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100; and for their squares, 100, 400, 900, 1600, 2500, 3600, 4900, 6400, 8100, 10000, from which we see that the square of one ten is 1 00, the square of two tens, 400, &c. : and hence, the square of tens will contain no figure of a less denomination than hundreds, nor of a higher name than thousands. Let us now take any number, as 78, and square it. We have 78 z:^ 70 + 8 ; that is, equal to 7 tens, or 70, plus 8 units. Seven tens, or 70 squared - - (70)2 _ 4900 twice the tens by the units is, 2 X 70 x 8 = 1120 square of the units is, - - - (8)^ = 64 lience, (78)2 — G084. Let us now reverse this process anc? ftad the square root of 6084. 110 ELEMENTS OF ALGEBRA. [CHAP. V 60 84 1 78 49 1184 1184 Since this number is composed of more than two places of figures, its roots will contain more than one. 60 84 But since it is less than 10000, which is the square of 100, the root will contain but two figures ; that is, units and tens. Now, the square of the tens must be found in the two left- hand figures which we will separate from the other two, by placing a point over the the place of units, and another over the place of hundreds. These parts, of two figures each, are called periods. The part 60 is comprised between the two squares 49 and 64, of which the roots are 7 and 8 : hence, 7 is the figure of the tens sought ; and the required root is composed of 7 tens and a certain number of units. The figure 7 being found, we write it on the right of the given number, from which we separate it by a vertical line : then we 7x2 = 148 subtract its square 49 from 60, which leaves a remainder of 11, to which we bring down the two next figures 84. The result of this operation is 1184, and this number is made up of twice the product of the tens by the units plus the square of the units. But since tens multiplied by units cannot give a product of a less name than tens, it follows that the last figure 4 can form no part of the double product of the tens by the units : this double product is therefore found in the part 118. Nov/, if we double the tens, which gives 14, and then divide 118 by 14, the quotient 8 is the units' figure of the root, or a figure greater than the units' figure. This quotient figure can never be too small, since the part 118 will be at least equal to twice the product of the tens by the units : but it may be too large ; for, the 118 besides the double product of the tens by the units, may likewise contain tens arising from the square of the units. To ascertain if the quotient 8 expresses the units, we write the 8 to the right of the 14, which gives 148, and then Ave multiply 148 by 8. Thus, we evidently form, 1st, the square of tlie units, and 2d, the double product of the tens by the units. This mul- tiplication being effected, gives for a product USl, a number equal CIIVP. v.] EXTRACTION OF THE SQUARE ROOT OF NUMBERS. Ill to the result of the first operation. Having subtracted the prod- uct, Ave find the remainder equal to : hence 78 is the root required. Indeed, in the operations, we have merely subtracted from the given number 6084, 1st, the square of 7 tens or of 70; 2d, twice ihe product of 70 by 8 ; and 3d, the square of 8 : that is, the ihree parts which enter into the composition of the square of 78. Remark. — The operations in the last example have been per- formed on but two periods. It is plain, however, that the same reasoning is equally applicable to larger numbers ; for, by chan- ging the order of the units, we do not change the relation in which they stand to each other. Thus, in the number GO 84 95, the two periods 60 84, have the same relation to each other, as in the number 6084 ; and hence, the methods pursued in the last example are equally ap- plicable to larger numbers. Hence, for the extraction of the square root of numbers, we have the following RULE. I. Separate the given number into periods of two figures each, be- ginning at the right hand : the period on the left will often contain but one figure. II. Find the greatest square in the first period on the left, and place its root on the right after the manner of a quotient in division. Subtract the square of the root from the first period, and to ihe remainder bring down the second period for a dividend. III. Double the root already found and place it on the left for a divisor. Seek how many times the divisor is contained in the divi- dend, exclusive of the right-hand figure, and place the figure in the root and also at the right of the divisor. IV. Multiply the divisor thus augmented, by the last figure of the root, and subtract the product from the dividend, and to the re- mainder bring down the next period for a new divide7id. V. Double the whole root already found, for a new divisor, and continue the operation as before, until all the periods are brought down. I. Remark. — If, after all the periods are brought down, there is no remainder, the proposed number is a perfect square. But if 112 ELEMENTS OF ALGEBRA. [CHAP. V there is a remainder, we have only found the root of the greatest perfect square contained in the given number, or the entire part of the root sought. For example, if it were required to extract the square root of 168, we should find 12 for the entire part of the root and a re- mainder of 24, which shows that 168 is not a perfect square. But is the square of 12 the greatest perfect square contained in 168? That is, is 12 the entire part of the root? To prove this, we will first show that, the difference between the squares of two consecutive numbers, is equal to twice the less number augmented by unity. Let a = the less number, and a -f 1 = the greater. Then (a + 1)2 = a^ + 2a + 1 and (a)2 = a^ Their difference is = 2a + 1 as enunciated. Hence, the entire part of the root cannot be augmented by 1, «»• less the remainder is equal to, or exceeds twice the root found, plus unity. But, 12 X 2 + 1 = 25 ; and since the remainder 24 is less than 25, it follows that 12 cannot be augmented by a number as great as unity : hence, it is the entire part of the root. The principle demonstrated above, may be readily applied in finding the squares of consecutive numbers. If the numbers are large, it will be much easier to apply the above principle than to square the numbers separately. For example, if we have (651)2 = 423801; and wish to find the square of 652, we have (651)2 ^ 423801 + 2 X 651 =: 1302 + 1 = l_ and (652)2 = 425104. Also, (652)2 = 425104 + 2 X 652 = 1304 + 1 = 1 (653)2 _ 426109. CHAP, v.] EXTRACTION" OF THE SQUARE ROOT OF NUMBERS. 113 II. Remark. — The number of figures in the root will always be equal to the number of periods into which the given number is separated. examples. 1. To find the square root of 7225. 2. To find the square root of 17689. 3. To find the square root of 994009. 4. To find the square root of 85678973. 5. To find the square root of 67812675. Of Incommensurahle Nvmhers. 118. If a number is not a perfect square, its square root is said to be incommensurahle, or irrational, because it cannot be ex- pressed in terms of the numerical imit. Thus, y 2 , v 5 , v 7 , are incommensurable numbers. They are also sometimes called radicals or surds. Two or more numbers are sa'-t ic oe pr.n'. wi'a ie'"p'"jT v each other, when there is no whole number except unity which will divide each of them without a remainder. Thus, the num- bers 3 and 5 are prime with respect to each other ; and so also are 4 and 7 and 9. In order to prove that the root of an imperfect power cannot be expressed by exact parts of unity, we must first show that, Every numher P, ichich ivill exactly divide the product A X B of two numbers, and which is prime with one of them, will divide the other. Let us suppose that P will not divide A, and that A is greater than P. Let us now find the greatest common divisor of A and P. If we represent the entire quotients by Q, Q', Q''^, he, and the remainders, respectively, by R, R', R^^, &c. ; we shall have A ||P_ Q ' P i|_R_ Q" R IIJR^ R',|R- hence, A = PQ + R, hence, P = RQ' + R', hence. R = R'Q'^ + R", hence. R'= \V'Q."'-\- R'" 114 ELEMENTS OF ALGEBRA. [CHAP. V. Now, since the remainders R, R^, R^^ &c., constantly dimin- ish, if the division be continued sufficiently far, we shall obtain a remainder equal to unity ; for the remainder cannot be 0, since by hypothesis A and P are prime with each other. Hence, we have the following equations : A = P Q +R P = R Q' + R' R = R' Q'^ + R'' R^ = R'^Q'^' + W Multiplying the first of these equations by B, and dividing by P, we have But, by hypothesis, — — — is an entire number, and since B and Q are entire numbers, the product BQ is an entire number. Hence, it follows that ^ is an entire number. If we multiply the second of the above equations by B, and divide by P, we have ^ BRQ^ , BR^ But we have already shown, that — -— is an entire number ; BRQ' . . u rr^x,. u • 1. BR' hence — — — is an entire number. Ihis being the case, - must also be an entire number. If the operation be continued until the number which multiplies B becomes 1, we shall have B X 1 — :g— equal to an entire number, which proves that P will di- vide B. In the operations above we have supposed A > P ; but if P > A, we should first divide P by A. Hence, if a number P will exactly divide the product of two num- bers, and is prime with one of them, it will divide the other CHAP, v.] EXTRACTION OF THE SQUARE ROOT OF FRACTIONS. 115 We see from what has preceded that, if P is prime with re- spect to any number as a, it will also he prime with respect to d? and the higher powers of a. For, if P will divide a^ = a X a, it must divide one of the fac- tors a or a. But this would be contrary to the supposition ; hence, P cannot divide a^. In the same way it may be proved that it cannot divide the higher powers of a. We will now show that the square root of an imperfect square cannot be expressed by a fractional nmnber. Let c be an imperfect square. Then if its exact root can be expressed by a fractional number, we can assume in which the fraction —r- is in its lowest terms : that is, in which a and b are prime with respect to each other. Now, if we square both members of the equation, Ave have ' = ¥^ in which c is an entire number ; and hence, if the equation is true, a^ must be divisible by h"^. But if a^ is divisible by 6^, the product a x a ^^ a^, must be divisible by b ; for the division would be effected by dividing twice by b. But we have seen that a^ is not divisible by h ; therefore, we cannot express the square root of an imperfect square by a fractional number. Extraction of the Square Root of Fractions. 119. Since the second power of a fraction is obtained by squar- ing the munerator and denominator separately, it follows that the square root of a fraction will be equal to the square root of the numerator divided by the square root of the denominator. a For example, V t? = -t-» ▼ 6'' a a a' 116 ELEMENTS OF ALGEBRA. [CHAP. V. But if the numerator and the denominator are not both perfect squares, the root of the fraction cannot be exactly found. We can, however, easily find the exact root to within less than one of the equal parts of the fraction. For this purpose, Multiphj both terms of the fraction by the denominator — this makes the denominator a perfect square. Then extract the square root of the perfect square nearest the value of the numerator, and place the root of the denominator under it — this fraction will be the approxi- mate root. 3 Thus, if it be required to extract the square root of — , we 15 multiply both terms by 5, which gives — : the square nearest 4 . . . 15 is 16: hence, — is the required root, and is exact to with- 5 in less than -— . 5 120. We may, by a similar method, determine, approximatively, the roots of whole numbers which are not perfect squares. Let it be required, for example, to determine the square root of an entire number a, nearer than the fraction — ; that is to say, to n find a number which shall differ from the exact root of a, by a quantity less than — . It may be observed that, an o = — n If we designate by r the entire part of the root of on', the number an^ will then be comprised between r^ and (r + 1 )2 ; and will be comprised between — - and ^ ~- ; and conse n^ n'' n^ quently the true root of a is comprised between r! and ^/^}l. r r + 1 that is, between — and . But the diflerence between n n 1 r these numbers is — : hence — will represent the square root n n CHAP, v.] EXTRACTION OF THE SQUARE ROOT OF FRACTIONS. 117 of a Avithin less than the fraction — . Hence to obtain the root : n Multiply the given number by the square of the denoj/iinator of the fraction which determines the degree of approximation : then ex- tract the square root of the product to the nearest unit, and divide this root by the denominator of the fraction. 1. Suppose, for example, it were required to extract the square root of 59, to within less than — . First, (12)" = 144; and 144 X 59 = 8496. Now, the square root of 8496 to the nearest unit, is 92 : hence 92 1 — = 7 A, which is true to within less than — . 12 ^-' 12 2 To find the vll to within less than — . ^ 15 3 To find the -v/223 to within less than — . ^ 40 Ans. 3 — -. 15 37 Ans. 14—-. 40 121. The manner of determining the approximate root in deci- mals, is a consequence of the preceding rule. To obtain the square root of an entire number within — , -— -, , &c., it is only necessary according to the preceding rule, to multiply the proposed number by (10)^, (100)^, (lOOO)^; or, which is the same thing, Annex to the number, two, four, six, (Sf-c, ciphers : then extract the root of the product to the nearest unit, and divide this root hy 10, 100, 1000, &c., which is effected by pointing off one, two, three, iSfC, decimal places from the right hand. EXAMPLES. 1. To find the square root of 7 to within 100' 118 ELEMENTS OF ALGEBRA. Having multiplied by (100)^, that is, having annexed four ciphers to the right hand of 7, it becomes 70000, whose root extracted to the nearest unit, is 264, which being divided by 100 gives 2.64 for the answer, which is true to 1 Too* 1 100' 1 within less than 2. Find the ^/29 to within 3. Find the >/ 227 to within 70000 4 [CHAP. V. 2.64 46 524 300 276 2400 2096 10000 304 Rem. Ans. 5.38. Ans. 15.0665. Remark. — The number of ciphers to be annexed to the whole number, is always double the number of decimal places required to be found in the root. 122. The manner of extracting the square root of decimal frac- tions is deduced immediately from the preceding article. Let us take for example the number 3.425. This fraction is 3425 equivalent to . Now 1000 is not a perfect square, but the denominator may be made such without altering the value of the , , 1 r. I.- • 34250 fraction, by multiplymg both the terms by 10; this gives or 34250 (ioo)5- Then extracting the square root of 34250 to the nearest unit, we find 185 ; hence — - or 1.85 is the required root to within — — . If greater exactness be required, it will be necessary to add to the number 3.4250 so many ciphers as shall make the periods of decimals equal to the number of decimal places to be found in the root. Hence, to extract the square root of a decimal fraction : Annex ciphers to the proposed number until the number of deci- mal places shall he equal to double the number required in the root. Then extract the root to the nearest unit, and point off from tht right hand the required number of decimal places. CHAP, v.] SQUARE ROOT OF ALGEBRAIC QUANTITIES. 119 EXAMPLES. 1. Find the -/ 3271.4707 to within .01. Ans. 57.19. 2. Find the -/ 31.027 to within .01. A71S. 5.57. 3. Find the -/ 0.01001 to within .00001. Ans. 0.10004. 123. Finally, if it be required to find the square root of a vul gar fraction in terms of decimals : Change the vulgar fraction into a decimal and continue the di- vision until the number of decimal places is double the number re- quired in the root. Then extract the root of the decimal by the last rule. EXAMPLES. 1. Extract the square root of — to within .001. This nuni Der, reduced to decimals, is 0.785714 to within 0.000001. The root of 0.785714 to the nearest unit, is .886 : hence 0.886 is the root of — to witliin .001. 14 \/ 2- 13 2. Find the \/ 2— to within 0.0001. Ans. 1.6931. 15 Extraction of the Square Root of Algebraic Quantities. 124. Let us first consider the case of a monomial. In order to discover the process for extracting the square root, let us see how the square of a monomial is formed. By the rule for the multiplication of monomials (Art. 41), we have (5^253^)2 _ 5^253^ X 5a^^c = 25a^^c^ ; that is, in order to square a monomial, it is necessary to .square its co-eficient, and double the exponent of each letter. Hence, to find the square root of a monomial, 1st. Extract the square root of the co-eficicnt and divide the ex- ponent of each letter bi/ two. 2d. To the root of the co-effLcievf annex each letter with its new exponent, and the result will he tli/: required root. 120 ELEMENTS OF ALGEBRA. [CHAP. V Thus, -/64a6^/4 =: 8a^^ ; for, 80^^*2 ^ q^W — Gia'^b^ and, -y/eSSa-'iV _ 25ab^c^ ; for, {25ab*c^y = 625a^h<^. 125. From the preceding rule, it follows, that, when a monomial is a perfect square, its numerical co-efficient is a perfect square, and the exponent of every letter an even number. Thus, 25a'*i2 is a perfect square, but 98ai* is not a perfect square ; for, 98 is not a perfect square, and a is affected with an uneven exponent. An imperfect square is introduced into the calculus by affecting it with the radical sign y , and written thus, y 98a6*. Quan- tities of this kind are called radical quantities, or irrational quan- tities, or simply radicals of the second degree. These expressions may sometimes be simplified. For, by the definition of the square root, we have ^/a' ^X y/ a = {-y/ af = a, yob X y ab ^[yuby =z ab, yabc X -y/abc = [yahc)'^ = abc, ■y abed X y abed = (yabcd)'^ = abed; and the same would be true for any number of factors. Again, {-/7. /T . /7. /7)2 = (77)2 . (^^Yy. . (y7')2 , (/T)2 ^ abed, by the rule for multiplying monomials (Art. 41). Now, since, [yabcdy =^ abed, and, (y a.yb.yc.-\/dy = abed ; it follows, that the quantities themselves are equal : hence, yf abed z=: yj a . yj b . y c . y d ; that is, The square root of the product of two or more factors is equal to the product of the square roots of those factors. This being proved, we can write y98ai* — ^496* X 2a - ^49^ X y/2a. But, yisT^ - 7^2 . hence, -/98ai^ = 7^2 y/2a. CHAP. V.J SQUARE ROOT OF ALGEBRAIC QUANTITIES. 121 In like manner, ^A5a-Pc^d = yf^aWc^ X ^hd = 3abc y/dbd. ■^864a"b^c^^ = ^/ I4iu^h*c^^ X 6bc — \2ab\^ -/oic. The quantity which stands Avithout the radical sign is called the co-cjicicnt of the radical. Thus, 7Z/-, 3ahc, and 12ub"c^, are co-efflcienls of the radicals. In general, to simplify a radical of the second degree : Divide the quantity under the radical sign by the smallest mono- mial, with reference to its co-eff,cients and exponents, that will give for a quotient a perfect square. Then, extract the root of the per- fect square and place it without the radical sign, under which, write the monomial used as a divisor. EXAMPLES. 1. To reduce ■y/loa^bc to its simplest form. 2. To reduce y 128i^a^J2 ^q ^^g simplest form. 3. To reduce y 32a^6^c to its simplest form. 4. To reduce -v/256a26^c^ to its simplest form. 5. To reduce -s/ lQ2Aa%'' c^ to its simplest form. 6. To reduce ■y/728a''b^c^d to its simplest form. 126. Since like signs in both the factors give a plus s-ign m the product, the square of — a, as well as that of + a, will be a^ : hence, the root of a^ is either -\- a or —a. Also, the square root of 25a2i* is either + 5ab^ or — dab"^. Whence we may con- clude, that if a monomial is positive, its square root may be af fected either with the sign + or — ; thus, -/9a* = ± 3a2, for, + Sa^ or — 3a^, squared, gives 9a*. The double sign ± with which the root is affected, is read plus or mimls. If the proposed monomial were negative, it would have no square root, since it has just been shown that the square of every quan- tity, whether positive or negative, is essentially positive. There- fore, V— 9, -/ — 4a2, ^/— Qd^b, are algebraic symbols which indicate operations that cannot be 122 ELEMENTS OF ALGEBRA. fCHAP. V. performed. They are called imaginary quantities, or rather, im aginary expressions, and are frequently met with in the resolution of equations of the second degree. 127. Let us now examine the law of formation of the square of a polynomial ; for, from this law, the rule is deduced for ex- tracting the square root. It has already been shown (Art. 46), that, (a -I- If = a2 + 2ah + IP- ; that is, The square of a Unomial is equal to the square of the first term plus twice the product of the first term by the second, plus the square of the second. The square of a pol}Tiomial, is the product arising from midti- plying the polynomial by itself once : hence, the first term of the product, arranged with reference to a particular letter, is the square of the first term of the polynomial arranged with reference to the same letter. Therefore, the square root of the first terra of such a product will be the first term of the required root. 128. Let us now extract the square root of the poljmomial 28^5 -j- 49a* + 4a6 + 9 + 42^2 + 12a3 , which arranged with reference to the letter a, becomes, 4a6 + 28a5 + 49a4 + 12a3 + 42a- + 9 2a3+ 7a2 + 3 4o6 4a3+ 7a2 R =28a5 + 49a*+12a3-i-42a2 + 9 7a2 28c5 + 49a* 28a5+49a* = (2r4-r')/ R' = - - 12a3 4-42a2 + 9 12a3 + 42a2 + 9 4a3+14a2+3 3 W= - - 12a2+ 42a2-|- 9 = (2» + r^y Now, since the square root of 4a^ is 2a'^, it follows that 2a^ is the first term of the required root. Designate this term by r, and the following terms of the root, arranged with reference to a, by /, 7^^, r'^^, &c. Now, if we denote the given polynomial by N, we shall have N = (r + r' + r'^+ r"'' + &c. ;)2 or, if we designate all the terms of the root, after the first, by i N = (r + 5)2 = 7-2 + 2rs + ^2 = r2 + 2r (t^ + r^' + r"'' + &c.) + ^2. CHAP. V.l SQUARE ROOT OF ALGEURAIC QUANTITIES. 123 If now we subtract r^ = 4a^, from N, and designate the re- mainder by R, we shall have R - N - 4a6 = 2r (r' + r'^ + r^'' + &c. ) + *^ ; in which the first term 2rr^ will contain a to a higher power than either of the following terms. Hence, if the first term of the first remainder be divided by twice the first term of the root, the quotient will be the second term of the root. If now, we place r -{- r' =z n and designate the remaining terms of the root, r^', r"\ &c., by s^y we shall have N = (« + sy - n2 + 2«/ + ^2 ; and R' = N - ra2 = (2r + 2/) [r" + r"' + &c.) + ^'2 ; in which, if we perform the midtiplications indicated in the sec- ond member, the term Irr" will contain a higher power of a than either of the following terms. Hence, if the, first term of the second remainder be divided by twice the first term of the root, the quotient will he the third term of the root. If we make r + r^ + r^" = ;/, and r''' + j-'^ + &c. = ^', we shall have N = («' +s"y = n'2 + 2ns" + s"^ ; and R'^ = N - n'3 = 2 (r + r^ + r") [f" + r'^ + &c.) + 5"^ ; from which we see, that the first term of any remainder, divided by twice the first term of the root, will give a new term of the re- quired root. It should be observed, that instead of subtracting n"^ from the given polynomial, in order to find the second remainder, that that remainder could be found by subtracting (2r + r')r' from the first remainder. So the third remainder may be found by sub- tracting (Jin + /^) 7^' from the second, and similarly for the re- mainders which follow. In the example above, the third remainder is equal to zero, and hence the given polynomial has an exact root. Hence, for the extraction of the square root of a polynomial we have the following 124 ELEMENTS OF ALGEBRA. [CHAP. V RULE. I. Arrange the polynomial with reference to one of its letters^ and then extract the square root of the frst term, which v:ill givi the frst term of the root. Subtract the square of this term fro/r the given polynomial. II. Divide the first term of the remainder by twice the first term of the root, and the quotient will be the second term of the root III. From the first remainder subtract the product of twice the first term of the root phis the second term, by the second term. IV. Divide the first term of the second remainder by twice the first term of the root, and the quotient will be the third term of the root. V. From the second remainder subtract the product of twice the first and second terms of the root, plus the third term by the third term, and the result will be the third remainder, from which the fourth term of the root may be found ; and proceed in a similar manner for the remaining terms of the root. EXAMPLES. 1. Extract the square root of the polynomial 49aH^ — 24ab^ + 25a* — 30a^ + 16&* First arrange it with reference to the letter a. 25a* — 30a^ -f 49a^^" — 24ab^ +166* 25a* R = - SOa^Z, + 49a2i2 _ — 30a^ + 9a%^ - 24ab^ + 16^** R' = + 40a262 _ + 40a2i2 _ - 24ab^ + 16i* -24ab^ + 16M R'' = 5a2 _ 2ab + 4b^ 10a2 — Sab — 3ab 30^36"T9a^6^ 10a2 — Gab + 4^2 4i2 40^2^2 :_ 240^3 + 166* 2. Find the square root of a* + 4a3a; + 6a2a;2 + 4ax^ + xK 3. Find the square root of a* — 2a3a; + Sa^x"^ — 2ax^ + x*. 4. Find the square root of 4x6 _^ i2a:5 -f 5x* — 2x^ + 7x^ — 2a: + 1. CHAP, v.] SQUARE ROOT OF ALGEBRAIC QUANTITIES. 125 5. Find the square root of 9a* — I2a^ + 28aW — 16aP + IGb*. 6. Find the square root of 25a^b^ — AOa^^c + IGaWc^ — \8a}P-c^ + SGZ^V — 30a*ic + 2\a^hc^ — 36a2Jc3 + 9a4c2. 129. We will conclude this subject with the following remarks*. 1st, A binomial can never be a perfect square. For, its root cannot be a monomial, since the square of a monomial will be a monomial ; nor can the root be a polynomial, since the square of the simplest polynomial, viz., a binomial, will contain at least three terms. Thus, an expression of the form a2 -(- J2 can never be a perfect square. 2d. A trinomial, however, may be a perfect square. If so, when arranged, its two extreme terms must be squares, and the middle term double the product of the square roots of the other two. Therefore, to obtain the square root of a trinomial, when it is a perfect square. Extract the roots of the two extreme terms, and give these roots the same or contrary signs, according as the middle term is posi- tive or negative. To verify it, see if the double product of the two roots is equal to the middle term of the trinomial. Thus, 9a^ — ASa^lj^ + Q\a%^ is a perfect square, for, y^ = 3a3, and -/G4^ = — Sah"^, and also, 2 x Sa^ x — Sai^ — _ 48a'«i2, the middle term. But 4a2 -f I4ai + 9^2 is not a perfect square : for, although Aa^ and + 9^2 are the squares of 2a and 2b, yet 2 X 2a X 3i is not equal to 14a J. 3d. When, in extracting the square root of a polynomial, the first term of any one of the remainders is not exactly divisible by twice the first term of the root, we may conclude that the proposed polynomial is not a perfect square. This is an evident consequence of the course of reasoning, from which the general rule for extracting the square mot was deduced. 126 ELEMENTS OF ALGEBRA. [CHAP. V 4th. When the polynomial is not a perfect square, the expres- sion for its square root may sometimes be simplified. Take, for example, the expression -y/a^b + AaW + 4aP. The quantity under the radical is not a perfect square : but it can be put under the form ab (a2 + 4ab + 4^2). Now, the factor within the parenthesis is evidently the square of a + 2b, whence we have ■y/a^ + 4a^^ 4- 4ab3 = (a + 2b) yfab. Of the Calculus of Radicals of the Second Degree, 130. A radical quantity is the indicated root of an imperfect power. If the root indicated is the square root, the expression is called a radical of the second degree. Thus, y/~^, 3 -/y, 7 ^/~2, are radicals of the second degree. 131. Two radicals of the second degree are similar, when the quantities under the radical sign are the same in both. Thus, 3^/ b and 5c y b are similar radicals ; and so also, are 9 y 2 and 7 -/Y^ Addition and Subtraction. 132. In order to add or subtract similar radicals, add or sub- tract their co-eff,cients, and to the sum or difference annex the com- mon radical. Thus, 3a y/~b + 5c y/~b — (3a + 5c) V^; and 3a y b — 5c yb = (3c — 5c) y b . In like manner, 7 V2a + 3 V^ = (7 + 3) V^ = 10 V^'f and 7 V'2a — 3 ^/2a = (7 — 3) -/ia = 4 ^/2a. Two radicals, which do not appear to be similar, may become so by simplification (Art. 125). CUAP. v.] RADICALS OF THE SECOXD DEGREE. 127 For example, >/ iSai"' -} b y/lba = \b y/Va + bh yfZa = 9b -/Sa , and 2-/ 45 — 3y/~E~ = 6 -/T — 3 y^ = 3 V~^ • When the radicals are not similar, their addition or subtraction can only be indicated. Thus, to add 3 y 5 to 5 y a , we wTite 5 ^/^ + 3 ^/T. Multiplication. 133. To multiply one radical by another, let us observe that ( V c X y/by = ah ; also that, [s/aby = ab ; hence, (V a X \/ by =z [-s/aby ; and consequently, ■\/ a X y b = yab : that is The product of two radicals of the second degree is equal to the square root of the product of the quantities under the radical signs When there are co-efRcients, we first multiply them together, and write the product before the radical sign. Thus 3 y/bab X 4 ^/20a = 12 y/\QOa^b = 120a -y/ b ' 2a yfbc X 3a yfhc — %a^ ^/¥^ = 6«^ic. 2a -/a2 + 52 X — 3a ^^^7^ — _ Qq^ (a2 4. 52). Division. 134. To divide one radical by another, let us observe that f^)'=i^ = ^: also, \y/b^ (Vby b that ( Y -j-j = -f- '• hence y a 7=^ — v T" • t^3,t is, ■vA ^ * T^e quotient of two radicals is equal to the square root of the quotient of the quantities under the radical signs. 128 ELEMENTS OF ALGEBRA. [CHAP. V. When there are co-efficients, write their quotient as the co-efi- rient of the radicals. For example, 5a I h 5a -y/T ^ 2b -/T— . And I2ac -y/Sbc-^ 4c y 2Z» = Say — — = 3a y 3c. 135. There are two transformations of frequent use in fniJirig the numerical values of radicals. The first transformation consists in passing the co-efficient of a radical under the radical sign. Take, for example, the ex- pression 3a y 5i. By applying the rules for the multiplication of radicals we may write, 3a ^/5b ^ ^/'^d^ X >/5h = y/^a^ x 5h = y/~\5^. Therefore, the co-efficient of a radical may be passed under the rai ical sign, as a factor, by squaring it. The principal use of this transformation, is to find an approxi- mate value of any radical, which shall differ from its true value, by less than unity. For example, take the expression 6y 13. Now, as 13 is not a perfect square, we can only find an approximate value for its square root ; and when this approximate value is multiplied by 6, the product will differ materially from the true value of 6 ylS- But if we write, 6 yii = -/62 X 13 = -v/36 X 13 = ^468, we find that the square root of 468 is the whole number 21, plus an irrational number less than unity. Hence, 6 y 13 = 21 plus an irrational number less than 1. In a similar manner we may find, 12 y 7 =31 plus an irrational number less than 1 136. Having given an expression of the form, a a or — =, P + V? P - V q in which a and p are any numbers whatever, and q not a per- fect square, it is the object of the second transformation to len- der the denominator a rational quantity. CHAP, v.] RADICALS OF THE SECOND DEGREE. 129 Tills object is attained by multiplying both terms of the frac- tion by p — y q , when the denominator is p + y y , and by p + \ q , Avhen the denominator is p — y q '■, and recollecting that the sum of two quantities, multiplied by their difference, is equal to the difference of their squares : hence, g ^( ? — Vq) __ (i{p — y/~q) _ gp — a \/T p + ^^q (p + VT) (p — Vq) p'^ -q p^ — q' a _ a (p -f- Vq) _ g(p + Vq) __ ap + a y/q p — V q {p- Vq) (p + VT) p^-q p'^ — q' in which the denominators are rational. As an example to illustrate the utility of this method of ap- proximation, let it be required to find the approximate value of 7 the expression -=rr. We write 3 — V 5 7 7(3 + y/~E) 21 + 7 V^ 3 — V5 9 — 5 4 But, 7 -y/ 5 = V'49 X 5 = ^245 = 15+ an irrational number less than unity. Therefore, 7 21 + 15-1- irr. number < 1 1 3 — V 5 4 4 hence, 9 differs from the true value by less than one Jourth. If we wish a more exact value for this expression, extract thi square root of 245 to a certain number of decimal places, add 21 to this root, and divide the result by 4. For another example, take 7 v^5 and find its value to within 0.01. We have, 7 ^/T _ 7 -/T( \/n — ^/T) _ 7 y/sB — 7 yTs V^il -\- ^/3 11 — 3 " 8 9 130 ELEME^fTS OF ALGKBRA. [CHAP. V. Now, 7 ^/bb = -/55 x"49'= ^2695 =n 51.91, williin 0.01, and 7-/l5 = -y/lS X 49 = ^735 ==27.11 - - - ; - therefore^, 1 ■/b' 51.91 —27.11 _ 24.80 _^ ,^ /- — == -= — — — J. 10. 1 1 + V 3 8 8 Hence, we have 3.10 for the required result. This is exact to within . 800 By a siDiilar process, it will be found that q I 9 / 7 ^_ "" y = 2,123, exact to within 0.001. 5 V 12 -6/5 Remark. — The value of expressions similar to those above, may be calculated by approximating to the value of each of the rad- icals which enter the numerator and denominator. But as the value of the denominator would not be exact, we could not deter- mine the degree of approximation which would be obtained, where- as by the method just indicated, the denominator becomes rational, and we always know to what degree the approximation is made. Examples in tJie Calculus of Radicals. We make the reductions in the examples which follow, accord ing to the methods indicated in Art. 125 ; though, it is sometimes necessary to multiply the quantity under the radical sign, instead of dividing it. 1. Reduce y 125 to its simplest form. We first seek the largest perfect square that will exactly di- vide 125. We try, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144. AVe find that 25 is the only one that will give an exact quotient : hence, '/l25 = -v/25 X 5 = 5 ^/l> . 2. Ke-Juce \/ to its simplest form. We observe that 25 will divide the numerator, and hencti ^^ 147 V 1 X 2 ,. / 2 — 5 47 ^ 147 CHAP. V.l EXAMPLES IN THE CALCULUS OF RADICALS. iol Since there is no perfect square which will divide 147, we must see if we can multiply it by any number which will give a perfect square for a product. Multiplying by 2 we have 284, which is not a perfect square. Then trying 3, we find the prod- not 441, whose square root is 21. Hence, we have Ai" / 2 X 3 r\ .5 r- 5 V - — - = 5 V = 5\/ xG= — Ve ^ 147 V 147 X 3 V 441 ^ 2\J^ 3. Reduce -y/oSa^a; to its most simple form. Ans. la yJ'Zx 4. Reduce y (a;^ — a^x^) to its most simple form. 5. Required the sum of -y/72 and ^J 128 . Ans. 14 V^ 6. Required the sum of y 27 and y 147 . Ans. 10 yy. 2 /27 7 Required the sum of V/ — - and \ — . 4 V o V 5Q 19 r-^ Ans. -— - y b 8. Required the sum of 2 yj a^h and 3 ^J^^hx^. 9. Required the sum of 9 ^243 and 10 "/3G3. /~3~ r^ 10. Required the difference of \ — and \ — . ^ V 5 V 27 4 r- Ans. — yl5. 45^ 11. Required the product of 5y 8 and 3^5. Anjs. 30^10. 2 / 1 3 /~7 12. Required the product of ~t-\I -— and — \ — . ^ ^ 3^8 4 V 10 Ans. l/i^ 13. Divide 6 -/lO by 3-/ 5. 14. What is the sum of ^J \%alr + h yjlba. 15. What is the sum of s/ \%a-P -f ^50^. Ans. {Sa^-b + Sai) ^/z^ih 133 ELEMENTS OF ALGEBRA. [CHAP. VT. CHAPTER VI. EQUATIONS OF THE SECOND DEGREE. 137. An Equation of the second degree is one in wliich the greatest exponent of the unknown quantity is equal to 2. If the equation contains two unknown quantities, it is of the second de- gree when the greatest sum of the exponents with which the unknown quantities are affected, in any term, is equal to 2. 138. Equations of the second degree are divided into two classes. 1st. Equations which involve only the square of the unknown quantity and known terms. These are called incomplete equations. 2d. Equations which involve the first and second powers of the unknown quantity and known terms. These are called complete equations. Thus, a;2 + 2x2 — 5 = 7 and 5a?2 — Sx^ — 4 = a, are incomplete equations ; and 8x2 — 5-p — 3jj2 -i^ a ■=: b 2x2 — 8x2 — a; — c = J, are complete equations. Of Incomi)lcte Equations. 139. The following is the most general form of an incomplete equation : viz., °^ c Q ' U we reduce this to an equation containing only entire terms, we have, 6ccx2 — Ghx^ — 7c rr Q>cd • CHAP, VI.] EQUATIONS OF THE SECOND DEGREE. 133 hence, x^ (6ac — 6i) = Qcd -\- 7c, ^ Qcd + 7c and, X- = — = m, bac — DO by substituting m, for the known terms which compose the sec- ond member. Hence, every incomplete equation can be reduced to an equation involving but two terms, of the form and from this circumstance, the incomplete equations are often called, equations involving two terms. There is no difficulty in resolving equations of this form ; for, we have a; = y wi . If m is a perfect square, the exact value of x can be found by extracting its square root, and the value will be expressed either algebraically or in numbers. If m is an algebraic quantity, and not a perfect square, it must be reduced to its simplest form by the rules for reducing radi- cals of the second degree. If wi is a number, and not a perfect square, its square root must be determined approximatively by the rules already given. But the square of any number is +, whether the number it- self have the -j- or — sign : hence, it follows that (+ v OT )^ = rn, and {— y m)'^ := m ; and therefore, the uidtnown quantity x is susceptible of two dis- tinct values, viz. X =z -\- y m , and x = — y m ; and either of these values being substituted for x will verify the piven equation. For, xXx = x'^ = -\-yrn X+ ym = m; ' and a:Xa: = a;2= — ym X — y m z::i m. The root of an equation is any expression which, being substi- tuted for the unknown quantity, will satisfy the equation ; that is, make the two members equal to each other. Hence, every incom- plete equation of the second degree has two roots which are numen- cally equal to each other; one having the sign plus, and the other the sisn minus. 134 ELEMENTS OF ALGEBRA. [CHAP. VI. 1. Let US take, as an example, the equation 3 ^ 12 24 ^ 24 which, by making the terms entire, becomes 8a:2 — 12 + IQx"^ — 1 — 24a'2 + 299, and by transposing and reducing 42a;2 = 378 and x"^ = = 9 ; 42 hence, a;=4-v9 = + 3; and ir^— y9 =—3. 2. As a second example, let us take the equation 3a2 = 5. Dividing by 3 and extracting the square root, we have T 1 15; 3 3 ' ' in which the values must be determined approxiraalively. 3. What are the values of x in the equation 11 (a:2 — 4) = 5(a;2 + 2). Ans. x = ±Z. 4. What are the values of x in the equation y /7i2 — a;2 m = n Ans. X ■=■ ± . — X Y 1 4- »^ Of Comj)lete Equations of the Second Degree. 140. The most general complete equation of the second degree can be expressed under the form oa;2 -\- hx — 3a; — c = d ; which may be put under the form ax"^ + {h — 3)x= d-\-c\ and by dividing by the co-efficient of ar^, we have ^ J -3 d+ c X^ -j X = . a a If now, we substitute 2p for the co-efficient of x, and represciii the value of the second member by q, we shall have, a;2 -\- 2px = q. CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 135 The reduction to the above form is made : 1st. By transposing all the terms involving x^ and a; to the first member, and all the known terms to the second. 2d. If the term involving x^ should be negative, the signs of all the terms of the equation must be changed lo render it posi- tive, and then divide both members by the co-efficient of x^. Hence, every complete equation of the second degree can be reduced to aji equation involving but three terms, and of the above form. The quantity q is called the absolute term. If we compare the first member of the equation cc? -\- 2px = q, with the square of a binomial {x -J- a)- = x"^ + Sax + a^ we see that it needs but the square of p to render it a perfect square. If then, p^ be added to the first member, it will become a perfect square ; but in order to preserve the equality of the members, p^ must also be added to the second member. Making these additions, we have x^ H- 2px + p2 = y -|- p2 J this is called completing the square, which is done by adding the square of half the co-efficient of x to both members of the equa- tion. Now, if we extract the square root of both members, we have, X -\- p = ± y/q + and by transposing p, we shall have i2 a; = — j9 + ^/q + p-, and a; = — p — yj q -f p^. Either of these values being substituted for x in the equation x^ + 2^^ = ? will satisfy it. For, we have from the first A'alue, ar2 = (— p + ^JY+fY _ pi _2py/q ^p-i -\.q ^ ^,2 and •Ipx =2p X i-p+ yjq + p2) == _ 2/ + 2p y/ q + f hence x"^ + 2pa: ~ q. 136 ELEMENTS OF ALGEBKA. [CHAP. VJ. For the second value, we have a-2 = ( — p — ^q -Y f-f — p^ +2p y/q + p'^ + q + p^ and 2px = 2p{-p-y/q + /) ^ _ 2;,2 _ 2p y/q + p2 ; hence, a;^ + 2pa; =: q ; and consequently, the values found above, are roots of the equa- tion. In order to refer readily, to either of these values, we shall call ihe one which arises from using the + sign before the radical, llie frst value of .r, or the first root of the equation ; and the other, the second value of r, or the second root of the equation. Having reduced a complete equation of the second degree tx) the form af2 -\- 2px =: q, we can write immediately the two values of the unknown quan- tity by the following RULE. The first value of the unknown quantity is equal to half the co- efficient of X taken with a contrary sign, plus the square root of the absolute term increased by the square of half the co-e£icient of x. The second value of the unJinoiun quantity is equal to half the to-ejident of x taken with a contrary sign, minus the square root vf the absolute term increased by the sqtiare of half the co-efficient of X. 1. Let us take as an example the equation a:2 — 7a: + 10 = 0. By transposing 10, we have a:2 —73; =: _ 10. Hence, a; = 3.5 + y/ — 10 + (3.5)2 _ 3,5 _|_ ^ 2.25 = 5, and a; = 3.5 - y/ — 10 -f (3.5)2 ^ 3,5 _ ^2.25 = 2. 2. As a second example, let us take the equation 5,1 3 2 ,273 --a:2 X A = 8 — — x — a:2-j . 6 2 4 3 12 K educing to entire terms, we have lOa'2 — 6x 4- 9 = 96 - Sx - 12:^2 ^ 273, CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 137 and transposing and reducing, 22x2 + 2a; = 360, and dividing both members by 22, „ , 2 3G0 22 22 ' hence, -=-^+\/^ + Q' and .= -l-\/- + (-y 22 V 22 \22/ It often occurs, in the solution of equations, that p^ and q are fractions, as in the above example. These fractions most gen- erally arise from dividing by the co-efRcient of x'^ in the reduc- tion of the equation to the required form. When this is the case, we readily discover the quantity by which it is necessary to mul- tiply the terms of q, in order to reduce it to the same denominator w\ith p"^ ; after which, the numerators may be added together and placed over the common denominator. After this operation, the denominator will be a perfect square, and may be brought from under the radical sign, and will become a divisor of the square root of the numerator. To apply these principles in reducing the radical part of the values of x, in the last example, we have /360 /ly _ / 360 X 22 1 _ /792 V 22 "*" \22/ ~ V (22P + C22)2 "~ V ~T 0+ 1 (22)2 ' (22)2 V (22)2 22 ^ 22 ' and therefore, the two values of x become, _ 1 89 _ 88 _ ^ ~ ~22 22 ~ 22 ~ "* ' d — _J__^__^^_ "^^ . *~~~22~'2""~~'22~~lT' either of which values being substituted for x in the given equa- tiim, will satisfy it. 3. What are the values of x in the equation cx2 — ac =1 ex — bx"^. 133 ELEMENTS OF ALGEBRA. [CHAP. VI. We have, by transposing and reducing, [a -{- b) x" — ex = ac; „ c (IC hence, x^ . a -\- a -\- and consequently, / ac 2{a-\-b)^^a + b'4(a + bf c I (IC c^ If now, we multiply both terms of y by 4 (a + b), it will be reduced to a common denominator with p^, and we shall have v/ ac c^ Aa^c -\- \ahc-\-c'^ y/ Ad^c -\- Aabc -{- c^ oTT "^ 4 (a + bf ^ ^ 4 (a + ^)^ ^ 2 (a + ^') ' c ± -sf Acre + 4aic + c^ hence, a- = ^-- — ; — -r . 2 (a + 6) 1-. Wha^ Lre the valuer tf a", in the 'iqaations, 6x3 _ 37^ _ _ 57. By reducing to the required form, we have 2_37 ^ _57_ ^ G ^ ~ ' 37 / 57 /37\2 hence, x ■= ■{ ±\/ — + ( — ) We observe, that if we multiply both terms of q by 2, and then by 12, that q and -p^ will have the same denominator ; hence. 12 V (12)2 ^ (12)2 But, 114x12^=1368; and (37)2=1369 . 37 /_ 1368 + 1369 37 1 hence, .= +-±V (l^j^— = + I^ ^ Tl^ ' . 37 , 1 38 19 ipnce, ar= -| =: — z= — , 12 12 12 6 ' .37 1 36 ^ and X z= A = — ==3. 12 12 12 CHAP. VI J EQUATIONS OF THE SKCOND DEGREE. 139 5. What are the values of x in the equation 4a2 _ 2x2 ^ 2ax = 18ab — I8b\ In this equation, the term which contains the second power of the unknown quantity is negative ; and since that term already stands in the first member of the equation, it can only oe ren- dered positive by changing the sign of every term of the equa- tion. Doing this, transposing, and dividing by 2. we have x"^ — ax = 2a' — 9ab + 9b^ ; whence, x =— ±\/2a^ — 9ab +9*2+-^ 2 ^ 4 9^2; and the root of the radical part is equal to -; 3b. Hence, a ,3a „,, , Cx= 2a — 3b. X = — d: ( 2b) ; hence, < 2 ^2 ' ' lx= a + 3^». EXAMPLES. X 4 1. Given, — 4 — a:2 — 2a: p- a;2 = 45 — 3x2, ^^ fi^j ^ (x= 7.12") 01 ( X = 7.12 ) Ans. < . / to within O.C ( X = — 5.73 3 2. Given, a:2 — 8x + 10 = 19, to find x. Ans. X = 9, and a? = — 1. 3. Given, x2 — x — 40 = 170, to find x. Arts. X = 15, and x = — 14. 4. Given, 3x2 + 2x — 9 = yg^ to find x. Ans. X = 5, and x = — 5|. 5. Given, 1x2 - ix + 7f = 8, to find x. 5 Ans. X =z Ik, and x = 6. Given, 77ix2 — 2mx y n = nx^ — mn, to find x. Y mn V ..... V fin Ans. X y m — y n y m -{- y n 140 ELEMENTS OF ALGEBRA. [CHAP. VI. « ^- 90 90 27 7. Given -— i= 0, to find x. X a; + 1 a; + 2 A A 5 Ans. a; = 4, x = — . cic 8. Given ax^ r = ex — hx^, to find x. a + . c ±: V c2 + 4ac Ans. X = 2{a-\-b) « ^. , , 3a2 6a2 + a& — 2Z-2 Pa; , , 9 Given abx^ H a; = , to find x. c c^ c 2a — b 3a -f 2b Ans. X = , a- = . ac be Tn^x 10 Given a^ -\- b"^ — 2bx -\- x^ = ——, to find a:. y 7r n / Ans. X = -r r (bn dz -J cP-rr^ + b'^m^ — a^n^). n* — nv- QUESTIONS. 1. To find a number such, that twice its square added to three times the number, shall be equal to 65. Let X represent the number. Then, 2x2 4- 3a; — 65, 3 23 3 23 13 4 4 ~~ 2" 3 ^ /65 , 9 whence, x= r — V":r + T7' = 4 ^2 lb there fijre, 3 23 x=3 1 =5, and x =z — 4 4 Both these values will satisfy the question, understood in its algebraic sense. We have, 2 X (5)2 + 3x5 = 2x25 + 15 = 05; / 13\2 13 169 39 130 ^^ and 2(-_)+3x--=--- = -=C5. Suppose we had stated the question thus : — To find a number such, that three times the number subtracted from twice its square, Khali give a remainder equal to 65. CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 141 If we denote the number by x we shall have 2a;2 — 3x = 65 ; 23 4 3 ^ /65 , 9 3 whence, x = -±\/- + - = - ^ ^ 3 23 13 , 3 23 iheretore, a; = = — , and x = = — 5, 4 4 2 4 4 values which differ from those found before, only in their signs. If the last enunciation be understood in its algebraic sense, the 13 — 5 equally with the -\ will satisfy both the enunciation and the equation. It is true that the second term — 3a; will be added to the first term ; for, the subtraction of 3 times — 5, will give +15. 2. A person purchased a number of yards of cloth for 240 cents. If he had received 3 yards less, for the same sum, it would have cost him 4 cents more per yard. How many yards did he purchase ? Let X = the number of yards purchased. 240 Then, the price per yard will be expressed by . If, for 240 cents, he had received 3 yards less, that is, x — 3 yards, the price per yard, under this hypothesis, would have been .p.se„.e. .. 2^, B., ., ..e e^nciaUon ..s .. eo. would exceed the first, by 4 cents. Therefore, we have the equa- tion 240 240 5 =4; X — 6 X whence, by reducing, a:^ — 3a; = 180, 3 ^ /9 . ,„^ 3 ± 27 and a: = -±\/x + lSO = — 2— ' therefore, a; = 15, and xz= — 12. The value a: = 15 satisfies the enunciation understood in its 240 arithmetical sense ; for, 15 yards for 240 cents, gives , or 1 6 cents for the price of one yard, and 12 yards for 240 cents, gives 20 cents for the price of one yard, which exceeds 16 by 4 142 " ELEMENTS OF ALGEBRA. [CHAP. VI. The — 12 will satisfy the question in its aloebraic sense, and considered without reference to its sign, will be the answer to the following arithmetical question : — A person purchased a number of yards of cloth for 240 cents : if he had paid the same sum for 3 i/ards more, it would have cost him 4 cents less per yard. How many yards did he purchase ? Remark. — In the solution of a problem, both roots of the equa- tion will satisfy the enunciation, understood in its algebraic sense. If the enunciation, considered arithmetically, admits of a double interpretation, when translated into the language of Algebra, the solution of the equation will make known the fact : and hence, while one root resolves the question in its arithmetical sense, the other resolves another similar question also in its arithmetical sense ; and both questions will be stated by equations of the same general form, having equal numerical roots with contrary signs. 3. A man bought a horse, which he sold for 24 dollars. At the sale, he lost as much per cent, on the price of his purchase, as the horse cost him. What did he pay for the horse ? Let X denote the number of dollars that he paid for the horse : then, X — 24 Avill express the loss he sustained. But as he lost x X per cent, by the sale, he must have lost -— upon each dollar, and upon X dollars he loses a sum denoted by — — ; we have then the equation a;2 = a; — 24, whence a;2 — 100a: = — 2400 ; 100 ' and a- = 50 ± -/— 2400 + 2500 = 50 rb 10. Therefore, a; = GO and x = 40. Both of these values satisfy the question. For, in the first place, suppose the man gave 60 dollars for the horse and sold him for 24, he then loses 36 dollars. Bui, from the enunciation he should lose 60 per cent, of 60, that is, of 60 = = 36 ; therefore 60 satisfies the enun- 100 100 elation. If lie pays 40 dollars for the horse, he loses 16 by the sale; CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 143 40 for, lie should lose 40 percent, of 40, or 40 x = 16; thcre- ^ 100 fore 40 verifies the enunciation. 4. A grazier bought as many sheep as cost him jC60, and after reserving 15 out of the number, he sold the remainder for £54, and gained 2s. a head on those he sold : how many did he buy ' Ans. 75. 5. A merchant bought cloth for which he paid jC33 15,y., which he sold again at £2 S.y. per piece, and gained by the bargain as much as one piece cost him : how many pieces did he buy ? Ans. 15. 6. What number is that, which, being divided by the product of its digits, the quotient is 3 ; and if 18 be added to it, the digits will be inverted ? Ans. 24. 7. To find a number such that if you subtract it from 10, and multiply the remainder by the number itself, the product shall be 21. Ans. 7 or 3. - S. Two persons, A and B, departed from different places at the same time, and travelled toward each other. On meeting, it ap- peared that A had travelled 18 miles more than B ; and that A could have gone B's journey in 15| days, but B would have been 28 days in performing A's journey. How far did each travel ? A 72 miles. B 54 miles. 9. A company at a tavern had X8 lbs. to pay for their reck- oning ; but before the bill was settled, two of them left the room, and then those who remained had 10.s. apiece more to pay than before: how many were there in the company? Aiis. 7. 10. What two numbers are those whose difference is 15, and*" of which the cube of the lesser is equal to half their product ? Ans. 3 and 18. 11. Two partners, A and B, gained $140 in trade: A's money was 3 months in trade, and his gain was S60 less than his stock ; B's money was S50 more than A's, and was in trade 5 months • what was A's stock? Ans. $100. 12. Two persons, A and B, start from two different points and travel toward each other. When they meet, it appears that A has travelled 30 miles more than B. It also appears that it M'ill take A Ans. 144 ELEMENTS OF ALGEBRA. [CHAP. VI 4 days to travel the road that B had come, and B 9 days to travel the road that A had come. What was tlieir distance apart when they set out? Ans. 150 miles. Discussion of Equations of the Second Degree. 141. Thus far, we have only resolved particular problems in- volving equations of the second degree, and in which the known quantities were expressed by particular numbers. We propose now, to explain the general properties of these equations, and to examine the residts which flow from all the sup- positions that may be made on the values and signs of the known quantities which enter into them. 142. It has been shown (Art. 140), that every complete equa- tion of the second degree can be reduced to the form x^ + 2px = q (1), in which p and q are numerical or algebraic quantities, whole numbers, or fractions, and their signs plus or minus. If we make the first member a perfect square, by adding p"^ to both members, we have x^ + 2j9a; -\- p"^ =. q -{- p^, which may be put under the form {x -^ pY = q -\- p^. Whatever may be the value of ^ -}- V^i ^^^ square root may be represented by m, and the equation put under the form {x + py- = 771^, and consequently, {x + pY — m' == 0. But, as the first member of the last equation is the difference between two squares, it may be put under the form (x + /) — n?) (a; + p + to) = 0, (2) in which the first member is the product of two factors, and the second 0. Now we can make the product equal to 0, and con- sequently satisfy equation (2), only in two diflerent ways : viz., making X -\- p — /« = 0, whence, a: = — p + »i, .ir, by making a; + p -f "^ = 0, w'honce, x ■=. -- p — m ; CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 145 and by substituting for m its value, we have X = — ;? + \q + Jp--, and r = — p — yy + V^- Now, either of tliese values being substituted for x in its cor- responding factor of equation (2), will satisfy that equation, and consequently, will satisfy equation (1), from which it was derived Hence we conclude, 1st. That every equation of the second degree has two roots, and only two. 2d. That every equation of the second degree may he decomposed into two binomial factors of the first degree with respect to x, having X for a frst term, and the two roots, taken with their signs changed, for the second terms. For example, the equation a;2 + 3a; — 28 = being resolved gives a: = 4 and a; = — 7 ; either of wliich values will satisfy the equation. We also have (x - 4) (x + 7) == x2 + 3a; — 28. If the roots of an equation are known, we readily form the bi- nomial factors and the equation. 1 . What are the factors, and what is the equation, of which the roots are 8 and — 9 ? X — 8 and a; + 9 are the binomial factors, and (a; — 8) ( a- + 9) = ;t2 -f a? — 72 = is the equation. 2. What are the factors, and what is the equation, of which the roots arc; — 1 and + 1. [x -{- l)(r — 1) =:a;2— 1 =0. 3. What are the factors and what is the equation, whose roots are 7 + J — 1039 , 7 — J — 1039 — — and — — . 16 16 — 8x2 _ 7a; 4. 34 = 0. 10 146 ELEMENTS OF ALGEBRA. [CHAP. VI. 1 1,3 If we designate llie two roots of any equation by x' ajid x" . we shall have 3?' = — p + -yj q + j)2^ and otf' =. — p ~- \q + p^ ; by adding the roots, we obtain, x' + x'' = — 2p ; •in, I by multiplying them together, x'x" = — q. Hence, 1st. The algebraic sum of the two roots is equal to the co-efficient of the second term of the equation, taken with a contrary sign 2d. The product of the two roots is equal to the absolute term, taken also with a contrary sign. 141. Thus far, we have regarded p and q as algebraic quanti- ties, without considering the essential sign of either, nor have we at all regarded their relative values If we first suppose p and q to be both essentially positive, then to become negative in succession, and after that, both to become negative together, we shall have all the combinations of signs which can arise ; and the complete equation of the second de- gree will, therefore, always be expressed under one of the four following forms : — a;2 + 2px = q (1), 'J? — 2px = q (2), ^2 ^ 2px = - q (3), x"^ — 2px = — q (4). These equations being resolved, give x= ~ p ± y/ q+ p^ (1), x~ +pr^^/ q+p-^ (2), x= —p ± y/ —q + p^ (3), a; = + p rt V — q-\- p^ (4). In order that the value of x, in these equations, may be found, either exactly or approximatively, it is necessary that the quan- tity under the radical sign be positive (Art. 126). Now, p^ being necessarily positive, whatever may be the sign of />, it follows, that in the first and second forms all the values CHAP. VJ.] EQUATIO.VS OK THK SECO.VD DEGREE. 1 t7 of X will be real. They will be determined exactly, when the quantity q -\- p'^ is a perfect square, and approximatively, when it is not so. Since q and p"^ are both positive, the numerical value of the radical expression ± -y/q + p'^ will be greater than p, and hence the second member of the equation will have the same sign as the radical. Therefore, in the first form, the first root of the equa- tion will be positive, and the second root negative. The positive root w^ll, in general, as already observed, alone satisfy the problem understood in its arithmetical sense ; the neg- ative value, answering to a similar problem, differing from the first only in this ; that a certain quantity Avhich is regarded as addi- tive in the one, is subtractive in the other, and the reverse. In the second form, the first value of x is positive, and the second negative, the positive value being the greater. In the third and fourth forms, the values of x will be imaginary when q > p^, and real when q < p"^. And since y — q -\- p'^ p^, that the conditions of the question will be incompatible with each other, and therefore, the values of x ought to be imaginary. Before showing this, it will be necessary to establish a propo- sition on which it depends : viz., If a given number be decomposed into two parts, and those parts multiplied together, the product will be the greatest possible when the parts are equal. Let a be the number to be decomposed, and d the difference of the parts. Then and and Now it is plain, that P will increase as d diminishes, and that it will be the greatest possible when ^ = ; that is, a a a^ . ^ . — X — = — IS the greatest product. 147. Now, since in the equation x^ — 2px = — q 2p is the sum of the roots, and q their product, it follows that q cannot be greater than p'^. The relations between p and q, there- fore, fix a limit to the value of q ; and if we assume, arbitrarily, q > p", we express by the equation a condition which cannot be fulfilled, and this contradiction is made apparent by the values of X becoming imaginary. Analogy would lead us to conclude that. When the value of the unknown quantity is found to be imagin- ary, the conditions expressed by the equation are incompatible tvith each other. Remark. — Since the roots, in the first and second forms, have contrary signs, the condition that their sum shall be equal to a given number 2p, does not fix a limit to their product : hence, in those two forms the roots are never imaginary. CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 149 148. We shall conclude this discussion by the following re marks : — 1st. If, in the third and fourth forms, we suppose q =i p^, the radical part of the two values of x becomes 0, and both the values reduce to a; = =F p : the two roots are then said to he equal. In fact, by substituting p^ for q in the equation, it becomes x^ ± 2px =: — p'^, whence a:2 ± 2px + p2 — 0, that is, (x ± pf = 0. Under this supposition, the first member becomes the product of two equal factors. Hence, the roots of the equation are equal, since the two factors being placed equal to zero, give the same value for x. 2d. If, in the general equation, X- + 2px = q, we suppose y = 0, the two values of x reduce to X = — p -\- p =1 0, and x =: — p — p = — 2p. Indeed, the equation is then of the form x2 + 2px =0, or x{x + 2p) = 0, which can only be satisfied, either by making X = 0, or a? + 2p = ; whence, x = 0, and x = — 2p ; that is, one of the roots is 0, and the other the co-efEcient of x, taken with a contrary sign. 3d. If, in the general equation x"^ ± 2px z=z ± q, we suppose 2p = 0, there will result a;2 = rfc y, whence, x = ± y dz q ; that is, in this case the two values of x are equal, and have con- trary signs, real in the first and second forms, and imaginary in the third and fourth. The equation then belongs to the class of equations involving two terms, treated of in Art. 139. 4th. Suppose we have at the same time j9 1= 0, 5' = ; the equa- tion reduces to x"^ z=i 0, and gives two values of x, equal to 0. 150 ELEMENTS OF ALGEBRA. [CHAP. VI, 149. There remains a singular case to be examined, which is often met with in the resolution of problems involving equations of the second degree. To discuss it, take the equation ax^ -\- bx = c, ,., . — b ± y/b'^ + 4ac which gives x =z . ^ 2a Suppose, now, that from a particular hypothesis made upon the given quantities of the question, we have = 0; the expression for X becomes _ b±b , I ^'^T' whence. ' ~' ] 2b ^==-¥' Let us first interpret the first root of x — -—. By multiplying the numerator and denominator of the second member of the equation b + -/^»2 _|_ 4ac X = by — b — ■\/b'^ + 4ac 2a we obtain b"^ — (£2 _(_ 4ac) — 4ac 2a (—b — y/b'^+ 4ac) 2a {— b — y/U^ + Aac) X hence, x = "^ r, by dividing by 2a — b — -y/Z)^ -|- 4oc c and consequently, x =z — , by making a = 0. Hence we see that the apparent indetermination arises from a common factor in the numerator and denominator. In regard to the second root 2b ^=--0"' we see that it is presented under the form of infinity. By making a = 0, in the equation ax"^ -\~ bx ^= c, it reduces to an equation of the first degree, bx = c. CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 151 It is therefore impossible that it can hive two roots ; and hence, such a supposition gives one of the vahies of x infinite. We have ah-eady seen (Art. 147), that imaginary values of the unknown quantity indicate the introduction, into the equation, of contradictory conditions. By considering the above discussion, and that of Art. 110, we would conclude, that a result which is in- finite, indicates the introduction into the equation of a condition thai is absolutely impossible. If we had at the same time a = 0, b — 0, c = 0, the proposed equation would be altogether indeterminate. This is the only case of indetermination that the equation of the second degree presents. We are now going to apply the principles of this general dis- cussion to a problem which will give rise to most of the circum- stances that are commonly met with in problems involving equa- tions of the second degree. Problem of the Lights. C" A C B C 150. Find upon the line which joins two lights, A and B, of different intensities, the point which is equally illuminated ; ad- mitting the following principle of physics, viz : The intensity of the same light at two different distances, is in the inverse ratio of the squares of these distances. Let the distance AB, between the two lights, be expressed bv c; the intensity of the light A, at the units distance, by a; that of the light B, at the same distance, by b. Suppose C to be the equally-illuminated point, and make AC =: x, whence BC=zc — x. By the principle we have assumed, the intensity of tI, at the unity of distance, being a, its intensity at the distances 2, 3, 4, . a a a -, ■,■ cic, will be —-, -— , — -, &c.: hence, at the distance x it will 4 9 16 be .expressed by — . In like manner, the intensity of B at tlif b distance c — x, is ; but, by the conduions, these two (c — xy 152 ELEMENTS OF ALGEBRA. [CHAP. VI. a' A C B c intensities are equal to each other, and therefore we have the equation a;2 (c — xy ' wliich can be put under the form (c — xy _ b x^ a ' Hence, = — j=^ ; whence, c -J a , . , . c \/ h 1st root IS, x=. - ■ — = -j^=, which gives, c — a? = ya+yo ya-|-yo ^, • cya i-i- — c yf h 2d root IS, X = — ^= -j=., which gives, c — xz=. j=^ ■y/ a — yi yf a — yi 1st. Suppose a'^ b. i. ' e first value of x is positive ; and since y^ (y a + y i), whence, ya 1 , , C' — ; and consequently, . — t= > — . ■y/~a + y/~b 2' ' -v/T + y* 2 Indeed, this ought to be the case, since the intensity of A was supposed greater than that of B. The corresponding value of c — x, as may be easily shown, is also positive, and less than one half of c ; that is, c < -/T+ V~b 2 The second value of x is also positive ; but since, CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 153 V a a — y 6 it will be greater than c ; and consequently, the required point will be at some point C\ on the prolongation of AB, and at the rijiht of the two lights. We may, in fact, conceive that since the two lights exert their illuminating power in every direction, there should be upon the prolongation of AB, another point equally illuminated ; but this point must be nearest that light whose intensity is the least. We can easily explain, why these two values are connected by the same equation. If, instead of taking AC for the unknown quantity x, we had taken AC\ there would have resulted BC =. X — c \ and the equation a b Now, as [x — c)2 is identical with (c — xy, the new equation is identical with that already established, which consequently should have given AC^ as well as AC. And since every equation is but the algebraic enunciation of a problem, it follows that, when the same equation enunciates several problems, it ought by its different roots to solve them all. When the line AC^ is represented by the unknown quantity x both members of the equation _ _ -c-y/T ■yj a — Y 6 are negative, as they ought to be, since a; > c. By changing the signs of both members, we have = BC. Y a — Y 6 2d. Let a < S. This supposition gives a positive value for c Y a y/~a + ypb^ md since ■s/a + yfb > y/~a -\- ^/ a, that is, > 2 ■s/~a 154 ELEMENTS OF ALGEBRA. [CHAP. VI, a'~ A C B / it follows that, c and consequently, '-''>-7r' and therefore, under this hypothesis, the point C, situated between ,4 and B, will be nearer to A than B, as indeed it ought, since the feebler light is at A. The second value of x, that is, c \/ a c y a ya — y6 y b — ya is essentially negative. How is it to be interpreted ? Let us suppose that we had considered C^^, at the left of A, as the point of equal illumination, and that we had represented AC' by -X. Then, BC' = BA + AC' ; that is, BC = c -\- (— x) = c — x; and the equation of the problem would be a b ^ . a h that IS, — - = and therefore, this equation ought to give the point C which lies to the left of A, as well as the points C and C which lie to the right. It should be observed, that we hav^e regarded — x, which rep- resents AC'\ as a mere symbol, without reference to the essential sign of X. Indeed, the essential sign of the unknown quantify is in general, only made knoion in the final result. If it appears, in the final result, that x itself is negative, the numerical value of BC"=zc + {—x) becomes BC'' = c-irX\ that is, BC will be equal to c plus the numerical value of x or to c minus its algebraic value. Hence, \ b — ya -^ b - a quantity which is essentially positive. CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 155 3d. Let a ::zz h. Under this supposition, the vahie of ar, and that of c — x, for the point C between A and i?, both reduce to — ; that is, when the lights are of equal intensity, the point of equal illumination is .'U the middle of the line AB. The value of x, and that of c — x, for the points C and C'\ vvliich lie on the prolongation of AB, both reduce to -\- c -J a — c -J b . ■ ^ ■ , or, to , that is, to innnitv ; ' which indicates, that the conditions of the question are absoluteli/ impossible. It is evident, indeed, that they are so ; for, when the intensity of the two lights is equal, no part lying on the prolon- gation of AB could be as much illuminated by the distant as by the nearer light : hence, the supposition of equal illumination, from which the equation of the problem is derived, is impossible ; and this is shown in the analysis by the corresponding values of the unknown quantity becoming infinite. 4th. Let a =z b, and c = 0. Under these suppositions, the value of x and of c — a;, for the point of equal illumination between A and B, both reduce to 0, as indeed they ought to do, since the points A, B, and C, are then united in one. The value of x, and of c — x, for the points C^ and C, re- duce to the indeterminate form Resuming the equation of the problem (a — 5) a;2 - 2 acx rCr we see that it becomes, under the above suppositions, 0.a:2 — 0.x = 0, A"hir h may be satisfied by giving to x any value whatever : hence, t is a case of indetermlnation. Indeed, since the two lights are if the same intensity, and are placed at the same point, they ought illuminate equalli/ every point of the straight line. 156 ELEMENTS OF ALGEBRA. [CHAP. VI. 5th. Let c = 0, and a and b be unequal. Under this supposition, both values of x, and both values of c — X, will reduce to ; and hence, there is but one point of the line that will be equally illuminated, and that is the point at which the two lights are placed. In this case, the equation of the problem reduces to {a—h)x^ = 0, which gives two values, a; = 0, and x =z 0. The preceding discussion presents a striking example of the precision with which the algebraic analysis responds to all the relations which exist between the quantities that enter into the enunciation of a problem. Examples involving Radicals of the Second Degree. / 2a2 1. Given, x -^ ^/ a'^ -\- x"^ =z . , to find x. y m2 _^ ja2 By reducing to entire terms, we have X yj d^ + ^"^ -\- c? -{- y?" ^=^ 2a^, by transposing, x y a^ -\- x"^ ~ a"^ — x"^, and by squaring, a^x"^ + a:* = a* — 2a-a;2 -f ar*, hence, Sa^ac^ = a*. and consequently, ^ /«2 2. Given, V "^ + *^ — V "^ -b^ = b, to find x. By transposing, \J — -{- b"^ =: y — — b^ -\- b ; and by squaring, — + ^^ = — — ^^ + 2(^ y — — i^ _[. J2. hence, 52 = 26 \/-^ - ^2, and 6 = 2\/-^-62; and by squaring, b"^ = — 4h^ ; CHAP. VI.] EQUATIONS OF THE SECOND DEGREE 157 and hence, x^ = , and x = dz 5^2 b V 5 ^ a /a^ — a;^ x 3. Given, 1- V = -j-, to find x. Ans. X = ±1 -y/'iab — b'^ 4. Given, s/^+I + ^^/If. = Ps/IZl, =v 1 "• a' to find X. Ans. X a {b:^\f X. Ans. X ■ ^2ay/ h 1+6 ft , *»/ /i^ T 5. Given, — r = &, to find x. a + y a^ — x^ ^ ^ . v a; + v^ — « "^« ^ ■■ 6. Given, -5-^=:= ^^^ = , to find x. ij X — yf X — a X — a a (1 ± nf Ans. a; = —. 1 ±2» 7. Given, -^^—,== 1- ^— = V ~' ^° ^""^ *' 7ln5. a? = ± 2 \/ai — V^. ind a:. a(l IF V^2&~-^62) „ _,. a + a: 4- v 2ax + a:^ , ^ , 8. Given, = Z>, to find a: a -\- X Ans. X ■y/2b - b^ Of Trhiomial Equations. 151. Every equation which can be reduced to the form a?'" + 2^0?" = q, in which m and n are positive whole numbers, and 2p and q, known quantities, is called a trinomial equation. Hence, a trinomial equation contains three kinds of terms : viz., terms which contain the unknown quantity afiected with two dif- ferent exponents, and one or more known terms. 158 ELEMENTS OF ALGEBRA. [CHAP. VI. If we suppose m = 2 and n= 1, the equation becomes x^ 4- 2px = q, a trinomial equation of the second degree. 152. The resolution of trinomial equations of the second degree, has already been explained, and the methods which were pursued are, with some slight modifications, applicable to all trinomial equa- tions in which m = 2n, that is, to all equations of the form a:-" + 2px'' = q. Let us take, as an example, the trinomial eijuation of the fourth degree, ex'' — dx^ — ax* +/= 7 + h. We have, x' -\ x'^ = ; c — a c — a and by substituting 2p for the co-efTicieiit of x"^, and q for the ab- solute term, we have x'^ + 2px'^ =z q. If now, we make x"^ = y, and consequently, x =: ± y y, we shall have y- + 2pi/ = q, and y = — p ± '\/ q -\- p^ : hence, x = ± \/ — p d= y i q + p2. We see that the unknown quantity has four values, since each of the signs -f- and — , which affect the first radical can be com- bined in succession with each of the signs which aflect the sec- ond ; but these values taken two and two are numerically equal, and have contrary signs. EXAMPLES. 1, Take the equation X* — 25.t2 = — 144. If we make x"^ = y, the equation becomes, y^ — 25y = — 144, vvhicli gives, y — 16, and y — 9. CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 159 Substituting these values, in succession, for y in the equation a;2 ^^ y^ and there will result, 1st. a;2 = 16, which gives a? = + 4 and at = — 4. 2d. x^ = 9, which gives x = -{- 3 and x = — 3. Hence, the four values are +4, — 4, +3, and — 3. 2. As a second example, take the equation X* — 7a:2 = 8. If we make x~ = y, the equation becomes, y2 _ 7y =: 8, which gives y r= 8, and y = — 1. Substituting these values, in succession, for y, and we have 1st. a;2 = 8, which gives x = -{- 2 y 2, and x = — 2 y 2. 2d. a;2 = — 1, which gives x = -\- y— 1, and x = — y— 1- The last two values of x are imaginary. 3. Let us take the literal equation X* — (2bc + 4a2) x-= — b^c^. By making x'^ — , we have y2 _ (25c + 4a2) y = — Pc^ ; whence, y ■=. he -\- 2a? ± 2a y ic -f- cP- ; and consequently, a; = ± \J he + 2a2 ± 2a ^/hc + a^, 4. Suppose we have, 2x — lyf~x = 99. If we make yf x = y, we have « = y^, and hence, 2y2 — 7y = 99 ; from which we obtain ^1 y = 9, and y = — -. 121 hence, a? = 81, and x = , 4 153. Before resolving the general case of trinomial equation?, it may be well to remark that, the nth root of any quantity, is /i.-i fxpression which mitltiplied hy itself n — 1 tiines will produce the ^ii'€7i quantity. 160 ELEMENTS OF ALGEBRA. [CHAP. VI. The method of finding the nth root has not yet been explained, but it is sufficient for our present purpose that we are able to in- dicate it. Let it be required to find the values of y in the equation yin + 2py- = q. If we make y" = x, we have y^n _ ^p.^ and hence, the given r.juation becomes x^ + Ipx z=z q, and hence, x = — p ±: yf q + p"^ ; that is, y" = — p zb y ^ ■\- p^. and y = \J — p ±i ■>/ q + p"^. If we suppose n = 2, the given equation becomes a trinomial equation of the fourth degree, and we have y = Y — p ± y? + /• 154. The resolution of trinomial equations of the fourth degree, therefore, gives rise to a new species of algebraic operation : viz., the extraction of the square root of a quantity of the form a ± y/~b, in which a and b are numerical or algebraic quantities. To illustrate the transformations which may be effected in ex- pressions of this form, let us take the expression 3 ± y 5. By squaring it we have (3 ± yT^f = 9 ± 6 V^ + 5 = 14 ± 6 ^/T: hence, reciprocally, \/14itG-y/5 =3dry5. As a second example, we have (/? rh-v/TT)^ = 7 ± 2^77+ 11 = 18±2 V^: hence, reciprocally, \/ \Q ±2 -J 11 = -i/T" ± -y/Tl- Hence we see, that an expression of the form 4^ may sometimes be reduced to the form a' ± y/ 1/ or yHi' ± y/17\ CHAP. VI.] EQUATION'S OF THK SECOND DEGREE. 161 and when this transformation is possible, it is advantageous to effect it, since in this case we have only to extract two simple square roots ; whereas, the expression ya ^ecjuires the extraction of the square root of the square root. 155. If we represent two indeterminate quantities by p and y- we can always attribute to them such values as to satisfy tin' (vquations p + ? = \/« + /y - . - (1), and p — q = \J a — ^fT - - - (2). Xhese equations being multiplied together, give J»2 — ^2 _ ^ ^2 — J .... (3). Now, if p and q are irrational monomials involving only single radicals of the second degree, or, if only one is irrational, it fol- lows that p2 and q^ will be rational ; in which case, p- — q"^, or its value, -y/ a^ — b, is necessarily a rational quantity, and conse- quently, a^ — b is a perfect square. Under this supposition, a transformation can always be effectp) from (4), we have 2/)2 = a -\- c, aTid 2q~ = a — c . ^ i'^V^— ' ^'"^ ? =\/^ and therefore, p — \/^^— !-- " 11 162 ELEMENTS OF ALGKIiRA. [CHAP. VI and consequently, \Ja + ^, or p+q=±\/"-4r^±\/~-. Sja-V^, or p-,= ±\/l + i^\r^; hence, \/^W?=±(\/^^ + \/M - (6), \/WT=:.(\/I±-'-V^O - and y„_V* = d=^V-^--V^-j - - (7). These two formulas can be verified ; for by squaring both mem bers of the first, it becomes r— a -\- c a — c ' cP- — c^ i fl + / 6 = — ^ + -^— + 2 y --p- = a + Va2 - c2 ; but, sj cp- — b = c, gives c- = a^ — h. Hence, a -f ^fT — a + ^J a^ — d^ + b — a-\-y/h. The second formula can be verified in the same manner. Remark. — 156. Formulas (6) and (7) have been deduced with out reference to any particular value of c ; and hence, they are equally true whether c be rational or irrational. If, however, c is irrational, they will not simplify the given expression, for each will contain a double radical. Therefore, in general, this trans- formation is not used, unless a^ — b is a perfect square. EXAMPLES. 1. Reduce y 94 + 4J -/s =\/ 94 + -y/ 8820, to its sim- j>lest form. We have a = 94, h = 8820, whence, c = ^^2 -b = y/sSZQ — 8820 = 4, a rational quantity ; therefore, formula (6) is applicable to this case, and we have or, reducing, = ± (y 49 + y 45) ; CHAP. VI.] EQUATIOXS OF THE SECOND DEGREE. 163 y 94 + 42 ^fb = ± (7 + 3 ^T). therefore, Indeed, (7 + 3 y/~bf = 49 + 45 + 42 ^fb = 94 + 42 yfb . 2. Reduce \/ «p + S/n^ — 2m s/ np + m^^ to its simplest form. We have a=i np -\- 2m^, and b = Am?' [np + m"^), d^ — 6 = ri^p-, and c = ■\/ a? — b =. np ; and therefore, formula (7) is applicable. It gives, / /np + 2m^ -{- np np + 2Tr? — np\ ±iv — ^ V- — ^ /' and, reducing, ± {y np + m^ — »?). Indeed, {ynp + »i^ — w)^ = «/) + 2m'^ — 2m yf np + rr?. 3. Reduce to its simplest form, y 16 + 30 y - 1 + Y 16 - 30 y - 1- By applying the formulas, we find y 16 + 30 y - 1 =5 + 3 y — 1, and y 16 — 30 ^ — 1 =5 — 3 yj — 1 : hence, y 16 + 30 v^^ 1 + y 16 — 30 ^ — I = 10. This last example shows very clearly the utility of the general problem ; because it j)roves that imaginary expressions combined together, may produce real, and even rational results. 4. Reduce to its simplest form. y 28 + 10 V^ . Ans. 5 + ^3^ 5. Reduce to its simplest form, y 1 + 4 y/^^. Ans. 2 + y/~^Z 6. Reduce to its simplest form, y be + 2b y/Tc — b"^ -r \J be --2h y/bc — Z-2. Ans. rt 2b ICri ELEMENTS OF ALGEBRA. [CHAP. VI. 7. Reduce to its simplest form, V' ab + 4c2 — (Z2 _ 2 y/Aahc^ — ahcP. Ans. yjab — -s/ Ac^ — tl^. Equations of the Second Degree involving two or more Unknown Quantities. 157. An equation involving two or more unknown quantities, is said to be of the second degree, when the greatest sum of the ex- ponents (f the nnhnown quantities, in any term, is equal to 2. Thus, 3X2 4^ _y y1 ^y f/y -(- G = , IXIJ \X A^ tj =- ^ , are equations of the second degree. Hence, every general equation of the second degree, involving two unknown quantities, may be reduced to the form a-f -\- hxy + ca;2 ■\- dy -\- fx -\- g z=z. (i, a, b, c, &c., representing known quantities, either numerical or algebraic. Take the two equations a y- -\- b xy -{- c x"^ -{- d y -{- f X -{- g =0, a'y'^ + h'xy + c'x"^ -\- d'y -\- f'x -(- ^ = 0. Arranging them with reference to x, they become cx^-\-{by■\■f)x^-ay'^-\-dy■\-g m 0, cV + (//y +/0 .T + fl>2 _f. j/y + ^ ^ ; from which we may eliminate or, after having made its co-effi- cient the same in both equations. By multiplying the first equation by c' , and the second by c, they become cc'x2 + (iy 4-/)c'a:+ (ay^ + Jy +^ ) or = ahp. Let a = Z> = 1 ; the values of x, and y, then reduce to X = s ±. y *^ — p, and y z= s ^ y/ s^ — p ; whence we see, that under this supposition, the two values of x are equal to those of y, taken in an inverse order ; which shows, that if s + ^s^ — p represents the value of at, s — -sj s^ — p will represent the corresponding value of y, and reciprocally. This relation is explained by observing, that, under the last supposition, the given equations become a; -(- y = 25', and xy :=z p ; and the question is then reduced to finding two numbers of which the sum is 2s, and their product p, or in other words, to divide a number 2s, into two such parts, that their product may he equal to a given number p. 2. Find four numbers in proportion, knowing the sum 2s of their extremes, the sum 2./ of the means, and the sum 4c2 of their si|uarcs. Let II, X, y, z, denote the four terms of the proportion ; the equations of the problem will be ] st condition, - - - u-\- z =l1s, 2d condition, - - - ar -f y = 2^, since they are in proportion, uz z=. xy, 4th condition, - - u"^ -\- x"^ -\- y"^ -\- z"^ = Ac"^. At first sight, it may appear difficult to find the values of tho unknown quantities, but with the aid of an unknown auxiliary, they are easily determined. Let p be the unknown product of the extremes or meanji- ; we shall then have and Cu + z=i2s,') ... . Cu = s+^/^—p, ] \ which give, ] .-- i- ( uz =1 p, ) Kzz=s — ys^— p. (x + y= 2s', } . C -r = 5^+ ^/s"' - p ] [ which give, j .^ V xy — p, J \ y = s — y/ s ^ — p. CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 167 Hence, we see that the determination of the four unknown quan- tities depends only upon that of the product p. Now, by substituting these values of w, x, y, z, in the last of the equations of the problem, it becomes + {s' - y/s'"^ — pf = 46-2 ; and by developing and reducing, 4i-2 4- 4.y'2 _ 4p — 4^2 ; hence, p z= s^ -\- s''^ — c^. Substituting this value for p, in the expressions for u, x, y, z, we find ^ M = 6- + Y <::2 — s''^, r X =: s' -\- ^/ c'^ — s^-, \ Z =^ S — Y c2 — ^-'2^ ^ jy —. g/ — y' c2 — .«2. These four numbers evidently form a proportion ; for we liave UZ =. [S + V c2 — s'"^) {s — Y c2 — ^'2) = ^2 — c"^ -\- .v'2, xy = (5' + Y ^^ — ^^) {^ — V <^^ — s^) =2 s''^ — c2 4- -i' ^• Remark. — This problem shows how much the introduction ot an unknown auxiliary facilitates the determination of the principal unknown quantities. There are other problems of the same kind, which lead to equations of a degree superior to the second, and yet they may be resolved by the aid of equations of the first and second degrees, by introducing unknown auxiliaries. 3. Given the sum of two numbers equal to a, and the sum of their cubes equal to c, to find the numbers -\- y —a -|- y-^ = c. Putting x ^: s -{- z, and y z:^ s — z, we have a = 2j, 2sH + 35^2 4. ^3 By the conditions < ^ z, and y C x^ = .s^ + Xy"^ z= s^ — ^s^z + Zsz"^ hence, by addition, x^ + y^ = 2.y3 + Qsz"^ = c ; , c - 2s^ , ^ /c - 2A-3 whence, z^ = — , and z = ±\/ — , OS * 6s x = s±i\l — ; and y — s::i^\j- 'c — 2.S-" or, e>s ' y--^ y Q^ 168 ELEMENTS OF ALGEBRA. [CHAP. V3 and by substituting for s its value, 2 V V 3a / 2 V 12a ' and y^_^V(-^f-)-T-V-T^- 4. Given, —— = 48, and —^ = 24, to find x and y. Y a; Y a; Dividing the first equation by the second, we have = Y y = 2, and hence y = 4. Whence, from the second equation we have, 4,T 4^^ =24, Y a; and consequently, y a: = 6, and x = 36. 5. Given, x + V •^'Z + V = 19 ) ^ , , _ o o > to find X and y. and a;' + a:;/ + y^ = 133 ) Dividing the second equation by the first, we have X — y/Ty -\-y = 1, but, X 4- -s/xy + y = 19: hence, 2a; + 2y = 26 by addition, or, a; + y = 1 3 ; and y^y + 13 = 19 by substituting in the 1st eq. ; or, -^xy = 6 and a;y = 36. The 2d equation is, x"^ -\- xy -\- y"^ ^ 133, and from the last, 3a'y =108; by subtracting a^ — 2a'y + y^ = 25 : hence, x — y ^= zt: 5. But, X -\- y =z 13 : hence, at = 9, or 4 ; and y = 4, or 9. CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 169 6. Find the values of cc and y, in the equations a;2 + 3a: + y = 73 — 2x1/ 1/"^ -\- 3y -\- X = 44. By transposition, the first equation becomes, a;2 + 2xy + 3a; + y = 73 ; to which, if the second be added, there results, a:2 + 2a-y + y2 + 4t + 4y = (a- + y)2 + 4 (a: + y) = 117 [f now, in the equation we regard a: + y as a single unknown quantity, we shall have x + y= —2 ±1 yil7+ 4; hence, a; + y=— 2 + 11 = 9, and a; + y=— 2 — 11 = — 13; whence, a? = 9 — y, and x = — 13 — y. Substituting these values of x in the second equation, we have y2 4- 2y = 35, for x = 9 — y, and y2 4- 2y = 57, for a? = — i3 — y. The first equation gives, y = 5, and y = — 7. and the second, y = — 1 + -/Hi' and y = — 1 — -/58. The corresponding values of x, are a; = 4, 0? = 16 ; a; = _ 12 — ^58, and a: = — 12 + -/sS 7' Find the values of x and y, in the equations x'^y'^ + ^y^ -\- xy =^ 600 — (y + 2) x'^y^ X -{- y'^ = 14 — y. From the first equation, we have a;2y2 -f (y2 -|- 2y) ar^y^ -|- xy"^ -f xy = 600, or, xY (1 + y^ + 2y) + a:y (1 + y) = 600, or, again, x'^y'^ (1 + y)^ -|- a:v (1 + y) = 600 ; 170 ELEMENTS OF ALGEBRA. ICHAP. VI which is the form of an equation of the second degree, by re- garding a?y(l +y) as the unknown quantity. Hence, / /2 (1 + y) = - A db V600 +i=-i±\/- '2401 4 ' and if we discuss only the roots which belong to the + value of the radical, we have 49 ^y(i4-y)=-i + -2-=24; and hence, o y + y^ Substituting this value of x in the second equation, we liave {y' + yy-U(y' + y) = -24; whence, y2 _|_ y __ 12, and y^ _j_ y _ 2. . From the first equation, we have y = ± — = 3, or — 4; and the corresponding values of x, from the equation 24 f + y From the second equation, we have y = 1, and y = — 2 ; which gives a: = 12. 8. Given, x-y + xy"^ = 6, and x^y^ + x^y^ = 12, to find J and y Ans. x=2 or 1, y = 1 or 2. ^ ^. f a:2 + a: + y = 18-y2 > 9. Given, < -^ -^ ^ to find a; and y. (^ xy = ) f X = 3, or 2 ; or -3± y/~2. Alls. < , — C y = 2, or 3 ; or — 3 =f V 3. QUESTIONS. 1. There are two numbers whose difference is 15, and lial!' their product is equal to the cube of the lesser number. Vvhai are the numbers? Ans. 3 and 18 CHAP. VI. 1 EQUATIONS OF THK SEnOND DEGREE. 171 2. What two numbers are those whose sum multiplied by the greater, is equal to 77 ; and whose difference, multiplied by the lesser, is equal to 12 ? Ans. 4 and 7, or § v2 and y y'2. 3. To divide 100 into two such parts, that the sum of their square roots may be 14. Ans. 64 and 36. 4. It is required to divide the number 24 into two such parts, that their product may be equal to 35 times their difference. Ans. 10 and 14. 5. The sum of two numbers is 8, and the sum of their cubes is 152. What are the numbers ? Ans. 3 and 5. 6. The sum of two numbers is 7, and the sum of their 4th powers is 641. What are the numbers? Ans. 2 and 5. 7. The sum of two numbers is 6, and the sum of their 5th powers is 1056. What are the numbers ? Ans, 2 and 4. 8. Two merchants each sold the same kind of stuff: the sec- ond sold 3 yards more of it than the first, and together, they re- ceived 35 dollars. The first said to the second, " I would have received 24 dollars for your stuff." The other replied, " And I would have received 12^ dollars for yours." How many yards did each of them sell ? C 1st merchant x =: 15 } fa:=5 (2d - - - y= IS) (y=8. 9. A widow possessed 13,000 dollars, which she divided into two parts, and placed them at interest, in such a manner, that the incomes from them were equal. If she had put out the first por- tion at the same rate as the second, she would have drawn for this part 360 dollars interest ; and if she had placed the second out at the same rate as the first, she would have drawn for it 490 dollars interest. What were the two rates of interest ? Ans. 7 and 6 per cent 172 ELEMENTS OF ALGEBRA. [CHAP. VII. CHAPTER VII. OF PROPORTIONS AND PROGRESSIONS. 159. Two quantities of the same kind may be compared lo- gether in two ways : — 1st. By considering how much one is greater or less than the other, which is shown by their difference ; and 2d. By considering how many times one is greater or less than the other, wliich is shown by their quotient. Thus, in comparing the numbers 3 and 12 together with re- spect to their difference, we find that 12 exceeds 3, by 9 ; and in comparing them together with respect to their quotient, we find that 12 contains 3, four times, or that 12 is 4 times as great as 3. The first of these methods of comparison is called Arithmetical Proportion ; and the second. Geometrical Proportion. Hence, Arithmetical Proportion considers the relation of quantities to each other, with respect to their difference; and Geometrical Proportion, the relation of quantities to each other, with respect to their quotient. Of Arithmetical Proportion. 160. If we have four numbers, 2, 4, 8, and 10, of which the difference between the first and second is equal to the difference between the third and fourth, these numbers are said to be in arithmetical proportion. The first term 2 is called an antecedent, and the second term 4, with which it is compared, a consequent. The number 8 is also called an antecedent, and the number 10, with which it is compared, a consequent. The CHAP. VII. 1 ARITHMETICAL PROGRESSION'. 173 fir.si and fourth terms are called the extremes ; and the second and third terms, the means. Let a, b, c, and e, denote four quantities in arithmetical pro- portion ; and d the difference between either antecedent and its consequent. Then, a — b = d, and a z= b -\- d ; also, c — e ^ d, and c = c — d. By adding the last two equations, we have a -\- e ^:z b -\- c : that is, If four quantities are in arithmetical proportion, the sum of the two extremes is equal to the sum of the two means Arithmetical Progression. 161. ^\Tien the difference between the first antecedent and con- sequent is the same as between any two consecutive terms of the proportion, the proportion is called an arithmetical progression. Hence, an arithmetical progression, or a progression by differences, is a succession of terms, each of which is greater or less than the one that precedes it by a constant quantity, which is called the common difference of the progression. Thus, 1, 4, 7, 10, 13, 16, 19, 22, 25, . . . and GO, 56, 52, 48, 44, 40, 36, 32, 28, . . are arithmetical progressions. The first is called an increasing progression, of which the common difference is 3 ; and the sec- ond, a decreasing progression, of which the common difference is 4. An arithmetical progression, is also called, an arithmetical series ; and generally, A series is a succession of terms derived from each other accord- ing to some fixed and known law. Let a, b, c, d, e, f . . . designate the terms of a progression by differences ; it has been agreed to write them thus : a.b.c.d.e.f.g.h.i.k . . . This series is read, a is to b, as b is to c, as c is to d, as d is to e, &c. This is a series of continued equi-differences, in Avliich 174 ELEMENTS OF ALGEBRA. [CHAP. VU. each term is ut the same time a consequent and antecedent, with the exception of the first term, which is only an antecedent, and the last, which is only a consequent. 162. Let d represent the common diflerence of the progression a . h . c . e . f . g . h . k, &c., which we will consider increasing. From the definition of a progression, it follows that, b = a -{- d, czzzb-[-d=za-\- 2d, e = c -f f^ = « + 3 J ; and, in general, any term of the series, is equal to the first term plus as many times the common difference as there are precedino terms. Thus, let I be any term, and n the number which marks the place of it. Then, the number of preceding terms will be deno- ted by n — 1, and the expression for this general term, will be / = a + (« — l)rl That is, any term is equal to the first term, plus the product of the common difference by the numhcr of preceding terms. If we make n =: 1 , we have I = a ; that is, the series will have but one term. If we make n = 2, we have I =: a -]- d; that is, the series will have two terms, and the second term is equal to the first plus the common diflerence. EXAMPLES. 1. If a = 3 and d —2, what is the 3d term? Ans. 7. 2. If a= 5 and d — 4, what is the Cth term? Ans. 25. 3. If a = 7 and (1 = 5, what is the 9th term? Ans. 47. The formula, 1 = a -\-{n — \)d, serves to find any term whatever, without determining all those which precede it. CHAP. VII.] ARITHMETICAL PROGRESSION. 175 Thus, to find the 50th term of the progression, 1 . 4 . 7 . 10 . 13 . 16 . 19, . . . we have, Z = 1 + 49 x 3 = 148. And for the 60th term of the progression, 1 . 5 . 9 . 13 . 17 . 21 . 25, . . . Ave have, Z = 1 + 59 x 4 = 237. 163. If the progression were a decreasing one, we should have I— a — {n — l)d. That is, any term in a decreasing arithmetical progression, is equal to the jirst term minus the product of the common difference hy the number of preceding terms. EXAMPLES. 1. The first term of a decreasing progression is 60, and the common difference 3 : what is the 20lh term ? 1 = a — {n — \)d gives / = 60 — (20 — 1) 3 = 60 — 57 = 3. 2. The first term is 90, the common difiference 4 : what is the 15th term ? Ans. 34. 3. The first term is 100, and the common diflference 2 : what is the 40th term ? Ans. 22. 164. A progression by differences being given, it is proposed to prove that, the sum of any two terms, taken at equal distances from the two extremes, is equal to the sum of the two extremes. Let a . b . c . e . f . . . . i . k . I, be the proposed progression, and n the number of terms. We will first observe that, if x denote a term wdiich has p terms before it, reckoning from the first term, and y a term which has p terms before it, reckoning from the last term, we have, from what has been said, X = a + p X rf, and y=:l — pxd; whence, by addition, x -\- y — a -\- I. 176 ELEMENTS OF ALGEBRA. [CHAP. VII. Now, to find the sum of all the terms, write the progression below itself, but in an inverse order, viz., a . h . c . e .f . . . . i . k . I. I . k . i c . b . a. Calling S the sum of the terms of the first progression, 2>S will be the sum of the terms in both progressions, and we sli;i!l have 2S^{a-{-J)-{-{b + k) + {c + i) . . . +{i + c) + {k + h) + {l + a). And, since all the parts a -j- I, b -{- k, c -\- i . . . . are equal to each other, and their number equal to n, by which we desig- nate the number of terms in each series, we have 2S -\-h (a + /)n, or S = {^-^) That is, the sum of the terms of an arithmetical progression, is equal to half the sum of the two extremes multiplied by the number of terms. EXAMPLES. 1. The extremes are 2 and IG, and the number of terms 8 what is the sum of the series ? S /a + h . „ 2 + 16 [-^—) X n, gives S = — ^ X 8 = 72. 2. The extremes are 3 and 27, and the number of terms 12 : what is the sum of the series? Ans. 180. 3. The extremes are 4 and 20, and the number of terms 10: what is the sum of the series ? Ans. 120. 4. The extremes are 8 and 80, and the number of terms 10: what is the sum of the series '' A?is. 440. 165. The formulas fa+ h ~2 I = a -\- [n — \) d and S = ( ] X n. contain five quantities, a, d, n, I, and S, and consequcmlly give rise to the following general problem, viz. : Any three of these five quantities being given, to determine the other two. CHAP. VII.] ARITHMETICAL PROGRESSION. 177 This general problem gives rise to the ten following cases : — No. Given. c, d, n a, d, I a, d, S a, n, I a, n, S a, I, S d, n, I d, n, S d, 1, S n,l,S I, S n, S S, d d, I n, d a, S a, I a, d Values of the unknown quantities 1= a-{- {n—l)d; S — ^n [2a -\- (ti — 1) d]. l-a (Z + a) (/ - a + J) -^+^' -^^^ 2d • d—2a± J(d-2ay + 8dS , )d. S=ln(a + l); d = l-a n — 1' 2(-S — on) j_2S n [n — 1) ' 71 2S , (l-^a)(l — a) » = — : — 7 ; d = a + I 2S — (l-\- a) a=zl -{n~l)d; S = ^n [21 - (?i - 1) d]. i 2S — n(n — l)d , 2S + n{n — l)d a = :: ; / = — 2n 2n 2/4- J± ^/{2l-j-d)^^-8dS 2d ; a = l — (ra — l)d. 2S , ^ 2(7iJ-S) a = / ; d = — — . n n [n — 1) The solution of these cases presents no difficulty. Cases 3 and 9 give rise to equations of the second degree ; but one of the roots will always satisfy the enunciation of the question in its arithmetical sense. If we resume the formula I ^ a -{- {n — I) d, we have, a z= I — (n — \) d\ that is, The first term of an increasing arithmetical progression, is equal to any following term, minus the product of the common differc7ice by the number of preceding terms. From the same formula, we also find l-a d = n — 1 ; that is, 12 178 ELEMENTS OF ALGEBRA. [CHAP. VIl In any arithmetical progression, the common difference is equal to the difference between the first and last terms considered, divided by the ?inmber off terms less one. 1. Two terms of a progression are 16 and 4, and the number of terms considered is 5 : what is the common difference ? The formula l~a . , 16-4 ^ a = gives d = = 3. n — 1 *= 4 2. Two terms of a progression are 22 and 4, and the number of terms considered is 10 : what is the common difference ? Ans. 2. 166. The last principle affords a solution to the following question : — To ffnd a number m of arithmetical means between two given numbers a and b. To resolve this question, it is first necessary to find the com- mon difference. Now we may regard a as the first term of an arithmetical progression, 6 as a subsequent term, and the required means as intermediate terms. The number of terms of this pro- gression which are considered, will be expressed by m -\- 2. Now, by substituting in the above formula, b for /, and m -\- 2 for n, it becomes b — a , h — a d = -, or m + 2 — \ m + 1 that is, the common difference of the required progression is ob- tained by dividing the difference between the given numbers a and b, by one more than the required number of means. Having obtained the common difference, form the second term of the progression, or the first arithmetical mean, by adding d, or , to the first term a. The second mean is obtained by aug- //i -|- 1 menting the first by d, &c. 1. Find 3 arithmetical means between 2 and 18. The formula b -a . , 18-2 ^ d = , gives a = = 4 ; nencn the progression is 2 . 6 . 10 . 14 . 18. CHAP. VII.] ARITHMETICAL PROGRESSION. 179 2. Find 12 arithmetical means between 12 and 77. The for- mula b — a . , 77 — 12 d = -— , gives d = — = 5 ; m -\- I Id hence the progression is 12 . 17 . 22 . 27 72 . 77. 167. Remark. — If the same number of arithmetical means are inserted between the terms of a progression, taken two and two, these terms, and the arithmetical means united, will form one and the same progression. For, let a . 6 . c . e ./ . . . . be the proposed progression, and m the number of means to be inserted between a and b, b and c, c and e From what has just been said, the common difference of each partial progression will be expressed by h — a c — b e — c m-f-l' m-j-l' m -\- \ which are equal to each other, since a, b, c, . . . are in pro- gression : therefore, the common difference is the same in each of the partial progressions ; and since the last term of the first, forms the first term of the second, &c., we may conclude that all of these partial progressions form a single progression. EXAMPLES. 1. Find the sum of the first fifty terms of the progression 2 . 9 . 16 . 23 . . . For the 50th term, Ave have Z = 2 4- 49 X 7 = 345. 50 Hence, -S = (2 + 345) x — = 347 X 25 = 8675. 2. Find the 100th term of the series 2 . 9 . 16 . 23 . . . A71S. 695. 3. Find the sum of 100 terms of the series 1 .3.5.7.9... Ans. 10000 4. The greatest term considered is 70, the common difference 3, and the number of terms 21 : what is the least term and the sum of the series ? Ans. Least term 10 ; sum of series 840. 180 ELEMENTS OF ALGEBRA. [CHAP. VII 5. The first term of a decreasing arithmetical progression i? 10, the common difference one third, and the number of terms 21 : reqni-red the sum of the series. Ans. 140. 6. In a progression by differences, having given the common difference 6, the last term 185, and the sura of the terms 2945 . find the first term, and the number of terms. Ans. First term = 5 ; number of terms 31. 7. Find 9 arithmetical means between each antecedent and con. sequent of the progression 2. 5. 8. 11. 14... Ans. d = 0.3. 8. Find the number of men contained in a triangular battalion, the first rank containing 1 man, the second 2, the third 3, and so on to the n"', which contains n. In other words, find the ex- pression for the sum of the natural numbers 1, 2, 3, . . . from 1 to n, inclusively. . ^ n(ra+l) o 9. Find the sum of the n first terms of the progression of un- even numbers 1, 3, 5, 7, 9 . . . Ans. S = n^. 10. One hundred stones being placed on the ground, in a straight line, at the distance of 2 yards from each other, how far will a person travel, who shall bring them one by one to a basket, placed at two yards from the first stone ? Ans. 11 miles, 840 yards. Geometrical Pro])orlion. 168. Ratio is the quotient arising from dividing one quantity by another quantity of the same kind. Thus, if A and B repre- sent quantities of the same kind, the ratio of yl io B is expressed B A' 169. If there be four magnitudes, A, B, C, and D, having such values that B _D^ A" C then A is said to have the same ratio to B, that C has to I) \ or, the ratio o^ A to B is equal to the ratio of C to D. Wlien CHAP. VII.] GEOMETRICAL PROPORTION. 181 four quantities have this relation to each other, they are said to be in proportion. Hence, proportion is an equality of ratios. To express that the ratio of ^ to 5 is equal to the ratio of C to D, we write the quantities thus, A : B : : C : D, and read, A is to B, as C is to D. The quantities which are compared together are called the terms of the proportion. The first and last terms are called the two ex- tremes, and the second and third terms, the two means. 170. Of four proportional quantities, the first and third are called the antecedents, and the second and fourth the consequents; and the last is said to be a fourth proportional to the other three taken in order. 171. Three quantities are in proportion when the first has the same ratio to the second that the second has to the third ; and then the middle term is said to be a mean proportional between the other two, 172. Quantities are said to be in proportion by inversion, or in- versely, when the consequents are made the antecedents and the antecedents the consequents. 173. Quantities are said to be in proportion by alternation, or alternately, when antecedent is compared with antecedent and con- sequent with consequent. 174. Quantities are said to be in proportion by composition, when the sum of the antecedent and consequent is compared either with antecedent or consequent. 175. Quantities are said to be in proportion by division, when the difference of the antecedent and consequent is compared either with antecedent or consequent. 176. Equi-multiples of two or more quantities are the products which arise from multiplying the quantities by the same number. Thus, m X A and m x B, are equi-multiples of A and B, the common multiplier being m. 177. Two quantities, A and B, are said to be reciprocally pro- portional, or inversely proportional, when one increases in the same ratio as the other diminishes. When this relation exists, eilher of them is equal to a constant quantity divided by the other 182 ELEMENTS OF ALGEBRA. [CHAP. VII 178. If we have the proportion A : B : : C : D, 7? T) we have — = — -, (Art. 169); and by clearing the equation of fractions, we have BC = AD ; that is, Of four proportional quantities, the product of the two extremes is equal to the product of the two means. 179. If four quantities, A, B, C, and D, are so related to each other that A X D = B X C, we shall also have, ^ = ^. A O and hence, A : B : : C : D ; that is If the product of two quantities is equal to the product of two other quantities, two of them may he made the extremes, and the other two the means of a proportion. 180. If we have three proportional quantities, A : B : : B : C, we have —-=:—-; A B hence, B"^ z=z AC ; that is, The square of the middle term is equal to the product of the two extremes. 181. If we have A : B : : C : D, and consequently, — = — , jri. O multiplying both members of the equation by -^, we obtain JO C D^ 1- B' and hence, A : C : : B : D ; that is. If four quantities are proportional, they will be in proportion bij alternation. D B F and c" 'a~e CHAP. VII. J GEOMETRICAL PROPORTION. 183 182. If we have A : B : : C : D, and A : B : : E : F, we shall also have B 'A D F hence, tt = ^7 and C : D : : E : F ;» that is, O E If there are two sets of proportions having an antecedent and con sequent in the one equal to an antecedent and consequent of the other, the remainitig terms will be proportional. 183. If we have A : B : : C : D, and consequently, — = — , "A G we have, by dividing 1 by each member of the equation, A C -— = —-, and consequently, B : A : : D : C ; that is, B D Four proportional quantities will be in proportion, wheji taken in- versely (Art. 172). 184. The proportion A : B : : C : D, gives A x D = B x C. To each member of the last equation add B x D. We shall then have (A + B) X D = {C+ D) X B; and by separating the factors, we obtain A-\-B : B : : C -\- D : D. If, instead of adding, we subtract B X D from both members, we have {A-B) X D = {C-D)xB; which gives A — B : B : : C — D : D ; that is, If four quantities are proportional, they will be in proportion hy composition or division. 185. If we have B__ D_ A~ C 184 ELEMENTS OF ALGEBRA. [CHAP. Vll. and multiply the numerator and denominator of the first member by any number m, we obtain = -— and mA : mB : : C : D ; that is, mA C Equal multiples of two quantities have the same ratio as the quan- t.Lties themselves. 186. The proportions A : B : : C : D, and A : B : : E '. F. give A X D = B X C, and A x F= B x E; adding and subtracting these equations, we obtain A{D±F)=B{C:hE), or A : B : : C± E : D±F\ that is. If C and D, the antecedent and consequent, he augmented or diminished by quantities E and F, lohich have the same ratio as C to D, tlie resulting quantities will also have the same ratio. 187. If we have several proportions, A : B : : C : D, which gives A X D = B X C, A : B : : E : F, " " A x F = B x E, A : B : : G : H, " « Ax H= B x G. &c., &c., we shall have, by addition, A{D-{- F+ H)=B{C -}- E + G); and by separating the factors, A : B : C + E+G: D + F-j-H; that is. In any riumber of proportions having the same ratio, any antece- dent will be to its consequent, as the sum of the antecedents to the sum of the consequents. 188. If we have four proportional quantities A : B : : C : D, we have — = — ■ ; and raising both members to any power, as the nth, we have 5" _ D» ~A^ ~~C^' auii consequently, A'^ : B" : : C" : D" ; that is, E : F : : G : H, CHAP. VII.] GEOMETRICAL PROGRESSION. 185 If four quantities are proportional, any like powers or roots will be proportional. 189, Let there be two sets of proportions, A : B : : C : D, which gives — = — , ^ AC E G' Muhiply them together, member by member, we have 7?P T)fT J^= QQ^ ^'hich gives AE : BF : : CG : DH; that is, In two sets of proportional quantities, the products of the corres- ponding terms will be proportional. Of Geometrical Progression. 190. In the proportions which have been considered, it has only been required that the ratio of the first term to the second should be the same as that of the third to the fourth. If we im- pose the farther condition, that the ratio of the second to the third shall also be the same as that of the first to the second, or of the third to the fourth, we shall have a series of numbers, each of which, divided by the preceding one, will give the same ratio. Hence, if any term be multiplied by this quotient, the product will be the succeeding term. A series of numbers so formed is called a geojnetrical progression. Hence, A geometrical progression, or progression by quotients, is a series of terms, each of which is equal to the product of that which precedes it by a constant number, which number is called the ratio of the progression. Thus, in the two series, 3, 6, 12, 24, 48, 96, . . . 64, 16, 4, 1, -1, 1, . . . 4 16 each term of the first contains that which precedes it tvnce, or is equal to double that which precedes it ; and each term of the second contains the term which precedes it one-fourth times, oi is a fourth of that which precedes it. These are geometrical pro- 186 ELEMENTS OF ALGEBRA. [CHAP. VII. gressions. In the first, the ratio is 2 ; in the second, it is \. The first is called an increasing progression, the second a dc' creasing progression. Let a, b, c, d, e, f, . . . denote numbers in a progression by quotients : they are written thus : a : b : c : d : e : f : g . . . and it is enunciated in the same manner as a progression by dif- ferences. It is necessary, however, to make the distinction, thai one is a series of equal differences, and the other a series of equal quotients or ratios. It should be remarked, that each term of the progression is at the same time an antecedent and a con- sequent, except the first, which is only an antecedent, and the last, which is only a consequent. 191. Let r denote the ratio of the progression a : b : c : d . . . ; r being > 1 when the progression is increasing, and r < 1 when it is decreasing. We deduce from the definition, the following equations : b = ar, c = br = ar"^, d ^= cr ^ ar'^, e =z dr ^ ar* . . . ; and, in general, any term n, that is, one which has n — 1 terms before it, is expressed by cr""!. Let I be this term ; we have the formula I = ar''-\ by means of which we can obtain any term without being obliged to find all the terms which precede it. That is, Any term of a geometrical progression is equal to the first term multiplied by the ratio raised to a power whose exponent denotes the number of preceding terms. EXAMPLES. 1. Find the 5th term of the progression 2 : 4 : 8 : 16, &c., in which the first terra is 2, and the common ratio 2. 5th term =2 x 2* = 2 X 16 = 32. CilAP. VII.] GEOMETRICAL PROGRESSION. 187 2. Find the 8th term of the progression 2 : 6 : 18 : 54 . . . 8th term = 2 X 3^ = 2 X 2187 = 4374. 3. Find the 12th term of the progression 64 : 16 : 4 : 1 : — . . 4 / 1 \^i 43 1 1 12th term = 64 ( — ) =-— = — = — — . V4/ 411 48 65536 192. We will now explain the method of determining tlie sura of n terms of the progression a : b : c : d : e : f : . . . : i : k : I, of which the ratio is r. If we denote the sum of the series by S, and the 7ith. term by /, we shall have S = a + ar + ar"^ . . . . + ar"~'^ + a?""~'. If we multiply both members by r, we have Sr = ar -{- ar^ + ar^ . . . + a?'""^ + ar" ; and by subtracting the first equation, ar^ — a Sr — S = ar"^ — o, whence, S = — ; T — \ and by substituting for a;-", its value Ir, we have Ir — a S = r — 1 That is, to obtain the sum of any number of terms of a pro- gression by quotients, multiply the last term by the ratio, subtract the first term from this product, and divide the remainder by the ratio diminished by unity. EXAMPLES. 1. Find the sum of eight terms of the progression 2 : 6 : 18 : 54 : 162 . . . : 2 X 3^ = 4374 S = i^''- = illH?-^ = 6560. r - 1 2 188 ELEMENTS OF ALGEBRA. [CHAP. VII. 2. Find the suin of five terms of the progression 2 : 4 : 8 : 16 : 32 ; . . . . r — 1 1 3. Find the sum of ten terms of the progression 2 : 6 : 18 : 54 : 162 . . . 2 X 3^ = 39366. Ans. 59048. 4. What debt may be discharged in a year, or twelve months, by paying $1 the first month, $2 the second month, $4 the third month, and so on, each succeeding payment being double the last ; and what will be the last payment 1 Ans. Debt, $4095 ; last payment, $2048, 5. A gentleman married his daughter on New-Year's day, and gave her husband Is. toward her portion, and was to double it on the first day of every month during the year : what was her portion ? Ans. £204 15^. 6. A man bought 10 bushels of wheat on the condition that he should pay 1 cent for the first bushel, 3 for the second, 9 for the third, and so on to the last: what did he pay for the last bushel and for the ten bushels 1 Ans. Last bushel, $196,83 ; total cost, $295,24. 193. When the progression is decreasing, we have r <^ I and / < ff ; the above formula for the sura is then written under the form 1 — r in order that the two terms of the fraction may be positive. By substituting ar"~'^ for I in the expression for S, it becomes ar'^ — a r^ « — '^^" b = — — -, or S = . r — 1 1 — r EXAMPLES. 1. Find the sum of the first five terms of the progression 32 : 16 : 8 : 4 : 2. 32 — 2 X — „, S = °-^ = ?. =21 = 63. 1 - r 1 1 CHAP. VII.] GEOMETRICAL PROGRESSION. 189 2. Find the sura of the first twelve terms of the progression 64:16:4:l:-:...:64y , or -3-^. 1 1 1 64 - — -— X — 256 ^^a — lr 65536 4 _ 65536 _ 65535 ^ ~ l^ir> = 3 ~ 3 - ^ + 196608' ~A We perceive that the principal difficulty consists in obtaining the numerical value of the last term, a tedious operation, even when the number of terras is not very great. 194. Remark. — If, in the formula ^^ T-\ ' we suppose r = 1, it becomes This result is a symbol of indetermination. It often arises from the existence of a coraraon factor (Art. 113), which becomes nothing by making a particular hypothesis on the quantities which enter the equation. If this common factor can be divided out, the expression will assume a determinate form. This, in fact, is the case in the present question ; for, the expression r" — I is divisi- ble by r — 1 (Art. 61), and gives the quotient r»-i + ?-""^ 4- ?-"-3 + . . . + r + 1 ; hence, the value of S takes the form S = ar''-^ + ar'^-'^ + ar'^-'^ + ... -\- ar ■\- a. Now, making r = 1 , we have S = a + o + a+ • • • +o = na. We can obtain the same result by going back to the proposed progression a : 6 : c : . . . : /, which, in the particular case of r = 1, reduces to a : a : a \ . . . '. a, the sum of which series is equal to na. J 90 ELEMENTS OF ALGEBRA. [CHAP. VIT The result — , given by the formula, may be regarded as in dicatinijf that the series is characterized by some particular prop erty. In fact, the progression, being entirely composed of equal terms, is no more a progression by quotients than it is a pro- gression by differences. Therefore, in seeking for the sum of a certain number of the terms, there is no reason for using the ibrmula a{r- -I) ^- r-l ' in preference to the formula {a + I)n ^ - 2 • which gives the sum in the progression by differences. 195. The consideration of the five quantities, a, r, n, I, and S, which enter into the formulas Ir — a I = ar"-i and S r-l' give rise to several curious problems. Of these cases, we shall consider here, only the most im- portant. We will first find the values of S and r in terms of a, Z, and n. The first formula gives r"~i = — , whence r = \/ — . a » a Substituting this value in the second formula, the value of S will be obtained. The expression furnishes the means for resolving the following question, viz. : To jiml m mean proportionals hettoeen two given numbers a and b ; that is, to find a number m of means, which will form with a and b, considered as extremes, a progression by quotients. To find this series, it is only necessary to know the ratio. Now, the required number of means being m, the total number CHAP. VII.] GEOMETRICAL PROGRESSION. 19] of terms consiJereJ, will be equal to m + 2. Moreover, \vc have I = b, therefore the value of r becomes * a tliat is, we must divide one of the given numbers (b) hy the other (a), then extract that root of the quoticiit whose index is one more than the required number of means. Hence, the progression is / / . / L* a : a \/ — : a \/ — ;^ : a \/ — - : . . . 0. * a ''a- * a-* Thus, to insert six mean proportionals between the numbers 3 and 384, we make to = G, M'hence 3S4 7 / = V128 = 2; 3 whence we deduce the progression 3 : 6 : 12 : 24 : 48 : 96 : 192 : 384. Remark. — When the same number of mean proportionals are inserted between all the terms of a progression by quotients, taken two and two, all the progressions thus formed will constitute a single progression. Of Progressions having an ivjlnite Number of Terms. 196. Let there be the decreasing progression a : b : c : d : e : f '. . . .y containing an indefinite number of terms. The formula a — ar^ 1 — r which represents the sum of n terms, can be put under the form a ar" S = 1 — r 1 — r Now, since the progression is decreasing, r is a proper frac- ii(m, and r" is also a fraction, which diminishes as n increases. Therefore, the greater the number of terms we take, the more uill X r" diminish, and consequently, the more will the ] — r 192 ELEMENTS OF ALGEBRA. [CHAP. VII partial sum of these terms approximate to an equality with the first part of S ; that is, to • . Finally, when n is takec 1 — r greater than any given number, or n = 00, tlien X r* 1 — r will be less than any given number, or will become equal to , and the expression will represent the true value of the 1 — r sjim of all the terms of the series. Whence, we may conclude, that the expression for the sum of the terms of a decreasing progression, in which the number of terms is infinite, is 1 —r This is, properly speaking, the limit to which the partial sums approach, by taking a greater number of terms of the progression. The number of terms may be taken so great as to make the dif- ference between the sum, and -^-, as small as we please, and 1 — r the difl'erence will only become nothing when the number of terms' taken is infinite. EXAMPLES. 1, Find the sum of 1111 . ^ . 1 : — : — : — : — to infinity, 3 9 27 81 ^ We have, for the sum of the terms, S = —- = ^ ^ 1 -r , _ J_ 2 3 2. Again, take the progression 1 : — : — : — : — : — : &c. . . , 2 4 8 16 32 We have S = — ^— = — ^—- - 2. What is the error, in each example, for » = 4, /i == 5, « — 6 ■ C!ii4P. VIIJ.l PERMUTA.TIONS AND COMBINATIONS. 1 i3 CHAPTER VIII. FORMATION OF POWERS, AND EXTRACTION OF ROOTS OF ANY DEGREE. CALCULUS OF RADICALS. INDETERMINATE CO-EFFICIENTS 197. The resolution of equations of the second de^ee supposes the process for extracting the square root to be known. In like manner, the resolution of equations of the third, fourth, &c. de- gree, requires that Ave should know how to extract the third, fourth, &c. root of any numerical or algebraic quantity. The power of a number can be obtained by the rules of mul- tiplication, and this power is subjected to a certain law of forma- tion, which it is necessary to know, in order to deduce the root from the power. Now, the law of formation of the square of a numerical or algebraic quantity, is deduced from the expression for the square of a binomial (Art. 116); so likewise, the law of a power of any degree, is deduced from the same power of a binomial. We shall therefore first determine the development of any power of a binomial. 198. By multiplying the binomial x-i-a into itself several times, the following results are obtained : (x -\- a) =z X -\- a, (x 4- a)2 = x'^ + 2ax + a^, (x + a)^ =: x^ -\- 3ax^ -f- 3a~x + a^i (x + a)* = a;* 4- 4ax3 -|- da'^x'^ + 4a^x + a*, (x + a)5 — x^ -}- 5ax^ + lOa-x^ + lOa^x^ -f 5a* x -f a\ By examining the developments, Ave readily discover the law according to which the exponents of x decrease and those of a increase, in the successiA^e terms ; it is not, howcA-er, so easy to 13 JV4 ELEMENTS OK ALGEBRA. [CHAP. V1.U. discover a law for the co-efficients. Newton discovered one, by means of which a binomial may be raised to any power, without first obtaining all of the inferior powers. He did not, however, explain the course of reasoning which led hin\ to the disc )very ; but the law has since been demonstrated in a rigorous manner. Of all the known demonstrations of it, the most elementary is that which is founded upon the theory of combinations. However, as the demonstration is rather complicated, we will, in order to simplify it, begin by resolving some problems relative to permuta- tions and combinations, on which the demonstration of the formula for the binomial theorem depends. Theory of Pcrmutatiojis and Combinations. 199. Let it be proposed to determine the whole number of loays in which several letters, a, b, c, d, &c., can be written one after the other. The result corresponding to each change in the posi- tion of any one of these letters, is called a permutation. Thus, the two letters a and b furnish the two permutations, ab and ba. abc acb cab bac bca .cba Permutations, are the results obtained by writing a certain num- ber of letters one after the other, in every possible order, in such a manner that all the letters shall enter into each result, and each letter enter but once. Problem 1. To determine the number of permutations of which n letters are susceptible. In the first place, two letters, a and b, evidently ( ab give two permutations. ( ba Therefore, the number of permutations of two letters is ex pressed by 1 x 2. Take the three letters, a, b, and c. Reserve r c either of the letters, as c, and permute the other } ab two, giving ' ba In like manner, the three letters, a, b, c, furnish six permutations. CHAP. VIII.] PERiMOTATIONS AXD COMBINATIONS. 195 cab acb ahc chn hca _bac Now, the third letter c may be placed before ah, between a and h, and at the right of ab ; and the same for ba : that is, in one of the first permuta- tions, the reserved letter c may have three different ■places, giving three permutations. Now, as the same may be shown for each one of the first permutations, it follows that the whole number of permutations of three letters will be expressed by, 1 X 2 X 3. If now, a fourth letter d be introduced, it can have four places in each one of the six permutations of three letters : hence, all the permutations of four letters ^vill be expressed by, 1 x 2 x 3 x 4. In general, let there be n letters, a, b, c, &c., and suppose the total number of permutations of n — 1 letters to be known ; and let Q denote that number. Now, in each one of the Q permuta- tions, the reserved letter may have n places, giving n permutations : hence, when it is so combined with all of them, the entire num- ber of permutations will be expressed by Q x n. Let n^2. Q will then denote the number of permutations that can be made with a single letter; hence, Q = 1, and in this particular case we have, Q x n = 1 X 2. Let 71 = 3. Q will then express the number of permutations of 3 — 1 or 2 letters, and is equal to 1x2. Therefore, Q X n is equal to 1 x 2 x 3. Let n = 4. Q in this case denotes the number of permutations of 3 letters, and is equal to 1 x 2 x 3. Hence, Q x n becomes 1 X 2 X 3 X 4 ; and similarly, when there are more letters. 200. Suppose we have a number m, of letters a, b, c, d, &c. If they are written one after the other, in classes of 2 and 2, or 3 and 3, or 4 and 4 ... in every possible order in each class, in such a manner, however, tliat the number of letters in each result shall be less than the number of given letters, we may de- mand the whole number of results thus obtained. These results are called arrangements. > Thus, ab, ac, ad, . . . ba, be, bd, . . . ca, cb, cd, . . . are ar- rangements of m letters taken 2 and 2 ; or in sets of 2 each. In like manner, abc, abd, . . . bac, bad, . . . acb, acd, . . . are arrangements taken in sets of 3. Arrangements, are the results obtained by writing a number m of Utters one after the other in every possible order, in sets of 2 and 196 ELEMENTS OF ALGEBRA. [CHAP. VIII. 2, 3 and 3, 4 and 4 . . . n and n ; m being > n ; lliat is, the number of letters in each set being less than the whole number of letters considered. If, however, we suppose n = m, the ar- rangements taken n and n, will become simple permutations . Problem 2. Having given a number m of letters a, b, c, d . . . to determine the total number of arrangements that may he formed of them by taking them n at a time ; m being supposed greater than n. Let it be proposed, in the first place, to arrange the three let- ters, a, b, and c, in sets of two each. First, arrange the letters in sets of one each, and for each set so formed, there will be two letters reserved : the reserved letters for either arrangement, being those which do not enter. When we arrange with reference to a, the reserved letters will be b and c ; if with reference to b, the reserved letters will be a and c, &c. Now, to any one of the letters, as a, annex, in suc- cession, the reserved letters b and c : to the second arrangement h, annex the reserved letters a and c ; and to the third arrangement, c, annex the reserved letters a and b : this orives , , CO And since each of the first arrangements is repeated as many times as there are reserved letters, it follows, that the arrange- ments of three letters taken two in a set, will be equal to the ar- rangements of the same number of letters taken one in a set, multi- plied by the number of reserved letters. het it be required to form the arrangement of four letters, a, b, c, and d, taken 3 in a set. First, arrange the four letters in sets of two : there fab will then be two reserved letters. Take one of the sets and write after it, in succession, each of the re- served letters : we shall thus form as many sets of three letters each as there are reserved letters ; and these sets differ from each other by at least the last letter. ° -^ Take another of the first arrangements, and annex in succession the reserved letters ; we shall again form as many different arrangements as there are reserved letters. Do the same for all of the first arrangements, and it is plain, that the whole number of arrangements L (/ c ab ac ha he ca ba ac c a ad da be ch hd dh CHAP. VIII.] PERMUTATIONS AND COMBINATIONS. 197 which will be formed, of four letters, taken 3 and 3, will he equal to the arrangements of the same letters, taken two in a set, mul- tiplied hy the number of reserved letters. Ill order to resolve this question in a general manner, suppose the total number of arrangements of m letters, taken n — 1 in a set, to be known, and denote this number by P. Take any one of these arrangements, and annex to it, in suc- cession, each of the reserved letters, and of which the number is m — [ii — 1), or m — ra + 1 : it is evident, that we shall thus form a number m — ra + 1 of ^^w arrangements of n letters, each differing from the other by the last letter. Now, take another of the first arrangements of w — 1 letters, and annex to it, in succession, each of the m — n -\- 1 letters which do not make a part of it ; we again obtain a number rn — » + 1 of arrangements of n letters, differing from each other, and from those obtained as above, at least in one of the n — \ first letters. Now, as we may in the same manner, take all the P arrangements of the m letters, taken n — 1 in a set, and annex to each in succession each of the m — n + 1 other letters, it follows that the total number of arrangements of m letters taken n in a set, is expressed by P {m — n -I- 1). To apply this in determining the number of arrangements of m letters, taken 2 and 2, 3 and 3, 4 and 4, or 5 and 5 in a set, make ra = 2 ; whence, m — n -\- \ ^^ m — 1; Pin this case, will express the total number of arrangements, taken 2 — 1 and 2 — 1, or 1 and 1 ; and is consequently equal to ' m ; therefore, the formula becomes m{in — 1). Let n = 3 ; whence, m — n+l=??j — 2; P will then ex- press the number of arrangements taken 2 and 2, and is equal to m[m—V)\ therefore, the formula becomes m{in — 1) (rn — 2). Again, take n = 4 : whence, m — w+l=m — 3: P will ex- press the number of arrangements taken 3 and 3, or is equal to m,{m — 1) (^ — 2) ; :.Lerefore, the formula becomes »i (m — 1) (»i — 2) (m — 3). 198 ELEMENTS OF ALGEBRA. [CHAP. VIII. Remark. — From the manner in which these results have been deduced, we conclude that the general formula for m letters taken n in a set, is m (ot — 1) (m — 2) (m — 3) . . . . (m — n + 1) ; that is, it is composed of the product of the n consecutive numbers comprised between m and m — n + 1, inclusively. From this formula, that of the preceding Art. can easily be deduced, viz., the development of the value of Q x n. For, we see that the arrangements become permutations when the number of letters entering into each arrangement is equal to the total number of letters considered. Therefore, to pass from the total number of arrangements of m letters, taken n and n, to the number of permutations of n letters, it is only necessary to make w = « in the above development, which gives n(« - 1) (« -2) (n- 3) 1. By reversing the order of the factors, and observing that the last is 1, the next to the last 2, the third from the last 3, &c., we have 1x2x3x4 {n-2){n-\)n, for the development of Q x ri. This is nothing more than the series of natural numbers com- prised between 1 and n, inclusively. 201. When the letters are disposed, as in the arrangements, 2 and 2, 3 and 3, 4 and 4, &c., it may be required that no two of the results, thus formed, shall be composed entirely of the same letters, in which case the products of the letters will be different j and we may then demand the whole number of results thus ob- tained. In this case, the results are called combinations. Thus, ab, ac, he, . . . ad, bd, . . . are combinations of the let- ters a, b, and c, Sic, taken 2 and 2. In like manner, abc, abd, . . . acd, bed, . . . are combinations of the letters taken 3 and 3. Combinations, are arrangements in which any two will differ from each other by at least one of the letters which enter them. Hence, there is an essential difference in the signification ol the words, permutations, arrangements, and combinations. CHAP. VIII.] PLRMUTATIOXS AND COMBINATIONS. 199 Problem 3. To determine the total number of different combina- tions that can be formed of m letters, taken n in a set. Let X denote the total number of arrangements that can be formed of m letters, taken n and n : Y the number of permuta- tions of n letters, and Z the total number of different cornbinations taken n and 7i. It is evident, that all the possible arrangements of m letter?, taken ra in a set, can be obtained, by subjecting the n letters of each of the Z combinations, to all the permutations of which these letters are susceptible. Now, a single combination of n let- ters gives, by hypothesis, Y permutations ; therefore Z combina- tions will give Y X Z arrangements, taken ,i and n ; and as X denotes the total number of arrangements, it follows that the three quantities, X, Y, and Z, give the relations X = Y X Z ; whence, Z = -:r=. But we have (Art. 200), X= P(m-n+ 1), and (Art. 199), Y = Q x n ; ^ P(m — n+l) P m — n+l therefore, Z = — ^— ^^— -' = -77 X ^!—. Q X n Q n Since P expresses the total number of arrangements, taken n — 1 and n — 1 , and Q the number of permutations of n — 1 P letters, it follows that — expresses the number of different cora- binations of m letters taken n — 1 and n — 1. To apply this to the case of the combinations of m letters taken 2 and 2, 3 and 3, 4 and 4, &c. P Make n = 2, in which case, — expresses the number of com- binations of m letters taken 2 — 1 and 2—1, or taken 1 and 1, and this number must be equal to m ; the above formula there- fore becomes m — 1 m (m — 1) m X or — ^ -. 2 1.2 P . Let n =: 3 ; — will express the number of combinations taken 200 ELEMENTS OF ALGEBRA. 'CHAP. VIII. mhn — 1 ) 2 and 2, and is equal to -— ; and the forunua becomes t m[m — 1) (w — 2) r2T3 * In like manner, we find the number of combinations of m. let- ters taken 4 and 4, to be m{m — \) (ot — 2) ( ct — 3) ^ L2^3^4 ' and, in general, the number of combinations of m letters taivcn n and n, is expressed by m (m — 1) (wz — 2) (m — 3) . . . {m — n -\- 1) L2.3.4 . . . {;j— l).n ' which is the development of the expression P(?n — n4- 1) Q Xn We may here observe that, if we have a series of numbers, decreasing by unity, and of which the first is m and the last m — p, m and j5 being entire numbers, that the product of these numbers will be exactly divisible by the continued product of all the natural numbers from 1 to p + 1 inclusively ; that is, w? (?M — I) (m — 2) (m — 3) . . . {m — p) 1 . 2 '. 3 '. 4 : . . . {JTl) is a whole number. For, from what has been proved, this ex- pression represents the number of different combinations that can be formed of m letters taken in sets of /> + 1 and p -{- I. Now this number of combinations is, from its nature, an entire number ; therefore the above expression is necessarily a whole number. Demonstration of the Binomial Theorem. 202. In order to discover more easily the law for the develop- ment of the with power of the binomial sc -\- a, let us observe the law of the product of several binomial factors, x -\- a, x -^ b, X + c, X -\- d . . . of which the first term is the same in each, and the second terms different. CHAP. VIII.] BINOiMTAl THEOREM. X -}- a X + b X + 1st product - a;2 4- a ab + b X -}- c 2d - - - x^ -{- a 'x^ + lib X -f abc + b + ac + c + be X -\- d 3d - - ' X* + a ,t3 + ab a^ + abc + b + ac + abd + c + ad -\- acd + d 4- + + be bd cd + bed 201 X + abed From these products, obtained by the common rule for alge- braic multiplication, we discov^er the following laws : — 1st. With respect to the exponents, we observe that, the ex- ponent of X, in the first term, is equal to the number of binomial factors employed. In each of the following terms to the right, this exponent diminishes by unity to the last term, where it is 0. 2d. With respect to the co-efficients of the different powers of a,', that of the first term is unity ; the co-efficient of the second term is equal to the sum of the second terms of the binomials ; the co-efficient of the third term is equal to the sum of the prod- ucts of the different second terms, taken two and two , the co- efficient of the fourth term is equal to the sum of their different products, taken three and tliree. Reasoning from analogy, we may conclude that the co-efficient of the term which has n terms be- fore it, is equal to the sum of the different products of the second terms of the m binomials, taken n and n. The last term of the product is equal to the continued product of the second terms of the binomials. In order to prove that this law of formation is general, suppose that it has been proved true for a number m of binomials ; let us 202 ELEMENTS Of ALGEBRA. [CHAP. VIII. see if it will continue to be true when the product is multiplied by a new factor. For this purpose, suppose to be the product of ?« binomial factors, AV"~" representing the term which has n terms before it, and ilfT'"~"+i the term which immediately precedes. Let a: + ^ be the new factor by which we multiply ; the prod- uct when arranged according to the powers of x, will be + B I a;"'-^ + C -{- Ak\ + Bk + . . . +N -\-Mk -hUk. From which we perceive that the law of the exi-)onents is evi dently the same. With respect to the co-efficients, we observe, 1st. That the co-efficient of the first term is unity ; and 2d. That A -\- k, or the co-efficient of a"', is the sum of the second terms of the m -j- 1 hiiiomials. 3d. Since, by hypothesis, B is the sum of the different products of the second terms of the m binomials, taken two and two, and since A x k expresses the sum of the products of each of the second terms of the m binomials by the new second term k ; there- fore, B -f Ak is the sum of the different products of the second, terms of the m + 1 binomials, taken two and two. In general, since N expresses the sum of the products of the second terms of the m binomials, taken n and n, and M the sum of their products, taken n — 1 and n — 1 ; if we multiply the last set by the new second term k, then N -J- Mk, or the co-effi- cient of the term which has n terms before it, will be equal to the sum of the different products of the second terms of the m -\- I binomials, taken n and n. The last term is equal to the continued product of the second terms of the m + 1 binomials. Therefore, the law of composition, supposed true for a number m of binomial factors, is also true for a number denoted hy m -{- I Hence, it is true for ?n + 2, &c., and is therefore general. 203. Let us now suppose, that in the product resulting fron he multiplication of the m binomial factors, X -\- a, X -\- b, X -\- c, X -}• d, . . . . CHAP. VIII.] BINOMIAL THEOREM. 203 we make, a =: fj — c ^^ d: . . . . we shall then have (x + a) (x + 6) (,T + c) =(x + a)'". The co-efficient of the first term, x"", will become 1. The co-effi- cient of a'"'"^ being a + i + c + J, . . . will be a taken 7?i times ; that is, ma. The co-efficient of cif~^, being ab -\- ac -\- ad . . . . reduces to a^ _j_ ^2 -|- a^ . . . that is, it becomes a^ taken as many times as there are com- binations of m letters, taken two and two, and hence reduces (Art. 201), to m — I 2 2 The co-efficient of a;'""^ reduces to the product of a^, multiplied by the number of different combinations of m letters, taken three and three ; that is, to m — 1 m — 5 &c. 2 3 In general, let us denote the term, which has n terms before it, by Nx"'~'^. Then, the co-efficient A'' will denote the sum of the products of the second terms, taken n and n ; and when all of the terms are supposed equal, it becomes equal to a" multiplied by the number of different combinations that can be made with m letters, taken n and n. Therefore, the co-efficient of the general term (Art. 201), is p (m - n + I) iV.= — ^—- a", Q X 71 from which we deduce the formula, m — 1 {x + a)'" = x'" -f max'"~^ + ?n . — _ w^ar'""^ 771 — 1 m— 2 , , P(7n — 7i + l) 4- m . . «3^'"-3 . . . -4 ^~-- a»j;'"-" . . . + a"". 2 3 Q . n The term P (m — n + \) 5^ ;^ L Qi^m-n Qn IS called the general ter7n, because by making n = 2, 3, 4, all of the others can be deduced from it. The term which im- mediately precedes it, is evidently. 201 ELEMENTS OF ALGEBRA. [CHAP. VIIl. expresses the number of combinations of m letters taken n — 1 and n — 1. Hence, we see, that the co-efficient Q X n P n equal to the co-efficient — of the preceding term, muhipUed by m — n -\- 1, the exponent of x in that term, and divided by «, the number of terms preceding the required term. P . Since — is the co-efficient of the preceding term, we may, by observing how the co-efficients are formed from each other, ex- press the co-efficient of the general term thus, _ m {in — ]) (in — 2) (m — 3) . . . {m — n -{- 2) (m — n + I) ^ ~ 1 r~2 '. 3 '. 4 T. {n — f) ; n '' The simple law, demonstrated above, enables us to determine the co-efficient of any term from the co-efficient of the preceding term. The co-rffi,cicnt of any term is formed by multiplying the co-cfp,- cient of the preceding term by the exponent of x in that term, and dividing the product by the number of terms which precede the re- quired term. For example, let it be required to develop (a; + of- From this law, we have, (x -f ay = a:« + Qax^ + Iba"^ x^ + 20d^x^ + 15a%2 _[_ q^^^ -\- a^. After having form.ed the first two terms from the general formula x^ -\- max'^'~'^ -{- , . . . multiply 6, the co-efficient of the second term, by 5, the exponent of x in that term, and then divide the product by 2, which gives 15 for the co-efficient of the third term. To obtain that of the fourth, multiply 15 by 4, the exponent of x in the third term, and divide the product by 3, the number of terms which precede the fourth ; this gives 20 ; and the co-efficients of the other terms are found in the same way. In like manner, we find (,^ _^ a)io = ^\Q ^ iQax^ + 45a2a;8 j^ \20a^x'' + 2l0a^x^ CHAP. VIII.] BINOMIAL THEORE.M. 203 204. It frequently occurs that the terms of the binomial arc affected with co-efficients and exponents, as in the following ex- ample : Let it be required to raise the binomial 3a"c — 2bd to the fourth power. Placing Ba^c = x and — 2bd = y, we have (a; + yY = x* -\- 4x^y -j- 6x^)/^ -j- Axy^ + y* ; and substituting for x and y their values, we have (3a2c - 2hdy = (Sa^c)* + 4 (3a2c)3 (_ 2bd) + 6 (Sa'^cf (- 2bd)^ + 4 (3a2c) (_ 2bdy + (- 2bdy, or, by performing the operations indicated, (3a^c — 2bdY = 81a^c* — 2l6a^c^d + 216a*cH^d^ — 96a'^cb^d^ + I6b*d\ The terms of the development are alternately plus and minus, as they should be, since the second term is — . 205. The poAvers of any polynomial, may easily be found bv the binomial theorem. For example, raise a -\- b -{• c to the third power. First, put b -\- c z= d. Then {a -{- b + cf = (a + df = a^ -\- 2aH + SarP + J3 . and by substituting for the value of d, (a + 6 + c)3 = a3 -f 2an + Sai^ _^ ^3 3a2c + 362c + Qahc + 3ac2 4- 35c2 -f c3. This development is composed of the cubes of the three ter??is, plus three times the square of each term by the first powers of the two others, plus six times the product of all three terms. To apply the preceding formula to the development of the cui:e of a trinomial, in which the terms are affected with co-efficieiif.s and exponents, designate each term by a single letter, and perfurm the operations indicated; then replace the letters introduced by their values 20G ELEMENTS OF ALGEBRA. [CHAP. VIII, From this rule, we will find that (2a2 _ 4ab + 3^2)3 ^ Sa<^ — 48a^b + \32a^P - 208a^^ + I98a-b^ — ]08ab^ + 27b^. The fourth, fifth, &c. powers of any polynomial can be developed m a similar manner. Conseqiiences of the Binomial Formnla. 206. The development of the binomial expression (x -\- a)'^ will always contain ?w + 1 terms. Hence, if we take that term of the development which has ?i terms before it, the number of terms after it will be expressed by m — n. Let us now seek the co-efficient of the term which has n terms after it, and which, consequently, has m — n terms before it. We obtain this co-efficient by simply substituting m — n for ra, in the last value of N in Art. 203. We then have, _ m{m- l)(m-2) . . (n + 2) (?z + 1) ~ 1 . 2 . 3 . . . (m — ?t — 1) (m — ?i)' As we can always take the term which has n terms before it, nearer to the first term than the one which has m — n terms be- fore it, we will examine that part of the co-efficient which is derived from the terms lying between these two. We may write m{m—\). . . { m — n\-\).{m — n).{m-n-\). . . (n + 2) . (n +1) 1 . 2 n («+l) • (w + 2) . . {in — n — \)\m-i{) Now, by cancelling the like factors in the numerator and de- nominator, we have ^^ m {in — 1) .... (/7j — n 4- 1) N = — ^^ : hence, 1.2 n In the development of any power of a binomial, the co-ejficients at equal distances from the two extremes are equal to each other. 207. If we designate by K the co-efficient of the term which has n terms before it, that term will be expressed by Za"a;"'-'' ; and the corresponding term counted from the last term of the series, will be Ka'^~~"x'\ Now, the first co-efficient expresses the number of different com- binations that can be formed with m letters taken n and n ; and the second, the number which can be formed when taken m — n CHAP. VIII.] CUBE ROOT OF NUMBERS. 207 and m — n ; we may therefore conclude that, the miniher of dif- ferent combinations of m letters taken n and n, is equal to the num- ber of combinations of m letters taken m — n and m — n. For example, twelve letters combined 5 and 5, give the same number of combinations as when taken 12 — 5 and 12 — 5, or 7 and 7. Five letters combined 2 and 2, give the same number of combinations as when combined 5 — 2 and 5 — 2, or 3 and 3 208. If, in the general formula, (x + a)'" — x'" -{■ max'"-'^ + m — ; «2^m-2 _^^ ^^^ we suppose x = I, a = 1, we have, m — 1 m — I m — 2 (1 + !)"• or 2"^=! + m + m — — + m — — . — — +, &c. That is, the sum of the co-ej^cients of all the terms of the for- mula for the binomial, is equal to the mth power of 2. Thus, in the particular case (x + ay = x^ + 5ax^ + lOa^.r^ + lOa^x- -r 5a*x + a\ the sum of the co-efficients 1 + 5+10+10 + 5+1 is equal to 2^ = 32. In the 10th power developed, the sum of the co-efficients is equal to 2^'^ = 1024. Extraction of the Cube Root of Numbers. 209. The cube or third poioer o^ a number, is the product which arises from multiplying the number twice by itself. The cube root, or third root of a number is either of three equal factors into which it may be resolved ; and hence, to extract the cube root, is to seek one of these factors. Every number which can be resolved into three equal factors that are commensurable with unity, is called a perfect cube; and any number which cannot be so resolved, is called an imperfect cube. The first ten numbers are roots, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10; cubes, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Eeciprocally, the numbers of the first line are the cube ruois of the numbers of the second 208 ELEMENTS OF ALGEBRA. [CHAP. Vllt. We perceive, by inspection, that there are but nine perfect cubes among all the numbers expressed by one, two, and three figures. Every other number, except the nine written above, which can be expressed by one, two, or three figures, will be an imperfect cube ; and hence, its cube root will be expressed by a whole number, plus an irrational number, as may be shown by a course of rea-' soning entirely similar to that pvu-sued in the latter part of Art. 118. 210. Let us find the difference between the cubes of two con- secutive numbers. Let a and a + 1, be two consecutive whole numbers; we have {a+ ly = a^ 4- 3a2+ 3a + 1 ; whence, (a + 1)^ — a^ = Sa^ -|- 3a -j- i. That is, the difference between the cubes of two consecutive whole numbers, is equal to three times the square of the least number, plus three times the number, plus 1. Thus, the difference between the cube of 90 and the cube of 89, is equal to 3 (89)2 + 3 X 89 + 1 = 24031. 211. In order to extract the cube root of an entire number, we will observe, that when the figures expressing the number do not exceed three, the entire part of the root is found by comparing the number with the first nine perfect cubes. For example, the cube root of 27 is 3. The cube root of 30 is 3, plus an irrational number, less than unity. The cube root of 72 is 4, plus an ir- rational number less than unity, since 72 lies between the perfect cubes 64 and 125. When the number is expressed by more than three figures, the process will be as follows. Let the proposed number be 103823. 103 823 47 48 47 64 ~8 48 384 47 ! 398.23 329 192 2304 188 2209 48 47 18432 15463 9216 8836 110592 103823 This number being comprised between 1,000, which is the cube CHAP. Vin.] CUBE ROOT OF NUMBERS. 201? of 10, and 1,000,000, which is the cube of 100, its root will be expressed by two figures, or by tens and units. Denoting the tens by a, and the units by h, we have (Art. 198), (a + hf = a?-\- Zd^b + 2ab'- + l"^. Whence it follows, that the ube of a number composed of tens and units, is made up of four Jistinct parts : viz., the cube of the tens, three times the product of the square of the tens by the units, three times the product of the tens by the square of the units, and the cube of the units. Now, the cube of the tens, giving at least, thousands, the last three figures to the right cannot form a part of it : the cube of tens must therefore be found in the part 103 which is separated from the last three figures. The root of the greatest cube con- tained in 103 being 4, this is the number of tens in the required root. Indeed, 103823 is evidently comprised between (40)'' or 64,000, and (50)^ or 125,000 ; hence, the required root is com- posed of 4 tens, phis a certain number of units less than ten. Having found the number of tens, subtract its cube, 64, from 103, and there remains 39, to which bring down the part 823, and we have 39823, which contains three times the square of the tens by the units, plus the two parts named above. Now, as the square of tens gives at least hundreds, it follows that three times the square of the tens by the units, must be found in the part 398, to the left of 23, which is separated from it by a point. Therefore, dividing 398 by 48, which is three times the square of the tens, the quotient 8 w^ill be the units of the root, or some- thing greater, since 398 hundreds is composed of three times the square of the tens by the units, together with the two other parts. We may ascertain whether the figure 8 is too great, by forming from the 4 tens and 8 units the three parts which enter into 39823; but it is much easier to cube 48, as has been done in the above ta- ble. Now, the cube of 48 is 110592, which is greater than 103823; therefore, 8 is too great. By cubing 47 we obtain 103823; hence the proposed number is a perfect cube, and 47 is its cube root. Remark I. — The units figure could not be first obtained, because the cube of the units might give tens, and even hundreds, and the tens and hundreds would be confounded with those which ari^e from other parts of the cube. 14 10 ELEMENTS OF ALGEBRA. [CriAP. VIII. Remark II. — The operations in the last example have been per- formed on but two periods. It is plain, however, that the same reasoning is equally applicable to larger numbers ; for, by chan- fj;ing the order of the units, we do not change the relation in which they stand to each other. Thus, in the number 43 725 658, the two periods 43 725, have the same relation to each other, as in the number 43725 ; and hence, the methods pursued in the last example are equally ap plicable to larger numbers. 212. Hence, for the extraction of the cube root of numbers, we have the following RULE. I. Separate the given number into periods of three figures each, beginning at the right hand: the left-hand period will often contain less than three places of figures. II. Seek the greatest cube in the first period, at the left, and set its root on the right, after the manner of a quotient in division. Sub- tract the cube of this figure of the root from the first period, arid to the remainder bring down the first figure of the next period, and call ihis number the dividend. III. Take three times the square of the root just found for a di' visor, and see how often it is contained in the dividend, and place the quotient for a second figure of the root. Then cube the figures of the root thus found, and if their cube be greater than the first two periods of the given number, diminish the last figure ; but if it be less, subtract it from the first two periods, and to the remaindei bring down the first figure of the next period, for a new dividend. IV. Take three times the square of the while root for a new divi- sor, and seek how often it is contained in the new dividend ; the quotient will be the third figure of the root. Cube the whole root., and subtract the result from the first three periods of the given num- ber, and proceed in a similar way for all the periods. Remark. — If any of the remainders are equal to, or exceed. three times the square of the root obtained plus three times this root, vlus one, the last figure of the root is too small and must be aug Tiented by at least unity (Art. 210). CHAP. VIII.] EXTRACTION OF ROOTS. 311 EXAMPLES. 1. V48228544 =364. 2. V27054036008 = 3002. 3. ^483249 — 78, with a remainder 8697. 4. ^91632508641 = 4508, with a remainder 20644129. 5. ^32977340218432 = 32068. To extract the v}^ Root of a ivhole Number. 213. The n''' root of a number, is one of the n equal factors into wliich the number may be resolved. If the factors are com mensurable with unity, the number is said to be a perfect power, if they are not commensurable with unity, the number is said to be an imperfect power. In order to generalize the process for the extraction of roots, we will denote the proposed number by N, and the degree of the root to be extracted by n. If the number of figures in N, does not exceed n, the rational part of the root will be expressed by a single figure : for the n"' power of 9 is the highest power which can be expressed by n figures. Form the ?^'* powers of all the numbers from 1 to 9 inclusive. Compare the given number with these powers, and if either of them is equal to the given number, it will be a perfect power ; if not, the root of the one next less will be that part of the required root wliich can be expressed by a whole number. When N contains more than n figures, there will be more than one figure in the root, which may then be considered as composed of tens and units. Designating the tens by a, and the units by b, we have (Art. 203), n — 1 N = {a -\- b)" = a" + na-'-^i + n — ; a^-^^ +, &c. ; that is, the proposed number contains the n^'^ power of the tens, plus n times the product of the n — l'** power of the tens by the units, plus a series of other parts which it is not necessary to consider. Now, as the n'^ power of the tens, cannot give units of an or- der inferior to 1 followed by n ciphers, the last n figures on the right, cannot make a part of it. They must then be pointed ofT 212 ELEMENTS OF ALGEBRA. [CHAP. VIII. and the root of the greatest n'^ power contained in the figures on the left should be extracted : this root Avill be the tens of the re- quired root. If this part on the left should contain more than n figures, the n figures on the right of it, must be separated from the rest, and the root of the greatest n'^ power contained in the part on the left extracted, and so on. Hence the following RULE. I. Divide the nmnber N into periods of n figures each, beginning at the right hand ; extract the root of the greatest n^^ power con- tained in the left-hand period, and subtract the n'*" poivcr of this figure from the left-hand period. II. Bring down to the right of the remainder derived from the left-hand period, the first figure of the next period, and call this number the dividend. III. Form the n — 1 power of the first figure of the root, mul- tiply it by n, and see how often the product is contained in the divi- dend : the quotient will be the second figure of the root, or something greater. IV. Raise the number thus formed to the xC-^ power, then subtract this result from the two left-hand periods, and to the new remainder bring down the first figure of the next period : then divide the niun- ber thus formed by n times the n — 1 power of the two figures of the root already found, and continue this operation until all the pe- riods are brought down. EXAMPLES. 1. What is the fourth root of 531441? 53 144i I 27 2* = 16 4 X 23 = 32 I 371 (27)* = 531441. We first divide off, from the right hand, the period of four figures, and then find the greatest fourth root contained in 53, the first period to the left, which is 2. We next subtract the 4th power of 2, which is 16, from 53, and to the remainder 37 we bring down the first figure of the next period.' AVe then divide 371 by CHAP. Vllt.] EXTRACTION OF ROOTS. 213 4 times the cube of 2, which gives 11 for a quotient : but this we know is too large. By trying the numbers 9 and 8, we find them also too large: then trying 7, we find the exact root to be 27. 214. Remark. — When the degree of the root to be extracted is a multiple of two or more numbers, as 4, 6, . . . ., the root can be obtained by extracting the roots of more simple degrees, successively. To explain this, we will remark that, {a^Y = a^ X a^ X a^ X a^ = a3+3+:^+3 ^ a^xi ^ ai2. and that in general (Art. 13), (a"')" = a"' X a'" X a"* X a"* . . . = a"*^" : hence, the n*** power of the m"* power of a number, is equal to ike mn'*" power of this number. Let us see if the reciprocal of this is also true. Let \J -yf a := a'' ; then raising both members to the n"^ power, we have, from the definition of the n^^ root, V a = «'« ; and by raising both members of the last equation to the m'* power a = (a'")'" = o/'"". Extracting the mn"^ root of the last equation, we have mil I , yj a ^= a \ " A and hence, since each is equal to a' . Therefore, the n'** root of the m"" root of any number, is equal to the mn''> root of that number. And in a similar manner, it might be proved that n / mn / y a = Y a. By this method we find that 1. V 256 = y -/ 256 = ^/l6 = 4. 2 V 2985984 = y V 2985984 = \/l728 = 12. 214 ELEMENTS OF ALGEBRA. [CHAP. VIII 3 V 1771561 = \J ^fvn\bE\ = 1 1 . 4. V 1679616 = Vl296 = -^^1296 = 6. Remark. — Although the successive roots may be extracted in any order whatever, it is better to extract the roots of the lowest degree first, for then the extraction of the roots of the higher degrees, which is a more complicated operation, is effected upon numbers containing fewer figures than the proposed number Extraction of Roots by Ajjproximation. 215. When it is required to extract the n"^ root of a number which is not a perfect power, the method already explained, will give only the entire part of the root, or the root to within unity. As to the number which is to be added, in order to complete the root, it cannot be obtained exactly, but we can approximate to it as near as we please. Let it be required to extract the n'^ root of the whole number a to within a fraction — ; that is, so near, that the error shall P be less than — . P We will observe that we can write «»" If we denote by r, the root of op" to within unity, the number a X p" ^" (^ "t" ^Y i— = a, will be comprehended between — and ; pn ' ^ pn pn therefore the "■^^c'^ = 512a^^c^. Therefore, in order to raise a monomial to any power, raise the co-efficient to that power, and multiply the exponent of each of (he letters hy the exponent of the power. Hence, to extract any root of a monomial, 1st. Extract the root of the co-efficient and divide the exponent of each letter hi/ the index of the root. 2d. To the root of the co 218 ELEMENTS OF ALGEBRA. [CHAP. VIII. efficient annex each letter with its new exponent, and the result will be the required root. Thus, y^Aa^^c^ = Aa^bc"^ ; \/\Qa%^\^ = laWc. From this rule we perceive, that in order that a monomial may De a perfect power, 1st, its co-efficient must be a perfect power; and 2d, the exponent of each letter must be divisible by the in- dex of the root to be extracted. It will be shown hereafter, how the expression for the root of a quantity, which is not a perfect power, is reduced to its simplest terms. 221. Hitherto, in finding the power of a monomial, we have paid no attention to the sign with which the monomial may be affected. It has already been shown, that whatever be the sign of a monomial, its square is always positive. Let n be any whole number ; then, every power of an even degree, as 2n, can be considered as the n'^ power of the square ; that is, {a^Y = a^n. Hence, it follows, that every power of an even degree, will be es- sentially positive, whether the quantity itself be positive or negative. Thus, (d= 2a^Pcy = + 16a^^^c\ Again, as every power of an uneven degree, 2n -\- 1, is but the product of the power of an even degree, 2«, by the first power ; it follows that, every power of an uneven degree, of a monomial, is affected with the same sign as the monomial itself. Hence, {-{- Aa^y ^ + 64a%^ ; and {— 4a^by = — 64a^^ From the preceding reasonings, we conclude, 1st. That when the degree of the root of a monomial is uneven, the root will be affected with the same sign as the monomial. Hence, V+ 8a3 = + 2a ; \/ - 8a^ = -2a; %/ - 32a^%^ = - 20^. 2d. When the degree of the root is even, and the monomial a posi- tive quantity, the root is affected either with the sign -\- or — . Thus, V81a*5i2= ±3aJ3; \/&l'a^ = ± 2a^ . c5d. When the degree of the root is even, and the monomial nega- tive, the root is impossible ; for, there is no quantity which, beint CHAP. VIII.] EXTRACTION OF ROOTS. 219 raised to a power of an even degree, will give a negative result. Therefore, 4/ 6/ r 8/ V — «. V — 0, W — c, are symbols of operation which it is impossible to execute. They are imaginary expressions (Art. 126), like V— a, y/—b, EXAMPLES. 1. \Vhat is the cube root of Sa%^c^'^'f. Ans. 2a'^bc* 2. What is the 4th root of 81 a^Pc^^ 1 Ans. 3a/;V. 3. What is the 5th root of — 32a^c^'^d^^ ? Ans. — 2ac^d^. 4. What is the cube root of — 125a^^c^ ? Ans. — baWc. Extraction of Roots of Polijnomiah. 222. Let us first examine the law of formation of any power of a polynomial. To begin with a simple example, let us develop (a + y + zy. If we place y -\- z =z u, we shall have, (a + w)3 = a3 + 3a^-u + 3au^ + u^ ; or by replacing u by its value, y -^ z, {a + y + zY = a3 + Sa^ {y + z) + 3a [y + zf + (y + zf ; or performing the operations indicated, (a + y + 2)3 = a3 + 3a2y + 3a^z + 3«y2 + Qayz + 3az^ + y3+ 3y'^z + 3yz^ + ^^• When the polynomial is composed of more than three terms, as a -}- y -\- z -{- X . . . . p, let, as before, u = the sum of all the terms after the first. Then, a -\- u will be equal to the given polynomial, and (a + w)3 = a3 -f 3dhi -|- 3mfi + rfi ; from which we see, that the cube of any polynornial is equal to the cube of the first term, plus three times the square of the first term multiplied by each of the remaining terms, plus other terms. If u does not contain a, it is plain that the exponent of a in each term, as a^, 3a'^u, &c., will be greater than in any of the following terms ; and hence, every term will be irreducible with the terms which precede or follovj it. 220 ELEMKNTS OF ALGEBRA. [CHAP. VIII. If u contains a, as in the polynomial a^ -{- ax -\- b, where u = ax -{- b, the terms will still be irreducible with each other, provided we arrange the polynomial with reference to the letter a. For, if the given polynomial be arranged with reference to a, the exponent of a in the first term will be greater than the exponent of a in XI : hence, its cube will contain a with a greater exponent than will result from multiplying its square by u. Also, the co-efficient of u multiplied by the first term of w, will contain a to a higher power than any of the following terms of the development, and hence, will be irreducible with them ; and the same may be shown for the subsequent terms. In order to extract any root of a polynomial, we will first ex- plain the method of extracting the cube root. It will then be easy to generalize this method, and apply it to the case of any root whatever. Let N be any polynomial, and R its cube root. Suppose the two polynomials to be arranged with reference to some letter, as a, for example. It results from the law of formation of the cube of a polynomial (Art. 222), that in the cube of R, the cube of the first term, and three times the square of the first term by the second, cannot be reduced with each other, nor with any of the following terms. Hence, the cube root of that term of N which contains a, af- fected with the highest exponent, will be the first term of R ; and the second term of R will be found by dividing the second term of N by three times the square of the first term of R. By examining the development of the trinomial a -\- y -\- z, we see, that if we form the cube of the two terms of the root found as above, and subtract it from N, and then divide the first term of the remainder by 3 times the square of the first term of i2, the quotient will be the third term of the root. Therefore, having arranged the terms of N, with reference to any letter, we have, for the extraction of the cube root, the following RULE. I. Extract the cube rout of the first term. II. Divide the second term of 1^ by three times the square of the first term o/" R ; the quotient will be the second term of R. CHAP. VIII.l EXTRACTION OF ROOTS. 221 [II. Having found the first two terms of R, form the cube of this binomial and subtract it from N ; after ivhicJi, divide the first term of the remainder by three times the square of the first term of R : the quotient will be the third term of R. IV. Cube the three terms of the root found, and subtract the cube from N : then divide the first term of the remainder by the divisor ilrcadij used, and the quotient will be the fourth term of the root: the remaining terms, if there are any, may be found in a similar manner. EXAMPLES. 1. Extract the cube root of x^—6x^-\-\5x'^—20x'^-\- Ibx^—Sx+l. a;"— 6j;H 15a;* - 20x3 + 15^2 — 6j; + 1 \yZ_2x-{-\ (a;2 — 2a7)3 = ar6—6a;5+ 12^-4— Sx"^ 3a;* 1st rem. Sx* — 12a;3 +, &c. (a:2— 2a;+l)3 = a;<5— 6a;5 + 15x* — 20a;3 + 15x2— 6x + 1. In this example, we first extract the cube root of x^, which gives a;2, for the first term of the root. Squaring x"^, and multi- plying by 3, we obtain the divisor 3a;* : this is contained in the second term — &x^, — 2a; times. Then cubing the root, and sub- tracting, we find that the first term of the remainder 3a;*, contains the divisor once. Cubing the whole root, we find the cube equal to the given polynomial. Hence, a?2 — 2a; + 1, is the exact cube root. 2. Find the cube root of a^6 + 6a;5 — 40x3 _|_ qq^ _ 54 3. Find the* cube root of 8x6 _ 12x5 + 30x* — 25a;3 + 30x2 _ 12a? + 8 223. The rule for the extraction of the cube root is easily ex- tended to a root with a higher index. For, Let a ■\- b -\- c -\- . . . f be any poljmomial. Let s = the sum of all the terms after the first. Then a -{- s ■= the given polynomial ; and (a + sy = a" -f na"~'^s + other terms. That is, the n*'' power of a polynomial, is equal to the n^^ power of the first term, plus n times the first term raised to the power 222 ELEMENTS OF ALGEBRA. ICHAP. VIII. n — 1, multiplied by each of the remaining terms, -\- other terms of the development. Hence, we see, that the rule for the cube root will become the rule for the n"^ root, by first extracting the n"^ root of the first term, taking for a divisor n times this root raised to the n — 1 power, and raising the partial roots to the n^'^ power, instead of' to the cube. EXAMPLES. 1. Extract the 4th root of 16a* — QQa^x + 2\Qa^x'^ — 216«x3 + 81x*. 16a*— 96a3a;+216a2a:2— 216ax3+81a;* 2a — 3a; (2a — 3.r)* = 16a*- 96a3a;4-216a2a;2— 216ax3 + 81a;* 4x(2a)3 = 32a3 We first extract the 4th root of 16a*, which is 2a. We then raise 2a to the third power, and multiply by 4, the index of the root ; this gives the divisor 22a?. This divisor is contained in the second term — 96a%, — Sa? times, which is the second term of the root. Raising the whole root to the 4th power, we find the power equal to the given polynomial. 2. What is the 4th root of the polynomial, SlaV + 16^*i* — 96a2c53d3 _ 2\QaH'^bd + 216a*c2i2(Z2, 3. Find the 5th root of 32x5 _ 80a;* + 80a:3 — AOx^ + lOx — 1. Calculus of Radicals. 224. When the monomial or polynomial whose root is to be ex- tracted, cannot be resolved into as many equal and rational fac- tors as there are units in the index of the root, it is said to be an imperfect power. The root is then indicated by placing the quantity under the radical sign, and writing over it at the left hand, the index of the root. Thus, the fourth root of 3ai2 -}- 9ac*, is written V3a62 + 9ac5. The index of the root is also called the index of the radical. It is plain that a monomial will be a perfect power, when the numerical co-efficient is a perfect poioer, and the exponent of each Utter exactly dvvisihle hy the index of the root. CHAP. VIU.] CALCULUS OF RADICALS. 223 By the definition of a root (Art. 213), we have (\/abc . . . y^ — abc . . . ; and by the rule for the raising of powers, and since the n'^ powers are equal, the quantities themselves are equal : hence, Y aic . . . . =\/ a x\/ b Xyc... that is, the n'** root of the product of any number of factors, is equal to the product of their n'** roots. 1. Let us apply the above principle in reducing to its sim plest form the imperfect power, y 54tt*6'^c2. We have ybAa^bH"^ = \/21aW X \/2ac^ = Sab y^Zac"^. 2. In like manner, \f^ = 2 y^; and y48a^b^c^ — 2ab^c yi.ac^ ; 3. Also, Vl92a^ki2= y64aV2 x yTd> = 2ac^ y^ab. In the expressions, Sab -y/2ac2, 2 y a^, 2ah'^c ySac^, each quan- tity placed before the radical, is called a co-ejjicient of the radical. 225. The rule of Art. 214 gives rise to another kind of sim- plification. . 6 / Take, for example, the radical expression, y Aa^ ; from this rule, we have ''4a2 \J y \a^. and as the quantity affected with the radical of the second de- gree, y , is a perfect square, its root can be extracted : hence. y~\^^ = y2a. In like manner, yS^aW = \J y SGaH'^ = yOab. In general, m /■ 224 ELEMENTS OF ALGEBRA. [CHAP, VIU that is, when the index of a radical is a multiple of any number w, and the quantity under the radical sign is an exact n^''- power, we can, without changing the value of the radical, divide its index by n, and extract the n*^** root of the quantity under the sign. This proposition is the inverse of another, not less important ; VIZ., the index of a radical may he multiplied hy any number, pro- vided we raise the quantity under the sign to a power of which this 'number is the exponent. For, since a is the same thing as y a", we have, a = -w Y a" 226. This last principle serves to reduce two or more radicals to a common index. For example, let it be required to reduce the two radicals ^/2a and y (c + b) to the same index. By multiplying the index of the first by 4, the index of the sec- ond, and raising the quantity 2a to the fourth power ; then multi- plying the index of the second by 3, the index of the first, and cubing a -i- b, the value of neither radical will be changed, and the expressions will become V^ = V 2*a* = V 16a4 ; and V(7+l) = V (« + ^)^- Hence, to reduce radicals to a common index, we have the fol- lowing RULE. Midtiply the index of each radical by the product of the indices of all the other radicals, and raise the quantity under each radical sign to a power denoted by this product. This rule, which is analogous to that given for the reduction of fractions to a common denominator, is susceptible of similar modifications. For example, reduce the radicals to the same index. Since 24 is the least common multiple of the indices 4, 6, and 8, it is only necessary to m\iUiply the first by G, the second by CHAP. VIII. J CALCULUS OF RADICALS. 225 4, and the third by 3, and to raise the quantities under each rad- ical sign to the 6th, 4tli, and 3d powers respectively, which ffives In applying the above rules to numerical examples, beginners very often make mistakes similar to the following : viz., in redii cing the radicals y 2 and y 3 to a common index, after having multiplied the index of the first, by tliat of the second, and the index of the second by that of the first, then, instead of multiply- ing the f.xponent of the quantity under the first sign by 2, and the exponent of that under the second by 3, they often multiply the quantity under the first sign by 2, and the quantity under the sec- ond by 3. Thus, they would have yr^ y-Z X 2 = y^, and V^'= V3~xl = Vq" Whereas, they should have, by the foregoing rule, y~2 = V (2)2 = \/T, and y/T = %/~{^ = \f^ Reduce y 2, y 4, y ?„ to the same index. Addition and Suhtraction of Radicals. 227. Two radicals are similar, when they have the same index., md the same quantity under the sign. Thus, 3 y ai and 7 -y/ ab ; as also, 3a- yi^' ^^^ ^^^ y^ are similar radicals. In order to add or subtract similar radicals, add or subtract their co-pfficients, and to the sum or difference annex the common radical Thus, 3 VT + 2 \/T= 5 VT"; also, 3 %/T- 2 VT= V^ Again, Ba^/T ± 2c \/T= (3a ± 2c) \fT. Dissimilar radicals may sometimes be reduced to similar radi- cals, by the rules of Arts. 224 and 225. For example, i V"48'aZ^ + b y/lba = \b y,f^ia + 55 yjYa = 9b y/da. [15 J 22G ELEJIENTS OF ALGEBRA. [CHAP. VIIl 2. V8tt3i+ 16a* — V^* + 2a/^3 = 2a V^ + 2a - i V^ + 2a; = (2a — b) yb + 2a. 3. 3 V^+2V2^= 3V^ + 2y2^= 5^2^- When the radicals are dissimilar and irreducible, they can only be added or subtracted, by means of the signs + or — . Multiplication mid Divisio7i. 228. We will suppose that the radicals haA'^e been reduced to a common index. Let it be required to multiply y o by y Z>. If we denote the product by P, we have and by raising both members to the n"^ power, (V^)" X ( VT)" = abzzz P" ; and by extracting the n^^ root, V^ X "y/T= P = \/^; that is, the product of the n"' roots of two quantities, is equal to the n'" root of their product. Let it be required to divide y a by V ^* If we designate the quotient by Q, we have Q; b and by raising both members to the n"* power, "« / — — — Vt » (V by b and by extracting the n"' root, that is, the quotient of the n"^ roots of two quantities, ts equal to the n'** root of their quotient. CHAP. VIII.] CALCULUS OF RADICALS. 22'/ Therefore, for the muhiplication and division of radicals, we have the following RULE. I. Reduce the radicals to a common index. II. If the radicals have co-ejicients, frst midtiphj or divide them separately. III. Multiply or divide the quantities under the radical sign by each other, and prejix to the product or quotient, the common radi' cal sign. , EXAMPLES. 1. The product of ^ c * a ^ cd _ 6a2(a2 + ^>2) 2 The product of 3a V^ X 2h V W^c = Gab \/ 32a*c = 12a^^/2c. 3 The quotient of 4. The product of 3a VTx 55 Vic = 15aJ X ^\/8b*cK 5. Multiply y/~2x V~^ by V y X V y Ans. ^YT. 6 Multiply 2 ^15 by 3 ^10. Ans. 6 V337500. 5 r 3 7. Multiply 4\/-|- by 2\/— . 10 .Ans. 8 V 256 228 ELEMENTS OF ALGEBRA. [CHAP. Vlil •' „ , 2 v^ X VT . , 8. Keduce — , . — ^-?=^ to its lowest terms. Ans. 4^V^28& o T, 1 . /V i X 2 VY . , 9. Reduce V .. ^ 7^-= to its lowest terms. 4 V 2 X V 3 10. Multiply ^2, \fT, and i/T' together. yln^. V648000. 7 rr 3 A— 1 1 . Multiply V -TT' V "o"' ^^^ * V 65 together. V 27 / — / — / — / — 12. Multiply (4\/y+5Vl) by (i/I- + 2 \/i-). 43 13 ^- 13. Divide yVy ^'X (>^+ ^ V y) 14. Divide 1 by VT+ V^ .Ani'. a — b 15. Divide \f^ + V^i" by y^ - \/T. ^ra*. Powers and Roots of Radicals. 229. By raising %/"«" to the ti"' power, Ave have (V7)" = V7x V« X V^- • • = v^. by the rule just given for the uiultiplication of radicals. Hence, for raising a radical to any power, we have the following CHAP. VIII.] CALCULUS OF RADICALS. 229 • RULE. Raise the quantity under the sign to the given power, and affect the result with the radical sign, having the primitive index. If it has a co-efficient, Jirst raise it to the given power. EXAMPLES. 1 . ( V 4^)^ = V{4aT = Vl6«' = 2« \/lfi. 2. (3 y2af = 3-^ . \f(2d)' = 243 ^32^ = 4S6a \/ Aa^. When the index of the radical is a muhiple of the power lo which it is to be raised, the result can be simplified. For, Y 2a = v/ Y 2a (Art. 214): hence, in order to square Y2a, we have only to omit the first radical, which gives Again, to square -y/ 3(5>, we have %/'ob = \/%/^: hence, Consequently, when the index of the radical is divisible hij the exponent of the power to which it is to be raised, perform the di- vision, leaving the quantity under the radical sign unchanged. Let it be required to extract the W^ root of the radical Y a. We have (Art. 214), m I ' _ • / " / "•« /~ V/ Y a = Y o. Hence, to extract the root of a radical, multiply the index of the radical by the index of the root to he extracted, leaving the quan- tity under the sign unchanged. This rule is nothing more than the principle of Art. 214, enun- ciated in an inverse order. 1. yVic^^'V^; and y\/Tc = %/Tc. When the quantity under the radical is a perfect power, of the degree of either of the roots to be extracted, the result can be siinplined. 230 ELEMENTS OF ALGEBRA. [CHAP. VIII Thus, larmer, y : In like m 230. The rules just demonstrated for the calculus of radicals depend upon the fact, that the n^'^ root of the product of several factors, is equal to the product of the n'^ roots of these factors This, however, has been proved on the supposition that, wlien the powers of the same degree of two expressions are equal, the expres- sions themselves are also equal. Now, this last proposition, which is true for absolute numbers, is not always true for algebraic ex- pressions ; for it is easily shown that the same number can have more than one square root, cube root, fourth root, (SfC. Let us denote the algebraic value of the square root of a by X, and the arithinetical value of it hj p; we have the equations oc^ = a, and x"^ = p"^, whence x =^ ± p. Hence we see, that the square of p, (which is the root of «), will give a, whether its sign be + or — . In the second place, let x be the algebraic value of the cube root of a, and p the numerical value of this root; we have the equations x^ = a, and x^ = p^. The last equation is satisfied by making x = p. Observing that the equation x'^ =/)•' can be put under the form ^3 — p3 — 0, and that the expression x^ — p'^ is divisible by a: — p (Art, 61), which gives the exact quotient, x'^ -{- px -{- p^, the above equation can be transformed into (x — p) (*^ + poc -{- p"^) = 0. Now, every value of x which will satisfy this equation, will satisfy the first equation. But this equation can be satisfied by supposing X — p = 0, whence xz=p; or by supposing x^ -\- px -^ p"^ = 0, from which last, we have P P I — ir / — 1 ^ -y — 3 \ CHAP. VIII.] CALCULUS OF RADICALS. 231 Hence, the cube root of a, admits of three different algebraic values, viz., P, p[ r^ j. and p( -^ j. Again, resolve the equation at* = p*, in which p denotes the arithmetical value of y a . This equation can be put under the form x^ —p^ — 0; which reduces to and this equation can be satisfied, by supposing x^ — ^- = 0, whence x :=. ±. p\ or by supposing 3-2 _j_ p2 __ 0, whence x ^ — v — p^^ — P y — ^ We therefore obtain four different algebraic expressions for the fourth root of a. As another example, resolve the equation an6 _p6 _ 0. This equation can be put under the form (a;3 — /)3) (a;3 + p3) _ q ; which may be satisfied by making either of the factors equal to zero. But j;3— p3 = 0, gives , /_idrV-3\ X =z p, and X =: p I ^ j. And if in the equation x^ -{- p^ = 0, we make p = — p', it be- comes x^ — p'^ = 0, from which we deduce . = ,', and ,=,.(:ii±./ZI), or, substituting for p' its value — p, /-l±/^3\ X = —p, and x = —p [^ -^ j. Therefore, the value of x in the equation x^ — p^ = 0, and con 232 ELEMENTS OF ALGEBRA. [CHAP. VIII. sequently, the Gth root of a, admits of six values. If we make a =z -_ , and a= 2 these values may be expressed by p, ap, ap, —p — ctp — a'p. We may then conclude from analogy, that in every equation of the form x^ — a = 0, or x"^ — p'" = 0, x is susceptible of m dif- ferent values ; that is, the m'^ root of a number admits of m different algebraic values. 231. If, in the preceding equations, and the results correspond- ing to them, vi'e suppose, as a particular case, a = 1 , whence p ■— \, we shall obtain the second, tnird, fourth, &c. roo'-s of unity. Thus + 1 and — 1 are the two square roots of unity, because the equation a;- — 1 = 0, gives a: = ±: 1. In like manner, -1 + V-S -1 ->/ — 3 + 1, - ^ 2 ' 2 ' are the three cube roots of unity, or the roots of x^ — 1 = ; and + 1, -1, + V"^ -/^ are the four fourth roots of unity, or the roots of a;* — 1 = 0. 232. It results from the preceding analysis, that the rules for the calculus of radicals, which are exact when applied to abso- lute numbers, are susceptible of some modifications, when applied to expressions or symbols which are purely algebraic ; these modifica- tions are more particularly necessary when applied to imaginary expressions, and are a consequence of what has been said in Art. 230. For example, the product of y — a by y — a, by the rule of Art. 228, would be yf — a X\/— a = \/+ a^. Now, -yfcP- is equal to ± a (Art. 139); there is, then, ap- jKtrently, an uncertainty as to the sign with which a should be all'ected. Nevertheless, the true answer is — a\ for, in order CHAP. VIII.] CALCULUS OF RADIC.A.LS. 233 to square \' m, it is only necessary to suppress the radical; but V — aXy — a =. (-y/ — 0)2 = — a. Again, let it be required to form the product y/ — a X \/ — ^• By ihe rule of Art. 228, we shall have ^/ — a X y/ — b = y/ -\-ab. Now, ^/ ah = zh /J (Art. 230), p being the arithmetical vaiue, of the square root of ah ; but the true result is — j) or — -y/ ah, so long as both the radicals y — a and y — 1^ are a fleeted with the sign +• a 1 ; and y — h For, y — a = -y/ 1 hence, y/ — a X s/ — b = -J a . yf — 1 X -y/ ~h X y = Jah X — 1 = — J~d). By similar methods we find the different powers of be as follows : — 1 to 1. / - 1 X y - 1 = (/- 1)^ = — 1. 3. (V^r = (/zTfr-.c/^T)^- = _ 1 X - 1 = + 1. Again, let it be proposed to determine the product of \/ — a by the y — b which, from the rule, would be \/ + ab, and con- sequently, would give the four values (Art. 230), + ya6, — y ab, -\- y ah . -^ — 1, — y ab . y — 1. To determine the true product, observe that V":rT=VT.V3T, and V^^=VT.V-T. ce \J - a .\/ —h= \/ab. y/ _ 1 But 234 ELEMENTS OF ALGEBRA. [CHAP. VIII. We will apply llie preceding calculus to the verification of the expression -1 + 7^^ 2 considered as a root of the equation a;^ — 1 = ; that is, as one of the cube roots of 1 (Art. 231). From the formula, {a + by =za^ + 30% + Sab^ +b'\ we have i^f^r ' - (-l)^+3(-l)^V-3 + 3(- 1) . (7- 3)2 + (V"^^ — 1 — -v/ — 3 The second value, , may be verified in the same manner. It should be remarked, that either of the imaginary roots is the square of the other ; a fact which may be easily verified. Theory of Exponents. 233. In extracting the /i'''' root of a quantity a"", we have seen that when ot is a multiple of n, we should divide the exponent m by n the index of the root. When m is not divisible by n, the operation of extracting the root is indicated by indicating the di' vision of the two exponents. Thus, m a notation founded on the rule for the exponents, in the extrac- tion of the roots of monomials. In such expressions, the numerator indicates the power to lohich the quantity is to be raised, and the denominator, the root to he extracted. Q t 2 . i 7, Therefore, y d^ = a^ ; and -y/ a' z:^ a*. If it is required to divide a'" by o", in which m and n are posi- tive whole numbers, we know that the exponent of the divisor should be subtracted from the exponent of the dividend, and we have CHAP, VIII.] THEORV OF EXPONKNTS. 235 If m > n, the division will be exact ; but when m < n, the di- vision cannot be effected, but still we subtract, in the algebraic sense, the exponent of the divisor from that of the dividend. Let p be the arithmetical difference between n and m ; then will n z:^ m 4- p, whence p- = a ? : «"" 1 1 1 but ■ — r- — — ; hence, a~P =. — . a"'+P aP aP Therefore, the expression a~P is the symbol of a division which has not been performed ; and its true value is the quotient repre- sented by unity, divided by a, affected with the exponent p, taken positively. Thus, -3 ^ 1 5 1 a •*=:—- ; and a~^ = — -. a^ aP Since, a~P = - — ; and = a?, we conclude that, aP a-P Any factor may be transferred from the numerator to the denom- inator, or from the denominator to the numerator, by changing the sign of its exponent. If it is required to extract the «"» root of — , we have a"* , n / , —m — =: a~"» ; hence, \/ — = V a~^ = a " ' a'" ^ a"^ The notation of fractional exponents, whether positive or nega- tive, has the advantage of giving an entire form to all expres- sions whose roots or powers are to be indicated. From the conventional expressions founded on the preceding rules, we have wt , n / 1 in o" = \/a"'; a-P = — ; and V — = a "• aP ^ a"* We may therefore substitute the second value in each expres- sion, for the first, or reciprocally. As aP is called a to the p power, when p is a positive whole number, so, by analogy, 7n _m a " , a~P, a " , m are called, respectively, a to the — power, a to the — p power 71 236 ELEMENTS OF ALGEBRA. [CHAP VIII. Ttl a to the power, in which algebraists have generalized the word power. It would, perhaps, be more accurate to say, a, ex- m ?n . , ponent — , a, exponent — p, and a, exponent — — ; using the word poiuer only when we wish to designate the product of a number multiplied by itself two or more times. AlidtipUcation of Quantities affected with any Exponents. 3 2 234. In order to multiply a^ by a^, it is only necessary to add the two exponents, and we have 3 2 Sj,! 1 — -, LA 5 V ^3 — n^ '^ — y/lS For, by Art. 233, a^ = Y a^ J and a^ ^= y/ a^ ; hence, a^ x a^ = y o? X y a"^ , reducing to a common index (Art. 226), and then multiplying, 3 2 1, . 19 a^ X a^ = V "'^ = '^^^• _3. 5 Again, multiply a * by a^. _3 * /Y !i g . We have, a * — V/ — r, and a^ = ^/ a^ ; hence, a-^ Xai = \/l-X ^^Jl/lx^V^oJ^/^^^lf, ^ „A , and consequently, _3 3 5 3,5 9 ,10 1_ -4 + 6 _ ^-12+12 = „I2. In general, multiplying a " by a ? ; we have m p a " X ai = a " ? = a» to its simplest terms. 1 2yT(3)2 ) Ans. V 3 384 ^ 3. Reduce V < iz: f > to its simplest terms. 4. What is the product of of 4. 0253 _{_ a^ii ^ ab-^ah^ -{- b^, by a 2 — ^3, Ans. a^ — h^ 5. Divide a^ — a^b~'^ — cflb + b^, by a' — b~^ . Ans. cP" — h. Remakk II. — 111 the resolution of certain questions, we shall be led to consider quantities affected with incommensurahle exj)onent&. Now, these incommensurable exponents may be treated in the same manner as those which are commensurable. For, let us observe, that an incommensurable quantity, such as v^ or ^/\\., may be determined approximatively, as near as we please : hence we can always conceive it to be replaced by an exact fraction, which shall differ from it by less than any assignable quantity. Having done this we can apply to the fraction which represents the incommensurable, the rules already demonstrated. 240 ELEMENTS OF ALGEBRA. [CHAP. VIU. Method of IndcteTminate Co-efficients. 238. The binomial theorem demonstrated in Art. 203, explains the method of developing into a series any expression of the form (a + 5)"', in which m is a whole and positive number. Algebraists have invented another method of developing alo-e- braic expressions into series, called the method by indeterminule co-efficients. This method is more extensive in its applications, can be applied to algebraic expressions of any nature whatever, and indeed, the general case of the binomial theorem may be de- monstrated by it. Before considering this method, it will be necessary to explain what is meant by the term function. Let a ^ b -\- c. In this equation, a, b, and c, mutually depend on each other for their values. For, a := b -\- c, b = a — c, and c = a — b. The quantity a is said to be a fu7iction of b and c, b a function of a and c, and c a function of a and b. And generally, when one quantity depends for its value on one or more quantities, it is said to be function of each and all the quantities on which it de- pends. 239. If we have an equation of the form, A-\- Bx -^ Cx^ + Dx^ + Ex^ + &c. = ; it is required to find the values of the co-efficients A, B, C, D, E, &c., under the following suppositions : 1st. That no one of the co-efficients is a function of x. 2d. That the series shall be equal to zero, whatever be the number of its terms ; and 3d. That it shall be equal to zero, whatever value may be at- tributed to X. Now, since the co-efficients are independent of x, their values cannot be affected by any supposition made on the value of .r : hence, if they be determined tor one value of x, they will be known for all values whatever. CHAP. VIII.] INDETERMINATE CO-EFFICIENTS. '241 Let us now make T = 0, which gives Bx + Cx2 + Dx^ + Ex* + &c. = ; and consequently, Jl = 0. If we divide by x, we have B + Cx + Dx"^ + Ex^ + &c. = , and by again making a: = 0, we have Cx + Dx2 + Ex^ + &c. = ; and consequently, B = Dividing again by x, we have C + Da: 4- iia;2 -f &c. = 0; and by again making a; = 0, we obtain Dx 4- Ex'i + &c. = 0, and consequently, C ^ ; and by continuing the process we may prove that, each co-efficient must be separately equal to zero. It should be observed, that A may be considered the co-efficient of x^. 240. The principle demonstrated above, may be enunciated un- der another form. If we have an equation of the form a -\- hx -{- cx^ -{- dx"^ + . . . = a'-\- h'x + c'x"^ + d'x:^ + . . . which is satisfied for any value whatever attributed to x, the co- efficients of the terms involving the same powers of x in the two members, are respectively equal. For, by transposing all the term.? into the first member, the equation will take the form A-\-Bx+ Cx-^ + Dx^ -\- Ex* + &c. = ; whence, a — a^=z 0, b — b'= 0, c — a;2 . . . 7 a;" ) . \ a a a / Remark. — It is here supposed that the degree of x in the nu- merator is less than it is in the denominator. If it was not, it would first be necessary to perform the division, arranging the quantities with reference to x, which would give an entire quo- tient, plus a fraction similar to the above. r^, . ,, . 1 -x-3x^ + 4x^ + x* . Ihus, m the expression —cT^ 5 — . gives 2 — 5a: -L 3x^ — x'^ X* + 4a!3 — 3a;2 — a; + 1 -f 7a;3 — 8x2 + 0; — a;3 + 33-2 — 5ar + 2 -a; -~7 ■ + 13a;2 — 34a; + 15. Performing the division, we find the quotient to be — x — 7. plus the fraction 13x2 — 34a? + 15 ^ 15 - 34a; + 13a;2 - - a;3 + 3x2 — 5a: + 2 " 2 — 5a: + 3x2 — ^^3 CHAP. VIII. 1 BINOMIAL THEOREM. 247 Demonstration of the Binomial Theorem for any Exponent. 244. It has been shown (Art. 61), that any expression of the form ^m _ ym^ ig exactly divisible by a; — y, when 77J is a positive whole number. That is, npTtl fl#ffl X — y J p y 3 The number of terras in the second member is equal to m ; and if we suppose x = y, each term will beco.me equal to x^—^ : hence, X"' — x" ,^771 ^m = mx" X — X We propose to prove that the quotient will have the same form when m is negative, and also, when ?n is a positive or negative fraction. First, when m is a whole number, and negative. Let 71 be a positive whole number, and numerically equal to m Then, ?ra = — n. By observing that — x~"y~''^ X (a;" — y") = x~" — y""", we have — = — x-'^y-" X ^^ ^— = — a;-2"«a:'»-^ = — 7«x~"~\ X — y X — y after making y = x ; and by restoring m, — 7*,^m 1 mx"^ \ 77? being a negative number. Second, let m he a positive fraction, or m =z — . ? 1 E Let x^ = V, whence, xi =.vp, and a: = u? ; i E and yi ^=:u, whence, y)"*. In order further to simplify the expression, let us make b — =zx; a then, the binomial to be developed will be of the form (1 4- a-)-. As this development must be expressed in terms of ar, and kno\vn quantities dependent for their values on 1 and m, we may assume (1 + x)"* = A 4- Bx + Cx2 + Z)x3 + Ex^ + &c. . . . (1), in which the co-efficients A, B, C, Sic, are independent of x, and functions of 1 and m. Now, since this equation is true for any value of x, if we make X =z 0, we have (1)'" = ^ = 1. Substituting this value in equation (1), we have (1 + a-)-" = 1 + J5x + ar2 + Dx^ + Ex^ + &c. . . . (2). Since the form of the above development will not be changed by placing y for x, we may write (1 + y)'" = 1 + 5y -(- Cf + Df 4- E?/ 4- &c. . . . (3). Subtracting equation (3) from (2), and dividing both member by X — y, we have ('+')''-i'±yT^B^'^^^+rJ-^^^D^^^^^h&c. . . .(4) X — y X — y X — y x — y Make I ■}- x = v, and 1 4- y = « ; whence, x — ij ^: v — u. Substituting these values in the first member of equation (4) and we have V- -n- ^ ^(x-y) ^ C^-^-^^ + D^^^^ + &o. .. .(5) V — u X — y x — y x — y If now, we make X z= y, whence, v = v. 250 ELEMENTS OF ALGEBRA. [CHAP. VIII we have, from Art. 244, v"» — u"* v'" — d" -1 = ot(1 4- a:)"»-i ; V — U V — V while the quotients in the second member become, respectivel)', 2 1, ".: ^ = 2a;2-i = 2x; X — y x'^ — y „ „ , J 1 J 0-v,2— 1 X — y X — y '^ ~y = 3a;3-i = 3x" ; ^ ~^ = 4x*-i = 4^;^ ; &c. X — y ^ — y Substituting these values in equation (5), and we have m{l+ xy-^ = + B + 2Cx + 3Dx^ + 4Ex^ + &c (6). Multiplying both members of this equation by I -\- x, and ar- ranging the second member with reference to x, and we have '0^"t5 m(l +x)'" = 5 + 2C + B X -\- SDlx"^ + AE + 2C I + 3D x'^ + &c If we now multiply equation (2) by m, Ave have jfi n j^ x'^m z= m -\- mBx + mCx^ + mDx^ + mEx^ + &c. If we place the second member of the last two equations equal to each other, we shall obtain an identical equation. Then, placing the co-efficients of the like powers of x equal to each other (Art. 240), we have m B = m, whence, B =z — ; ^ B(m — 1) m(m — 1) 2C+ B=mB, whence, C= ^ '— ^ '• 3D+2C=mC, whence, D 2 1.2' C(m— 2) m{m—l)(m—2) 4E-\-3D=TnD, whence, E = 3 1.2.3 D(m— 3) m{m—l)(?n—2){m~3) 4 1.2.3.4 &c., &c. Substituting these values of A, B, C, D, &c., in equation (2), we obtain m(m — 1) „ m (m — I) (m — 2) , {l+xr=.l+mx+ ^\ ^ '^^ + 1 \ a '^ 3—^ ^' m{m-l)(m-2){m-3) ^^ ^^ ^1.2.3.4 ^ CilAP. VIII. 1 BINOMIAL THEOREM. 251 If we now replace x bv its value — , we have a / h \"^ b m{m—\)b'^ m(m — I) (m — 2) b^ (l+-) =l + m--\- ^ ^ ^ + - , o o -+&C. \ a / a 1.2 a-^ 1.2. 3 o'^ Finally, multiplying by a"*, we obtain m (m — 1) ,,. (a -f b)'" = a"* + ma^-^b -\ ^^ -' O^-^b^ 1 * A^ 7n im — ^) (fn — 2) + ^i 2 3 "" ^ ' a development which is of the same form as the one obtained in Art. 203, under the supposition of m being a positive and whole number. Ajii^Ucations of tlie B'moinial Theorem. If in the formula (a; + «)"* = (a m — \ a? m — \ m — 2a^ \ 1 +OT. — + m.—-—.— + w2.—-— .-——.— + . . . a; 2 a"' 2 3 a;'^ / 1 . - n / we make m = ■ — ■, it becomes (,r + «) " or V a; + a = n -i-l 1-1 i-3 J(,+l.Ji + l.iL^.4 + l « _.iL. «V . , .) \ n X n 2 x^ n 2 Z :r I or, reducing, \J x -^ a — i/, ,1 a 1 K— 1 a2 1 n_l 2re— I a3 \ >r" IH -. . . 1 . . . .. ..) \ 11 X n 2n x^ n 2n 3« x^ I The fifth term can be found by multiplying the fourth by — — — and by — , then changing the sign of the result, and so on. Remark. — If in this formula, we make n = 2, n = 3, n = 4, &c., the development will become the approximate square root, cube root, fourth root, &c., of the binomial {x -\- a) \ and by as- signing values at pleasure to x and a as well as to ti, we can find any root whatever of any binomial. If m is negative, or frac- tional, there will be no limit to the number of terms to which the series may be carried. Such a series is called an infinite series. 252 ELEMENTS OF ALGEBRA [CHAP. VIII, The binomial formula also serves to develop algebraic expres- sions into series. Take, for example, the expression , we have 1 — z _L_ = (1 _ 2)-i. In the binomial formula, make ?7i = — 1, x:=z\, and a = — Zi it becomes (1 _ ^)-i = 1 _ 1 . (_ ;r) - 1 . ^— . (- :^? -1-1 -1-2 . -3 or, performing the operations, and observing that each term is composed of an even number of factors affected with the sign — (1 — Z)-^ = = 1 + 2 + ^2 + ^3 + S* + 05 + . . . . The same result will be obtained by applying the rules for di- vision (Art. 55). 1 1st remair ider - +:^ 2d - - - - +z^ 3d - - - - +23 4th - - - - + X* + . 1 -]- z + z^ + z^ -\- z* + Again, take the expression — —, or 2(1—2)" We have 2(1 — z)-^ = 2[l-3.{-z)-3.^^.{-zY-3.=^.--f^.{-^y-.]; or 2 (1 - 2)-3 ^ 2 (1 + 32 + 6^2 + iQz^ + 152* + . . .) To develop the expression y 22 — 2^, which reduces to Vi^f] — ^)^ we first find = 1 2 2"^ 2 6 36 618 CHAP. VIII.] BlNOiMlAL THEORKM. hence 253 V2._.»=vs(i-|^-^^'-^-'-.H EXAMPLES. 1. To find the value of r — -— -- = (a + &)-2 in an infinite (a + by iries. 2. To find the value of in an infinite series. r -\- X x^ x^ x'^ Ans. r — oc -\ + -^, &c. 3. Required the square root of -^ in an infinite series cc^ x' 4. Required the cube root of — -„ -— in an infinite series. 40x6 1 / 2x2 5^4 40x6 Remark. — AVhen the terms of a series go on decreasing in value, the series is called a decreasing series ; and when they go on increasing in value, it is called an increasing series. A converging series is one in which the greater the number of terms taken, the nearer will their sum approximate to the true value of the entire series. When the terms of a decreasing and converging series are alternately positive and negative, we can, by taking a ^iven number, determine the degree of approximation. For, let a — h -\- c — d -\- e — f -\- . . ., &c., be a decreasing series, b, c, d, . . . being positive quantities, and let x denote the true value of this series. Then, if n denote any number of terms, the value of x will be found between the sum of the n and n + 1 terms. For, take any two consecutive sums, a — b -{- c — d -\- e — f, and a — b -{- c — d -{- e — f-{- g In the first, the terms which follow — /", are 254 ELEMENTS OF ALGEBRA. ICHAP. VIII. but since the series is decreasing, the differences of the consecu- tive terms g — h, k — I, . . . are positive numbers ; therefore, in order to obtain the complete vaUie of a?, a positive number must be added to the sum a — b-\-c — d-{-e—f. Hence, we have a — b -\- c — J + e — y< X. In the second series, the terms which follow + g, are — h-{- k, — I -^ m . . . . Now, the differences — h -\- k, — I -{- m . . ., of the consecutive terms, are negative ; therefore, in order to ob tain the sum of the series, a negative quantity must be added to a — b-{-c — d + e — /+ g, or, in other words, it is necessary to diminish it. Consequently, a ^ b -\- c — d -\- e — f -j- g "^ x. Therefore, x is comprehended between the sum of the n and n + 1 terms. But the difference between these two sums is equal to g; and since x is comprised between them, their difference g must be greater than the difference betv.'een a: and either of them ; hence, the error committed hrj takmg n terms, a — b + c — d + e — f, of the series, for the value of x, is numerically less than the follow- ing term. Summation of Scries. 246. An interesting, and at the same time useful application of the principles involved in the summation of series, is found in determining the number of balls or shells contained in a given pile, Let ABC be a triangular pile of balls, having eight balls on each of the three equal lines, AB, BD, and AD, and also, eight balls in height along the line CB. Now, the proposed pile consists of 8 horizontal courses, and the number of shot in each course, is the sum of an arithmetical series of which the first term is 1, the last term the number of courses from C, and the number of terms, also the number of courses from C. There- CHAP. VIII.] STIMMATIOX OF SERIES. •^55 fore we have 1st course is equal to (1 + 1) x 1= 1 2(1 (1 + 2) X 1 = 3 Sd (I + 3) X 11= 6 4tli (1 + 4) X 2 = 10 5th (1 + 5) X 21 = 15 6th (1 + 6) X 3 =21 7th (1 + 7) X 31 =28 8th (1 + 8) X 4 = 36. Hence, the num 3er of shot in the pile will be equal to the sura of the series 1, 3, 6, 10, 15, 21, 23, 36; in which any term is found by adding 1 to the number of the term and multiplying the sum by half the number of terms. Thus, if we suppose the horizontal layers to be continued down, and denote the number of any layer from the top by n, we shall have nin + 1) 1, 3, 6, 10, 15, 21, ... . -^^-; and the sum of this series Avill express the number of balls in a triangular pile, of which n denotes the number in either of the bottom rows. If the general term of any increasing series of numbers involves n to the m'^ degree, the sum of the series will not involve n to a higher degree than (m + 1). For, the sum of such series can- not exceed n times the general term, and hence, cannot involve n to a higher degree than m + 1. Let us therefore assume n{n-\r\) 1 + 3 + 6+10 + 15 = A + Bn+Cn'^ + Dn\ in which the co-efficients A, B, C, and D, are not functions of n. In order that these co-efficients may be determined, we must find four independent equations involving them. If we make n=l, we have A + B + C + D = 1 =1 (1), « =2, gives A + 2B + 4C+ 8Z)=l+3 =4 (2), n = 3, " A+3B+ 90 + 277)= 1 +3 + 6 =10 (3), n=4, " A + 4^+ 16C+ 64D= 1 +3 + 6 +10 = 20 (4). ^^'ob E LEMENTS OF ALGEBRA. [CHAP, vni Now, by a series o f subtractions we have Equation (2) — (1), giv as B + 3C + 7D = 3 . . . (5), " (3)-(2), " 5 + 5C+ 19Z) = 6 . . . (6), " (4)-(3). ' B + 7C +37D = 10 . . . (7), " (^^)-(5), ' 2C + 12D = 3 . . . (8), " (7)-(6), ' 2C+ 18D = 4 . . . (9), " (9)-(8), ' 6D = 1 ; hence, D = -- ; o also, 2C+ 18D = 4, gives C = — : > to 2 ' B + 7C-\-37D = 10. " B = i-; A-\-B^ C + D = 1, " ^==0. Hence, 1+3 + 6+10 + n(n+ 1) }n + -^«2 + -l«3 = -^ (2 + 3« + n2) n(n + !)(« + 2) 1.2.3" Let us suppose that we have a pile of balls whose base is a square, two sides of which, EF, FH, are seen in the figure, and that it terminated by a single ball at G. Now, the number of balls in the upper course will be expressed by P, in the sec- ond course by 2^, in the third course by 3^, &c. Hence, the series 12 + 22 + 32 + 42 + 52 -r . . . n2, will express the number of balls in a square pile, of Avhich the lumber of courses, and consequently the number of balls in one of the lower rows is, n. To find the sum of this series, assume 1 + 4 + 9 + 16 + . . . ifl = A -\- Bn + Cii^ -\- Dn\ CHAP. VIII.] SUMMATION OF SERIES. 257 from which we find A+B+C-{-D = l = 1 , A -{-2B + 4C+ 8D — 1 + 4 = 5 A + 3B + 9C + 27D =1 + 4 + 9 ^14 A-\- 4B + IGC + 64D = 1+4 + 9+ 16 = 30 and from these four equations, we find, by continued subtractions D = h C = i, B and A = ; hence, 1+4 + 9+16 + 25 7*2 = -J-« + J-?i2 + inS = ^{2n" + 3n+ 1) ^n (n + \){2n + 1) ~ 1 . 2 . 3 Let us now^ suppose that we have a rectangular or oblong pile of shot, as represented in the figure below. Suppose we take off from the oblong pile the square pile EFD. We then see that the oblong pile may be formed by adding to the square pile a series of triangular strata, each containing as many balls as are contained in one of the faces of the square pile ; and the number of the triangular strata -will be one less than the num- ber of balls in the top row. Therefore, if n denote the number of horizontal courses, the number of balls in one triangular strata n(n + \) will be expressed by ; and if m+1 denotes the whole til number of balls in the top row, the number of triangular strata will be denoted by m\ and the number of balls in all these strata bv X m. 17 258 ELEMENTS OF ALGEBRA. [CHAP. VIII. But since the number of balls in a square pile, whose side con- tains n balls is n{n+ l)(2n+ 1) 1 . 2 ~ 3 ' the number of balls in an oblong pile, whose top row contains m + 1 balls, and depth n balls, will be expressed by w(» + 1) (2ra + 1) n {n + 1) 1.2. 3 + 2 ^ "^ _ n{n-\-\) (1 + 271 + 3to) - 2 ^ ■ 3 • If we denote the general sum by S, we shall have the follow ing formulas for the number of shot in each pile. Triangular, S = ~^^ \^—L-J. = — . ^^ ^ („ + i + i) o ^ n (n + \) (2n + 1) 1 n(n+l) , . Square, S = -A_ L .-^-g^— -' = -j . -^g"— •(« + «+ 1). Rectangular, n(n+l) (2n +l+3m) 1 n{n + l) * — —- 2 • g- = 3 • — 2 -l{n+m) + {m + n) + {mi-\)]. Now, since is the number of balls in the triangular face of each pile, and the other factor, the number of balls in the longest line of the base plus the number in the side of the base opposite, plus the parallel top row, we have the following RULE. Add to the number of balls in the longest line of the base, the number in the parallel side opposite, and also the number iji the top parallel row; then multiply this sum by one third the number in tri- angular face. EXAMPLES. 1. How many balls in a triangular pile of 15 courses? Ans. G80 2. How many balls in a square pile of 14 courses? and how many will remain after 5 courses are removed ? Ans. 1015 and 960. CHAP. VIII.] SUMMATION OF INFINITE SERIES. 259 3. In an oblong pile the length and breadth at bottom are respect- ively 60 and 30 : how many balls does it contain ? Ans. 23405. 4. In an incomplete rectangular pile, the length and breadth at bottom are respectively 46 and 20, and the length and breadth at top 35 and 9: how many balls does it contain? Ans. 7190 Summation of hijinite Series. 247. An infinite series is a succession of terms unlimited in number, and derived from each other according to some fixed and known law. The summation of a series consists in finding an expression of a finite value, equivalent to the sum of all its terras. Different series are governed by different laws, and the methods of finding the sum of the terms which are applicable to one class, will not Apply universally. A great variety of useful series mav be summed by the following formula : Assume — 7 — = — 7^^— — r : n n -\- p n [n -{- p) then, ^—-1.(1. ?_). n [n -\- p) p \n n -\- p/ If now, by attributing known values to p and q, and different values in succession to n, the expression — ; r shall repre- n{n-\- p) sent a given series ; then, the sum of this series will bo equal (0 — multiplied by the difference between the two new series P of which — and — - — are the general terms. Hence, if the dif- n n -\- p ference of the sums of these series be known, and the value of — be known, we can find the value of the series — ; — ; — -, by p n{n+p) the formula *S = — (s' — s") even if we do not know the value P of he now series — and n n -\- p EXAMPLES. 1 Required the sum of the series 1 1 1 \- &c., to infinity 1.2 1.2.3 1 .3.4^ 1.4.5 ' 260 ELEMENTS OF ALGEBRA. [CHAP. VIII We see that if we make q = I, and p = 1, and n = 1, 2, 3 4, &c., in succession, that the first member of the formula, 9 n (tj + pY will, in succession, represent each term of the series ; while un (ler the same supposition, the second member will become, for w terms of the series. 111 1 '+T + T + T--- T L \2^34 n+l/_ ' 71+1 n + 1 If now, we suppose n = oo, the value of the sum of the series will become equal to 1. 2. Required the sum of n terms of the series To adapt the formula to this series, we make q = I, p =: 2, and n = 1, 3, 5, 7, &c.; we then have, for the sum of n terms, 111 1 1 +-7r + -^ + ^ 3 5 7 2tt — 1 = 1 - (y + -, + -=r • 7: r + 1 _ 1 • 1 i 2w + 1 7 ' 2 2n , 1 5 7 2« — 1 2/1 + 1 -, and — of this sum = 2ra + r p 211+1 If now, we suppose n = oo, the value of the series becomes equal to one half. 3. Required the sum of n terms of the series -\ 1 1 1- &c., to infinity. 1.4 ' 2.5 ' 3.6 ' 4.7 Here p= 3, q = I, n = 1, 2, 3, 4, &c. : hence, 1 — < 3 111 1 2 3 4 n 1 + 1 ' n + 2 ' n + 3 . /I 1 1 1.1.1. \4 5 6 71 n -\- I n -i = 1 rii - r-i- + -— + -^) 1 = - 3 1-6 \n + 1 ^n + 2 ^« + 3/J 18' when n = (30. CHAP. VIII ] SUMMATION OF INFINITE SERIES. 2fil 4. Find the sum of n terms of the series 2 3 4 5 6 , . 3-.T - 577 + 779 " 9^1 + n:i3 + ^'^•' '"^ '"^"^'>' r 2 3 4_ _5_ _ n+l I "3 b 'Y ~~^ '^ ' ' ' ' "^ 2ra+ 1 [ -( 2_ 34_ ^ n _„-|-l 5" " y "9" " ■ ■ ' ■ ~ 2n 4- 1 ^ 2;i + 3 which becomes, 2 n + 1 T ~ 2« + 3 (1 _1 4- 1 _ ± 1). If the number of terms used is even, the upper siwn will apply, the quantity within the parenthesis will become + 1, and the sum of the n terms before dividing by p, is 1 n -I- 1 1 1 = — -, wtien 1 = CE. 3 2n + 3 6 J f 71 is odd, the lower sign is used, and the quantity within the parenthesis reduces to zero, and we have 2 n+\ 1 = — -, when n = oo. 3 2« + 3 6 Then, since p =2, the sum of the series when n =: oo, is ---. 5. Required the sum of the series 4 4 4 4 4 1 1 1 1 h &c., to infinity. 1 5 ^ 5.9 ^ 9.13 ^ 13. 17 ^ 17 .21 ^ ^ Ans ] 262 ELEMENTS OF ALOEBRA. [CHAP. IX. CHAPTER IX. CONTINUED FRACTIONS, EXPONENTIAL (JUANTITIES, LOGARITHMS, AND FORMULAS FOR INTEREST. 248. Every expression of the form 1 1 1 1 .1 . 1 a + -r, a -\ r '^ + in which a, b, c, d, &c., are positive whole numbers, is called a continued fraction. Hence, a continued fraction has for its numerator the unit 1, and for its denominator a whole number, plus a fraction which has 1 for its numerator and for its denominator a whole number plus a frac- tion, and so on. 249. The resolution of an equation of the form gives rise to continued fractions. Suppose for example, a = 8, 6 = 32. We then have 8^ = 32, in which a; > 1, and x <^2. Make a; = 1 + — , y in which y > 1, and the proposed equation becomes, after chan- ging the members, 32 = 8* y = 8 X 8y, whence, \_ 82' = 4 and consequently, 8 = 4". CHAP. IX.] CO.NTINUED FRACTIONS. 263 It is plain, that the value of y lies between 1 and 2. Suppose 1 i+i - and we have, 8=4 ^=4x4^; \_ hence, 4^=2, and 4 = 2^ or 2 = 2 113 But, y=l+_^14-_=:_; 1 1 2 5 and a:=:l+ — z=l+ — =1 + — = — ; y 2 3 3 and this value will satisfy the proposed equation. For, 8^ = 83" = V"P = V(2^ = yW? = 2^ = 32. 250. If we apply a similar process to the equation 10^ = 200, we shall find 1 1 1 a=2H ; y = 3H ; s = 3-| . y z u Since 200 is not an exact power, a; cannot be expressed either by a whole number or a fraction : hence, the value of x will be incommensurable, and the continued fraction will not terminate, but will be of the form 1 I 1 x = 2H ==2H = 2 + " 3 + i 3+ ' ^ o 1 u + — , 149 2 149 3 1 • 1 therefore the value oi the fraction is comprised between -— and — . If we wish a nearer approximation, it is only necessary to op- , n ■ 19 11 ' 1 • /. • 65 erate on the fraction — as we did on the given traction 65 ° 149 and we obtain 19 1 65 -S' 65 1 149 ^+19 hence, If now, we neglect the part — , the denominator 3 will ht- letss than the true denominator, and -7- will be larger than the nuin- 1 ber which ought to be added to 2 ; hence, 1 divided by 2 H will be less than the true value of the fraction ; that is, if we stop at the first reduction and omit the fractional numbers, the result will be too great ; if at the second, it will be too small, &c. Hence, generally, if we stop at an odd reduction, and neglect the fractional part, the result will be too great ; but if we stop at an even reduc- tion, and neglect the fractional part, the result will be too small CHAP. IX.] CONTINUED FRACTIONS 265 Making two more reductions iu the last example, we have 65 1 149 1 2H .... 1st reduction, too great;' 3H - 2d " too small; 2 H 3d " too great ; 2 H 4th " too small ; 1 4- — 5th " too great. 252. The separate fractions 1 1 1 a \ 1 « 4- -r. a + 6' '1 c are called approximating fractions, because each affords, in suc- cession, a nearer value of the given expression. The fractions — , — -, — , &c., are called integral fractions, a c o J When the expression can be exactly expressed by a vulgar frac- tion, as in the numerical examples already given, the integral. fractions — , -— , — , &c., will terminate, and we shall obtain an a c expression for the exact value of the given fraction by taking them all. We will now explain the manner in which any approximating fraction may be found from those which precede it. 1. — ----= — 1st app. fraction. 2. — - - - - = — — ; — - 2d app. fraction 1 flO + 1 '^ + T 3. - - = 7—. 'r-^ 3d app. fraction 1 (ab + 1 Ic +a a b ab + 1 bc-\- 1 a + b + {ab + l)c +a T c By examining the third approximating fraction, we see, that its numerator is formed by multiplying the numerator of the prece- ding fraction by the denominator of the third integral fraction, and 266 ELEMENTS OF ALGEBRA. [CHAP. IX adding to the product the numerator of the first approximating fraction : and that the denominator is formed by muhiplying the denominator of the last fraction by the denominator of the third integral fraction, and adding to the product the denominator of the first approximating fraction. We should infer, from analogy, that this law of formation is general. But to prove it rigorously, let — , — , — , be any three approximating fractions for which the law is already established. Since c is the denominator of the last integral fraction, we have from what has already been proved, Let us now add a new integral fraction — to those already deduced, and suppose -^ to express the next approximating frac- P S tion. It is plain that -=r ^^^^ become — by simply substituting 1 , tor c, c -f — - : hence, a S _^V ' ^t)"^^ _ {Qc-\- P)d + Q _ Rd-\- Q Hence, we see that the fourth approximating fraction is deduced from the two immediately preceding it, in the same way that the third was deduced from the second and first ; and as any fraction may be deduced from the two immediately preceding in a similar manner, we conclude that, the numerator of the n^*^ approximating fraction is formed hy multiplying the numerator of the preceding fraction by the denominator of the n^** integral fraction, and adding to the product the numerator of the n — 2 fraction ; and the denom- inator is formed according to the same law, from the two preceding denominators. 253. If we take the difference between any two of the con- secutive approximating fractions, we shall find, after reducing them to a common denominator, that the difference of their numerators? CHAP. IX.] CONTINUED FRACTIONS. 267 will be equal to ± 1 ; and the denominator of this difference will be the product of the denominators of the fractions. Taking, for example, the consecutive fractions — , and — r -, a ao -\- I we have 1 b ab+l —ab +1 and a ab + 1 a(ab-{-l) a{ab + 1)' b bc+ } — 1 ab + 1 (ab + 1) c -\- a ~ {ab -i- 1) [{ab + 1 ) c + aj To prove this property in a general manner, let P_ Q_ R^ R' Q" R'' be three consecutive approximating fractions. Then P Q PQ'-P'Q and P' Q' P'Q' Q R R'Q - R( Q' R' Q'R' But Rz=iQc-\- P and R' z=z Q'c + P' (Art. 252). Substituting these values in the last equation, we have Q R {Q'c + P')Q-{Qc-\-P)i:r or reducing Q' R'~ R'Q' Q R P'Q~ PQ' Q' R' ~ R'Q' From which we see, that the numerator of the difference -p^ — ^ Q Q Q R is equal, with a contrary sign, to that of the difference . Q' R' That is, thf difference between the numerators of any two consecu- tive approocima.'ing fractions, when reduced to a common denominator, is the same with a contrary sign, as that yahich exists between the last numerator and the numerator of the fraction immediately fol- lowing. But we have already seen that the difference of the numerators of the 1st and 2d fractions is equal to + 1 ; also that the differ- ence between the numerators of the 2d and 3d fractions is equal to — 1 ; hence, the difference between the numerators of the 3d and 4th is equal to + 1 ; and so on for the following fractions 268 ELEMENTS OF ALGEBRA [CHAP. IX. Since the odd approximating fractions are all greater than the true value of the continued fraction, and the even ones all less (Art. 251), it follows, that when a fraction of an even order is subtracted from one of an odd order, the difference should have a plus sign ; and on the contrary, it ought to have a minus sign, when one of an odd order is subtracted from one of an even. 254. It has already been shown (Art. 251), that each of the approximating fractions corresponding to the odd numbers, exceeds the true value of the continued fraction ; while each of those cor- responding to the even numbers, is less than it. Hence, the dif- ference between any two consecutive fractions is greater than the difference between either of them and the true value of the con- tinued fraction. Therefore, stopping at the n'^ fraction, tht; result will be true to within 1 divided by the denominator of the 7t"' fraction, multiplied by the denominator of the fraction which fol- lows. Thus, if Q' and R^ are the denominators of consecutive fractions, and we stop at the fraction whose denominator is Q', the result will be true to within ■z^;-^}- But since a, b, c, d, &c., are entire numbers, the denominator R^ will be greater than Q' and we shall have 1 1 nence, if the result be true to within , it will certainly be true to within less than the larger quantity 1 Q'2 ' •hat is, the approximate result which is obtained, is true to within unity divided by the square of the denominator of the last approxi- mating fraction that is employed. If we take the fraction , we have 347' 829 1 = 2 + , 347 3 + CHAP IX.] EXPONENTIAL QUANTITIES. 269 Here, we have in the quotient the whole number 2, which may either be set aside, and added to the fractional part after its value shall have been found, or we may place 1 under it for a denominator, and treat it as an approximating fraction. Resolution of the Equation a' = h. 255. An equation of the form 0^= h, is called an exponential equation. The object in resolving tliis equation is, to find the exponent of the power to which it is ne- cessary to raise a given number a, in order to produce another given number b. Suppose it were required to resolve the equation 2^ = 64. By raising 2 to its different powers, we find that 2^ = 64 ; hence, a; = 6 will satisfy the conditions of the equation. Again, let there be the equation 3* = 243, in which x — b. In fact, so long as the second member Z» is a perfect power of the given number a, it may be obtained by raising a to its sue cessive powers, commencing at the first. Suppose it were required to resolve the equation 2^= 6. By making x = 2, and x = 3, we find 22 = 4 and 2^ = 8 ; from which we perceive that the value of x is comprised between 2 and 3. Make then, x = 2 -\ -, in which a/ > 1 . ar Substituting this value in the given equation, it becomes, 2 '=' = 6, or 22x2^ = 6; hence — 6 3 ox/ _ _^ _ . ^ - 4 - 2' and by changing the terms and raising both members to the a/ power, \2/ 270 ELEMENTS OF ALGEBRA. [CHAP. IX. To determine x\ make successively 3/ =.\ and 2 ; we find therefore, a^ is comprised between 1 and 2. Make, of = \ -{ , in which a;'' > 1. x' ( 3 \^' By substituting this value in the equation (— j = 2, {\Y^=^: hence, | x (|)^^ = 2, and consequently, ( — ) = — . 4 3 The hypothesis x"=.\, gives — < — ; and of x" = 2, gives -r- > — ; . b 9^2 therefore, x'^ is comprised between 1 and 2. Let a/'= I -\ — ; then, 4\i+-^ 3 , 4 /4\-^, 3 (-) .'"=-; hence, - x (-)""= ^ .QvV// 4 Whence, [-) = -. 9 \^'^' 4 If we make x"^ — 2, we have 9\2 81 4 Vs) ~ "eT ^ "3"' and if we make x^''' = 3, we have /9\3_ 729 4 _ \T/ ~ 5T2 T ' therefore, a/^^ is comprised between 2 and 3, Make a^^' = 2 -| ■, and we have /9\2+-!- 4 - 81/9\-i- 4 {_) siy= — ; hence, — ( — )x>v=: — . \8/ 3 '64\8/ 3 /256\it'v 9 and consequently, 1'243"/ ~ "s"" CHAP. IX.] EXPONENTIAL QUANTITIES. 271 Operating upon this exponential equation in the same manner as upon the preceding equations, we shall find two entire num- bers, k and h-\- \, between which x^^ will be comprised. Making 1 and x^ can be determined in the same manner as a;'^, and so oi;. Making the necessary substitutions in the equations X xf X ■ a;'^ we obtain the value of x under the form of a continued fraction x=l1-\ 1 ^+ 1 1 + - 1 2H Hence, we find the first three approximating fractions to be 1 _!_ 3 T' ¥' y and the fourth is equal to l2il±i=l(A„.2K). 5x2 + 2 12 '^ '' which is the true value of the fractional part of x to within ^'^ -T7T {^^t- 254). (12)2 144 Therefore, 7 31 1 x = 2 + — — — =: 2.58333 + to within 12 12 ■ 144 ' and if a greater degree of exactness is required, we must take a greater number of integral fractions. EXAMPLES. X = 2.46 to within 0.01. 0;= 0.477 " 0.001. X- ~ 0.25 " 0.01 3* = 15 10* z=. 3 5* = 2 3 272 ELEMENTS OF ALGEBRA. [CHAP. IX Theory of Logarithms. 256. If we suppose a to preserve the same value in the equation a^ — N, and N to become, in succession, every possible positive number, il is plain that x will undergo changes corresponding to those made in N. By the method explained in the last Article, we can de- termine, for each value of N, the corresponding value of x, either exactly or approximatively. Any number, except 1, may be taken for the invariable num- ber a ; but when once chosen, it is supposed to remain the same for the formation of one entire series of numbers. The exponent x of a, corresponding to any value of N, is called the logarithm of that number ; and the invariable number a is called the base of that system of logarithms. Hence, The logarithm of a number, is the exponent of the power to which It is necessary to raise an invariable number, called the base of the system, in order to produce the number. The general properties of logarithms are independent of any particular base. The use that may be made of them in nu- merical calculations, supposes the construction of a table, con- taining all the numbers in one column, and the logarithms of these numbers in another, calculated from a given base. Now, in calculating this table, it is necessary, in considering the equation a^ = N, to make N pass through all possible states of value, and to de termine the value of x corresponding to each of the values of N, which may be done by the method of Art. 255. 257. The base of the common system of logarithms, or as they are sometimes called, Briggs' logarithms, from their inventor, is uhe number 10. If we designate the logarithm of any number by log. or I, we shall have (10)0= 1- hence, (10)1= 10; hence, (10)2= 100; hence, (10)3= 1000; hence, (10)*= 10000; hence, &c., log. 1 = 0; log. 10 = 1; log. 100 = 2, log. 1000 = 3 log. 10000 &c. = 4. CHAP. IX.] THEORY OF LOGARITHMS. 273 Hence, in the common system, the logaruhm of any numt/cr between 1 and 10, is > and < 1. The logarithm of any nu!;;- ber between 10 and 100, is > 1 and < 2; the logarithm of any nmtiber between 100 and 1000, is > 2 and < 3 ; and so on. Hence, the logarithm of any number expressed by two figures. and which is not a perfect power of the base of the sysleni. will be equal to a whole number plus an approximating fraction, the approximate value of which fraction is generally expressed decimally. The integral part of a logarithm, is called the index or char- acteristic of the logarithm. By examining the several powers of 10, we see, that if a num- ber is expressed by a single figure, the characteristic of its logarithm will be ; if it is expressed by two figures, the characteristic of its logarithm will be 1 ; if it is expressed by three figures, the characteristic will be 2 ; and if it is expressed by n places of figures, the characteristic will be n — ] units. The following table shows the logarithms of the numbers, from 1 to 100. N. 1 Log. 0.000000 N. 26 Log. ] 1.414973 1 N. 51 hog. N. 76 Log. 1.707570 1.880814 2 0.301030 27 1.431364 52 1.716003 77 1.886491 3 0.477121 28 1.447158 i 53 1.724276 78 1.892095 4 0.602060 29 1.46-2398 54 1.732394 79 1.897627 5 "6 0.698970 0.778151 30 3l 1.477121 1 1.491362 1 55 56 1.740363 80 87 1.903090 1.748188 1.908485 7 0.845098 32 1.505150 1 57 1.755875 82 1.913814 8 0.903090 33 1.518514 i 58 1.763428 83 1.919078 9 0.954243 34 1.531479 j 59 1.770852 84 1.924279 ' 10 1.000000 35 1.544068 60 1.778151 85 1.929419 IT 1.041393 36 1.556303 6] 1.785330 86 1.934498 12 1.079181 37 1.568202 62 1.792392 87 1.939519 13 1.113943 38 1.579784 63 1.799341 88 1.944483 14 1.146128 39 1.591065 64 1.806180 89 1.949390 15 1.176091 j 40 1.602060 I 65 1.812913 90 1.954243 16 1.204120 41 1.612784 i 66 1.819544 91 1.959041 ! 17 1.230449 42 1.623249 67 1 .826075 92 1.963788 18 1.255273 43 1.633468 68 1.832509 93 1.968483 19 1.278754 44 1.64345:^ 69 1.838849 94 1.973128 20 2T 1.301030 1.322219 45 46 1.653213 70 7l 1 845098 1.8.512.58 i 95 96 1.977724 1.662758 1.982271 22 1.342423 1 47 1.672098 ■ 72 1.857333 97 1.986772 23 1.361728 48 1.68I24I 73 1.863323 98 1.991226 I 24 1.380211 49 1.690196 74 1.869232 99 1.995635 i 25 1.397940 ! 50 1.698970 1 75 1.875061 100 2.000000 274 ELEMENTS OF ALGEBRA, [CHAP. IX. The chavacterislic being always one less than the number of places of figures in the number, is not written down in the table of logarithms for numbers which exceed 100. Thus, in search- ing for the logarithm of 2970, we should find in tho table oppcsite 2970, the decimal part .472756. But since the number is ex- pressed by four figures, the characteristic of the logarithm is 3. Hence, log. 2970 = 3.472756, and by the definition of a logarithm, the equation a^ = N, gives 103.472756 _ 2970. Multiplication and Division by Logarithms. 258. Let a be the base of a system of logarithms, and sup- pose the table to be calculated. Let it be required to multiply together a series of numbers by means of their logarithms. De- note the numbers by N, N', N" , N"\ &c., and their correspond- ing logarithms by .r, x' , x" , x"\ &c. Then, by definition (Art. 256), we have aF — N, a"' - N\ a^" = A^'', a^'" = N"' . . . &c. Multiplying these equations together, member by member, and applying the rule for the exponents, we have a^+x'+x^'+x^'^ . . . = N X N' X N" X W" . . But since a is the base of the system, we have x^ x' ^x" -^ x'" . . . = log. [N, W, N'\ N'" ....); that is, the sum of the logarithms of any number of factors, is equal to the logarithm of tlic -product of those factors. 259. Suppose it were required to divide one number by another Let N and iV^ denote the numbers, and x and x' their logarithms We have the equations ax — N and ax' = N' ; qX jV" hence, by division = a^~x' =: -—- ; / N \ or X - y = log. N- log. N' = log. ^— j, that is, the difference between the logarithm of the dividend and the logarithm of the divisor, is equal to the logarithm of the quotient CHAP. IX.] THEORY OF LOGARITHMS. 275 Consequences of these Properties. A multiplication can be performed by taking the logarithms of the two factors from the tables, and adding them together ; this will give the logarithm of the product. Then finding this new logarithm in the tables, and taking the number which corresponds to it, we shall obtain the required product. Therefore, by a sim- ple addition, we jind the product arising from a mullipUcation. In like manner, when one number is to be divided by another, subtract the logarithm of the divisor from that of the dividend, then find the number corresponding to this difference ; this will be the required quotient. Therefore, hy a simple subtraction, we obtain the quotient arising from a division. Formation of Powers and Extraction of Roots. 260. Let it be required to raise a number N to any power de- m noted by — . If a denotes the base of the system, and x the n '' logarithm of iV, we shall have a^ = N, or N =z a^; tn whence, by raising both members to the power — , n m m TIT * iV" = a" . Therefore, log. I iV " ) = — . a; = — . log. N. If we make ra = 1 ; there will result, m . log. N = log. iV"» ; an equation which is susceptible of the following enunciation : If the logarithm of any number be multiplied by the exponent of the power to which the number is to be raised, the product will be equal to the logarithm of that power. 261. Suppose, in the first equation, m = I ; there will result. 1 — — log. N = log. iV" = log. V^; that is, n 77(6 logarithm of any root of a number is obtained, by divt ling (he logarithm of the number by the index of the root. 276 ELEMENTS OF ALGEBRA. [CHAP. IX.. Consequences. To form any power of a number, take the logarithm of this number from the tables, multiply it by the exponent of the power ; then the number corresponding to this product will be the required power. In like manner, to extract the root of a number, divide the log- arithm of the proposed number by the index of the root ; then the number corresponding to the quotient will be the required root. Therefore, by a simple multiplication, loe can raise a qua7i- tity to a power, and extract its root hy a simple division. 262. If we make the different exponents of 10 negative, the powers corresponding thereto will be decimal fractions. Thus, (10)-'= — 1=0.1; hence, log.0.1 =-1; (10)-2=: -— =0.01; hence, log. 0.01 =—2; Cl0)-3 = -J— = 0.001 ; hence, log. 0.001 =-3; >. ) 1000 » ' o CIO)-* = = 0.0001 ; hence, log. 0.0001 = — 4. ^ ^~ 10000 ^ &c., &c., &c. The logarithm of any fraction between oi;e and one tenth, as four tenths, for example, may be expressed thus, ^''^' (lo) ^ ^''°' (lo ^ '^ ) ^ ^*'^" To '^ ^"^'"^ = - i + log. 4. For the fractions between one hundredth and one tenth, as six hundredths, for example, we have For the fractions between one thousandth and one hundredth, as eight thousandths, for example, we have log. (-^) = log. f-i— X 8 ) = — 3 + log. 8. *= VlOOO/ " \1000 / ° Now, instead of performing the subtractions indicated above, we unite the decimal part of the logarithm to the negative charac- teristic. Thus, log. 0.4 = — 1 + log. 4 =: — 1.G02060 ; log. 0.06 = —2 + log.6 = —2.778151 ; log. 0.008 = — 3 + log. 8 = — 3.903090. CHAP. IX.] THEORY OF LOGARITHMS. 277 Adopting this method of writing the logarithms, we see that the logarithm of a decimal fraction may be found from the tables, by uniting to the logarithm of its numerator, regarded as a whole num- ber, a negative characteristic greater by unity than the number of ciphers between the decimal point and the frst signifcant fgurc. To demonstrate this in a general manner, let a denote the nii- [nerator of a decimal fraction, and b its denominator. From the nature of decimals, we shall have b =2 (lO)"*, in which m will denote the number of ciphers in the denomina- tor. Hence log. — - = log. \- — — ) zzz log. a — m log. 10 r= log. a — m. * 6 *= \(10)'"/ o o o Or, in other words, the logarithm of a whole number will be- come the logarithm of a corresponding decimal, by adding to it a negative characteristic containing as many units as there are ciphers in the denominator of the decimal fraction. Hence, the table of logarithms whose base is 10, will give the logarithms of all decimals, as well as of the integral numbers. GENERAL EXAMPLES. 1 . What is the square of 7 ? Log.7 Exponent of the power Number corresponding, 49 2. What is the 6th power of 2 ? W-2 Exponent of the power - Number corresponding, 64 3. What is the cube root of 64 ? Log. 64 .... Then, - - - . - Number corresponding, 4 4. What is the 4th root of 81? 5. What is the 5th root of 32 ? = 0.845098 z= 2_ 17690196. 0.301030 6 1.806180. = 1.806180 3) 1.806180 0.602060. Ans. 3. Ans. 2. 278 ELEMENTS OF ALGEBRA. [CHAP. IX 6. Log. (a . i . c . J . . . .) = log. a -f log. b + log. c . . , . 7. Log. f— - — ) = log. a + log.Z»4-log. c— log.(f— log. e. 8. Log. (a'" .b^.cP. . . .) = m log. a -\- n log. b -\- p log. c + . . . . 9. Log. (a2_ a;2) := log. (a + a;) + log. (a — x). 10. Log. y(a2 _ a;2) = J log. (a + a;) + 1 log. (a — x). 11. Log. (a^ X V^) = 3|log. a. 263. Let us resume the general equation and suppose a to be the base of a system of logarithms. Then, 1st, we have a^ = iV = a, whence, log. a = 1 ; 2(1 " a" zz: 1, whence, log. 1 = ; that is, whatever be the base of the system, its logarithm taken, in that system, is equal to 1, and the logarithm of 1 is equal to 264. Let us suppose, in the equation a' zi: N, . that • a > 1 . Then, if we make iV= 1, we shall have aO= 1. If we make iV< 1, we must have a-' = N, or — = iV < 1 . a* If now, N diminishes, x will increase, and when N becomes 0, we have a-* = — = 0, or c^ = 00 (Art. Ill); but no finite power of a is infinite, hence a; = oo : and therefore, the logarithm of in a system of which the base is greater than vnity, is an infinite number and negative. 265. Again, take the equation a== = N, and suppose the base a < 1. Then making, as before, N =z 1, we have a° =^ 1. CHAP. IX.] LOGARITHMIC SERIES. 279 If we make N less than 1, we shall have a^ — N <: 1 . Now, if we diminish N, x will increase; for, since a < 1, its powers will diminish as the exponent x increases, and when A^ = 0, X must be infinite, for no finite power of a fraction can be 0. Hence, the logarithm of in a system of which the basf is less than unity, is an infinite number and positive Logarithmic and Exponential Series. 266. The method of resolving the equation a-" = b, explained in Art. 255, gives an idea of the construction of log- arithmic tables ; but this method is laborious when it is necessary to approximate very near the value of x. Analysts have discovered much more expeditious methods for constructing new tables, or for verifying those already calculated. These methods consist in the development of logarithms into series. Taking again the equation c- =y, it is proposed to develop the logarithm of y into a series involving the powers of y, and co-eflicients independent of y. It is evident, that the same number y will have a different log- arithm in ditferent systems, that is, for different values of the base a; hence, the log. y, will depend for its value, 1st, on the value of y ; and 2dly, on a, the base of the system of logarithms. Hence, the development must contain y, or some quantity depen- dent on it, and some quantity dependent on the base a. To find the form of tl.is development, Ave will assume log. y = A + By + Cy^ -\- Dy3 -{-, &c., m which A, B, C, &c., are independent of y, and dependent on the base a. Now, if we make y = 0, the log. y becomes infinite, and is either negative or positive, according as the base a is greater or less than unity (Arts. 264 & 265). But the second member un- der this supposition, reduces to A, a finite number : hence, the development cannot be made imder that form 280 ELEMENTS OF ALGEBRA. [CHAP. X. Again, assume log. y = Ay + By"^ + Cy^ + Dy^ +, &c. If we make y = 0, we have log. y = ± OD, that is, =b oo = 0, which is aljsurd, and hence the development cannot be made un- der the last form. Hence we conclude that, the logarithm of a number catinot be developed in the j)owers of that number. Let us place, in the first member, 1 + y for y, and we have log. (1 + y) = Ay + By^ + Cy^ + Dy* + &c. . . . (1), making y = 0, the equation is reduced to log. 1=0, which does iiot present any absurdity. In order to determine the co-efficients A, B, C, . . . we shall follow the process of Art. 243. Since equation (1) is true for any value of y, it will be true if we substitute z for y, and we may write lug. (1 +z) = Az + Bz^ + Cz^ + Dz* + . . . (2). Subtracting equation (2) from (1), we obtain \og.{l^\-y)-\og.{\+z)=A{y-z)i-B{y^-z^~)+C{y^-z^)-{-..{3). The second member of this equation is divisible by y — z. Let us see, if we can by any artifice, put the first under such a form that it shall also be divisible by y — z. We have, log. (1 + y ) - log. (1 + ^) = log. ( |-|-f ) = log- (i + f^)- But since — can be regarded as a single number u, we \ + z can develop \og.[\-\-u), or log. (l + ^— \ in t-l^e same man- ner as log. (1 + y), which gives Substituting this development for log.(l+y)-log.(i+^), in the equation (3), and dividing both members by y — z, it be- comes \ -{- z [l -^ zf (1 -r zy = A + B{y ^ z) + C {y-^- + yz + z-^) + . . . CHAP. IX.] LOGARITII.AIIC SERIES. 281 Since this equation, like the preceding, is true for all A^alues of y and z, make y = ^, and there will result — — = ^ + iBy + 3Cy2 + xBy'^ + 5£y* + . . . 1 +y whence, by making the terms entire, and transposing, y + 3C + 2Z? y2 4- 4Z) + 3C + M) y* + .=., B=-4. c=--=^-4. p ( + J. + 2S Placing the co-efficients of the different powers of y equal to zero, we obtain the series of equations ^_A = 0, 25 + A=0, 3C + 2S=0, 4D4-3C==0 whence, 3C _ ^ ~~\ ~ ~T" The law of the series is evident ; the co-efficient or" the n'^ term is equal to ^: — , according as n is even or odd : hence, we ob- n tain for the development, AAA log. (1 +y) =vly-— y2 + -_y3__y4 . . . Hence, although the logarithm of a number cannot be developed in the powers of tliat number, yet it may be developed in the powers of a number less by unity. By the above method of development, the co-efficients B, C, D, E, &c., have all been determined in functions of A ; but A re- mains entirely undetermined. This indeed should be so, since A depends for its value on the base of the system, to which any value may be assigned. Denote by a/ that part of the second member of equation (4) which involves y, and suppose a to be the base of the system in which the log. (1 + y) is taken, and we have a^^'=14-y, or ^a/ = log. (1 + y). But the log. (1 + y) depends for its value on two things : \iz., on the number of units in y, and on the base of the system in which the logarithm is taken. The series denoted by x' is ex- 282 ELEMENTS OF ALGEBRA. [CHAP. IX pressed in y, and hence depends for its value on y alone. Bu A being independent of y, its value must depend on the base of the system ; and hence, The expression for the logarithm of any number is composed of tvo factors, one dependent on the number, and the other on the base cf the system in which the logarithm is taken. The factor which depends on the base, is called the modulus of the system of log- arithms. 267. If we take the logarithm of 1 + y in a new system, and denote it by F. (1 -f y), we shall have r.(l+y) = A'(y-| + ^-| + -|^-&c.) (5), in which A' is the modulus of the new system. If we suppose y to have the same value as in equation (4), we shall have V.{\ +y) : 1.(1 + y) : : A' : A; for, since the series in the second members are the same, they may be omitted. Therefore, The logarilh7ns of the same number, taken in two different systems, are to each other as the 7noduli of those systems. 268. Having shown that the modulus and base of a system of logarithms are mutually dependent on each other, it follows, thai if a value be assigned to one of them, the corresponding value of the other must be determined from it. If then, we make the modulus A^=\, the base of the system will assume a fixed value. The system of logarithms resulting from such a modulus, — d siich a base, is called the Naperian System. This was the first system known, and was invented by Baron Napier, a Scotch mathematician. With this modification, the proportion above becomes F.(l+y) : 1.(1 +y) : : I : A, and .4.1'.(1 +y) = 1.(1 +y). Hence we see that. The Naperian logarithm of any number, multiplied by the modu- lus of another system, will give the logarithm of the same number in that system. CHAP. IX.] LOGARITHMIC SERIES. 283 The modulus of the Naperian System being unity, it is found most convenient to compare all other systems with the Naperian ; and hence, the modulus of any system may be defined to be, The number by which it is necessary to multiply the Naperian logarithm in order to obtain the logaritJnn of the same number in the other system. 269. Again, A X l'.(l + y) = 1.(1 + y) gives ■ .'.(! + y)=!:S - p = 100 — 98,643 = $1,35 7. Compound Interest. Compound interest is when the interest on a sum of money be- coming due, and not paid, is added to the principal, and the in- terest then calculated on this amount as on a new principal. To find the amount of a sum p placed at interest for t yean, compound interest being allowed annually at the rate r. At the end of one year the amount will be S =p +pr = p{\ -{- r). CHAP. IX.] FORMULAS FOR INTEREST. L'l'l Since compound interest is allowed, this sum now becomes the principal, and hence at the end of the second year the amount will be S'=p(l + r)+pr{l +r)=p(l + r)2. Regard /» (1 + r)^ as a new principal; we have, at the end of the third year, S''=p(l +r)2+pr(l-hr)2=p(l + r)3 ; and at the end of t years, S=p{l-\-rY (5). And from Article 2G0 we have \og.S = \og.p + t\og.(l +r); and if any three of the four quantities S, p, t, and r, are given, the remaining one can be determined. Let it be required to find the time in which a sum p will double itself at compound interest, the rate being 4 per cent, per annum. We have from equation (5), S=p{\+rY. But by the conditions of the question, S = 2p = p{\-\-ry: hence, 2 = (1 + r)', _ log.2 _ 0.301030 ^^ ^ ~ log. (1 + r) ~ 0.017033 = 17.673 years = 17 years, 8 months, 2 days. To find the Discount. The discount being the difference between the sum (S and p, we have 292 ELEMENTS CF ALGEBRA [CHAP. X. CHAPTER X. t GENERAL THEORY OF EQUATIONS. 274. The most celebrated analysis have tried to resolve equa- tions of any degree whatever, but hitherto tlieir efforts have been unsuccessful with respect to equations of a higher degree than the fourth. However, their investigations have conducted them to some properties common to equations of every degree, which they have since used, either to resolve certain classes of equations, or to re duce the resolution of a given equation to that of one more simple In this chapter it is proposed to make known these properties, and their use in facilitating the resolution of equations. The development of the properties of equations of any degree, leads to the consideration of polynomials of a particular nature, and entirely different from those considered in the first chapter. These are expressions of tlie form Ax"" + Bx'"-' + Ca:'»-2 -\- . . . ^ Tx-\- U, in which m is a positive whole number ; but the co-efficients A, B, C, . . . T, U, any quantities whatever, that is, entire or fractional, commensurable or incommensurable. Now, in algebraic division, as explained in Chapter II., the object was this, viz. : having given two polynomials, entire with reference to all the let- ters and particular numbers involved in them, to find a third poly- ncmial of the same kind, ichich multiplied by the second shall jjro- duce the first. But when we have two polynomials, Ax^ + Ex-"-! + Ca;"'-2 -\. . , . ^ Tx -\- U, A'a^ -\- 5^r"'-i I- C'.T"'-2 Jf. . . . J^ Tx ^ U, which are necessarily entire only with respect to x, and in which the cO'Cfficients A, B, C . . ., A', B' , C . . ., are any quan- tises whatever, it may be proposed to find a third polynomial, of CHAP. X.] GENERAL THEORY OF EQUATIONS. 293 the same form and nature as those that are given, which multiplied bij the second vnll re-produce the first. 275. Ordinary polynomials, that is, polynomials which are en- tire with reference to all the exponents and co-efficients, are called rational and entire polynomials. Polynomials which are only en- lire with reference to the letter x, and whose co-efficients are any (iuantities whatever, are called entire functions of x. 276. Every complete equation of the rn"' degree, m being a pos- itive whole number, may, by the transposition of terms, and by the division of both members by the co-efficient of x", be put un- der the form a;'" + Px'"-! + Qx'"-2 -{-.,, ^ Tx + U =0, P, Q, R . . ., T, U, being co-efficients taken in the most general algebraic sense. Ani/ expression, which substituted in place of x satisfies the equa- tion, that is, renders its first member equal to 0, is called a root of the equation. 111. As every equation may be considered as the translation into algebraic language of the relations which exist between the given and unknown quantities of a problem, we are naturally led to suppose that, every equation has at least one root. We will admit this principle, which we shall have occasion to verify here- after for most equations. We will now demonstrate some of the principal properties of a general equation. First Property. 278. In every general equation of the form a;m ^ Px^-^ + Qj''"-2 + . . . J^Tx-\- TJ =Q, the first member is divisible by the difference between the un- known quantity x and a root of the equation ; that is, 7/' a is a root of the equation, the first member will be exactly divisible by x — a ; and reciprocally, if a divisor of the form x — a will exactly divide the first member, a will be a root of the equation. Let us suppose the first member of the proposed equation to be divided by x — a, and the operation continued until all the terms 294 ELEMENTS OF ALGEBRA [CHAP. X involving x are exhausted : the remainder, if there be any, will then be independent of x. If we represent the remainder by R, and the quotient obtained y Q', we may write a:'" + Px^-^ .... ^^ Tx -\- TJ = Q' {x - a) -\- R. Now, since by hypothesis, a is a root of the equation, if we substitute a for x, the first member of the equation will reduce to zero; the term Q' {x. — a) will also reduce to 0, and consequently, we shall have R=0. But since R does not contain x, its value will not be affected by attributing to x the particular value a : hence, the remainder R was originally equal to zero, and consequently, the first member of the equation ^m _|. p^m-l _|_ Q^m-2 , . , . -\. Tx + U — 0, is exactly divisible by x — a. Reciprocally, if a; — a is an exact divisor of the first member of the equation, the quotient Q' will be exact, and we sluill have R — 0: hence, jnm _j_ p^m-\ . . , ^ Tx + U = Q' {x — a). If now, we suppose x = a, the second member will reduce to zero, consequently, the first will reduce to zero, and hence a will be a root of the equation (Art. 276). It is evident, from the na- ture of division, that the quotient Q' will be of the form ^m-l _f. p'xm-2 _^ 7^/ + C// _ 0. 279. It follows from what has preceded, that in order to dis- cover whether any polynomial is exactly divisible by the binomial X — a, it is sufficient to see if the substitution of a for a; will reduce the polynomial to zero. Reciprocally, if any polynomial is exactly divisible by x — a, then we know, that if the polynomial Tje placed equal to zero, a will be a root of the equation. The property which we have demonstrated above, enables us to diminish the degree of an equation by unity when we know one of its roots, by a simple division ; and if two or more of the roots are known, the degree of the equation may be still further diminished by continuing the division. CHAP. X.] GENERAL THEORY OF EQUATIONS. 295 EXAMPLES. 1. One of the roots of the equation X* — 25a;2 + 60a: — 36 = is 3 : what is the equation containing the other roots ? X* - 25x2 -(- 60x - 36 ||a; - 3 X* — 3^3 a;3 + 3a:2 -1674^12 + 3x3 _ 25x2 3x3 _ 9^2 — 1 6x2 + 60x — 16x2 + 48x 12x — 36 12x— 36 Ans. x^ -f 3x2 — ]6x + 12 = 0. 2. Two roots of the equation X* —.12x3 _^ 48a;2 _ 68x -f 15 = are 3 and 5 : what is the equation containing the other two ? Ans. x2 — 4x -(- 1 = 0. 3. One of the roots of the equation x3 — 6x2 + llx — 6 = is 1 : what is the equation containing the other roots ? Ans. x^ — 5x + 6 = 4. Two of the roots of the equation 4x* — 14x3 — 5x2 -f- 31x + 6 = are 2 and 3 : find the equation containing the other roots. Ans. 4x2 _f_ 5a; _j_ 1 -_ 0. Second Property. 280. Every equation involving but one unknown quantity, has as many roots as there are units in the exponent which denotes its de- gree, and no more. Let the proposed equation be x'» + Px'"-! + Qx'"-2 + . . . 4- Tx + C/ = 0. Since every equation is supposed to have at least one root (Art. 277), if we denote that root by a, the first member will be divisi- ble by X — a, and we shall have the equation x« + Pa-""-^ -}-... ={x — a) (x"*-! + P^x^-2 +...)••• (H 296 ELEMENTS OF ALGEBRA. [CHAP. X But if we place we obtain an equation which has at least one root. Denote this root by b, we have (Art. 278), a™-i + P'x^-"^ -\- . . . =.{x ~b) (x'"-2 + P''x^-^ +...). Substituting the second member, for its value in equation (11 and we have, ^m _^ Pa;m-i J^ _,=,{^x — a){x — h) (a;'"-2 + P'' a;'"-3 +...).. (2) . Reasoning upon the polynomial as upon the preceding polynomial, we have ^m-2 ^ P'^x^-^ + . . . = (a; — c) (a;'"-3 + P"'oi?^-*- +...), and by substitution, , x'« 4- Px'^"^ + . . . = {x— a){x — b){x — c) (a''"-^ + P^'x'^-^) (3) 281. Observe, that for each binomial factor of the first degree with reference to x, the degree of x in the polynomial is diminished by unity ; therefore, after m — 2 factors of the first degree have been divided out, the exponent of x will be reduced to m — (m — 2) = 2 ; that is, we shall obtain a polynomial of the second degree with reference to x, which can be decomposed into two factors of the first degree (Art. 142), of the form x — k, x — I. Now, supposing the m — 2 factors of the first degree to have already been indicated, we shall have the identical equation, x'^-i- Pa;'"-i+ . . . =(x ~ a){x — b){x — c) . . .{x — k) (x — I) zzzO; from which we see, that the Jirst member of the proposed equation may be decomposed into m binomial factors of the first degree. As there is a root corresponding to each binomial divisor of the first degree (Art. 278), it follows that the m binomial factors of the first degree, x — a, x — b, x — c . . ., give the m roots, a, b, c . . ., of the proposed equation. But the equation can have no other roots than a, b, c . . . k, I For, if it had a root a'', different from a, b, c . . . I, it would have a divisor x — a'', different from x — a, x — b, x — c . . . x ■ I, which is impossible. Therefore, finally, CHAP. X.] COMPOSITION OF EQUATIONS. 297 Every equation of the \x\^^ degree has m roots, and can have no more. 282. In equations which arise from the multiplication of equal factors, such as {x - ay (x — by (x — cy {x — d) = 0, ihe number of roots is apparently less than the number of units in the exponent which denotes the degree of the equation. But this is not really so ; for, the above equation actually has ten roots, four of which are equal to a, three to b, two to c, and one to d. It is evident that no quantity a\ different from a, b, c, d, can verify the equation ; for, if it had a root a', the first member would be divisible by a? — a', which is impossible. Consequence of the second Property. 283. It has been shown that the first member of every equation of the m^^ degree, has m binomial divisors of the first degree, of the form X — a, X — b, X — c, . . . X — k, X — I. If we multiply these divisors together, two and two, three and three, &c., we shall obtain as many divisors of the second, third, &,c. degree, with reference to x, as we can form different com- binations of m quantities, taken two and two, three and three, &c. Now the number of these combinations is expressed by m — 1 m — 1 m — 2 2 ' 2 3 Hence, the proposed equation has m — 1 (Art. 201). 2 divisors of the second degree ; rn — 1 m — 2 2 devisors of the third degree ; m — 1 m — 2 2 3 4 divisors of the fourth degree ; and so on 298 ELEMENTS OF ALGEBRA. [CHAP. X Composition of Eqi/atio7is. 284. If in the identical equation a:"* + Px'"-^ + ... = (x — a) {x — b) {x — c) . . . {x — I), •we perform the multiplication of four factors in the second meiii- ber, we have, — abc — abd — acd — bed — a x^ + ab ~b + ac — c + ad -d + be + bd + ed X -j- abed ~ V =0. If we perform the multiplication of the m factors of the second member, and compare the terms of the two members, we shall find the following relations between the co-efficients P, Q, R, . . T, U, and the roots a, b, e, . . . k, I, of the proposed equation, viz.. ^a — b — c...—k — l = P, or a-{- b +e + . . . + k+ I = - P ; ab + ae + . . . -\- kl = Q, — abc — abd ... — ikl ^^ R, or ahe -\- abd -\- ikl = — R ; dt abed . . , kl = U, or abed . . . kl = ± U. The double sign has been placed in the last relation, because the product — a x — b x — c... x — I will be plus or minus according as the degree of the equation is even or odd. Hence, 1st. The algebraic sum of the roots, taken with contrary signs, is equal to the co-efficient of the second term ; or, the algebraic sum of the roots themselves, is equal to the co-efficient of the second term taken with a contrary sign. 2d. The sum of the products of the roots taken two and two, with their respective signs, is equal to the co-efficient of the third term. 3d. The sum of the products of the roots taken three and three, with their signs changed, is equal to the co-efficient of the fourth term ; or the co-efficient of the fourth term, taken with a contrary sign, is equal to the sum of the products of the roots tjiken three and three ; and so on. CHAP. X.] COMPOSITION OF EQUATIONS. 299 4th. The product, of all the roots, is equal to the last term ; that is, the product of all the roots, taken with their respective signs, is equal to the last term of the equation, taken with its sign, when the equation is of an even degree, and with a contrary sign, when the equation is of an odd degree. If one of the roots is equal to 0, the absolute term will he 0. The properties demonstrated (Art. 143), with respect to equa tions of the second degree, are only particular cases of the above Consequences. 1. If the co-efficient of the second term of an equation is equal to zero, the term will not appear in the equation ; and the sum of the positive roots is equal to the sum of the negative roots. 2. Every commensurable root of an equation is a divisor of the last or absolute term. EXAMPLES IN' THE FORMATION OF EQUATIONS. 1. Form the equation whose roots are 2, 3, 5, and — 6. We have, by simply indicating the multiplication of the factors, (x — 2){x — 3) {x — 5) (a; + 6) = 0. But the process may be shortened by detaching the co-efficients thus : 1— 2 |-3 — 3 +~& 1 — 5+ 6 (-5 5 + 25 — 30 1 - 10 + 31-30 1+6 6 - 60 + 186 - 180 1 - 4 — 29 + 156 — ISO. Hence, the required equation is x^ - 4x3 _ 29x2 _|. 156a; _ 180 = 0. 2. What is the equation whose roots are 1, 2, and — 3? Ans. a?3 — 7x + 6 = 0. 3. What is the equation whose roots are 3, — 4, 2 + y 3, and 2 — ^/~^l Ans. x* — 3x3 _ 15^.2 _|_ 49^, _ 12 = 0. 4. WTiat is the equation whose roots are 3 + y 5, 3 — y 5, and — 6 7 Ans. x^ — 32x + 24 = 0. 300 ELEMENTS OF ALGEBRA. [CHAP, X. 5. What IS the equation whose roots are 1 , — 2, 3, — 4, 5, and — 6 ? Ans. x<^ + 3x^ — 41a;i — STcc^ ^ 400^2 + 444a; — 720 = 0. 6. What is the equation whose roots are .... 2 + y — 1, 2 - /-T and - 3 ? Ans. x^ — x""- - 7x + 15 = 0. Of the greatest Common Divisor. 285. The greatest common divisor of two polynomials is the greatest polynomial, with reference to its exponents and co-efh- cients, that will exactly divide the proposed polynomials. If two polynomials be divided by their greatest common divisor, the quotients will be prime ivilh respect to each other ; that is, they will no longer contain a common factor. Hence, Two polynomials are prime with respect to each other when they have not a common factor. >» Let A and B be two polynomials, D their greatest common divisor, and A', B', the quotients after division. Then ^ = A^ and ~ = B'; and consequently, A = A' X D, and B = B' x D. Now, if A^ and 5' have a common factor d, then d x D would be a common divisor of the two polynomials and greater than D, either with respect to the exponents or the co-efficients, which would be contrary to the supposition. Again, since D exactly divides A and B, every factor of D will have a corresponding factor in both A and B." Hence, 1st. The greatest common divisor of two polynomials contains as factors, all the prime factors common to the two polynomials, and dons not contain any others. 2SG. We will now show that the greatest common divisor of two polynomials will divide their remainder after one of them iias been divided by the other Let A and B be two polynomials, D their greatest common divisor, and suppose A to contain the highest exponent of the let- ter with reference to which the polynomials A and B are arranged CHAP. X.] GREATEST COMMON DIVISOR. 301 Having divided A by B, suppose we have a quotient Q and a remainder R. We may then write A = B X Q + R. If now, we divide both members of the equation by D, we have ^_ 5_ R and since we suppose A to be divisible by D, the first member of the equation will be entire, and consequently, the second member must also be entire, since an entire quantity cannot be equal to a fraction. But since D also divides B, the first term of the sec- ond member is entire, and consequently, the second term is also entire, and therefore, R is exactly divisible by D. We will now show that if D will exactly divide B and R, that it will also divide A. For, having divided A by B, as before we have A = B X Q + -R, and by dividing by D, we obtain A B ^ R But since we suppose B and R to be divisible by D, and know Q to be an entire quantity, the second member of the equation is entire : hence, the first member is also entire, that is, A is ex- actly divisible by D. ?i Hence, 2dly. The greatest common divisor of two polynomials, is the same as that which exists between the least polynomial and their remainder after division. Remark. — If either of the polynomials A or 5 have a factor A^ common to all its terms, but not common to the other polynomial, the common divisor will be found in that part of the polynomial which is multiplied by the factor A\ 287. From these principles, we have, for finding the greatest common divisor of two polynomials, the following RULE. I. Take the first polynomial and suppress all the monomial Jactors common to each of its terms. Do the same with the second polyno- mial, and if the factors so suppressed have a common divisor, set it aside as forming a part of the common divisor sought. 302 ELEMENTS OF ALGEBRA. [CHAP. X II. Having done this, prepare the dividend in such a manner that its Jirst term shall be divisible bij the first term of the divisor ; then perform the division, and suppress in the remainder all the factors that are common to the co-tffcients of the principal letter. Then tahe this remainder as a divisor, and the sccojid polynomial as a dividend, and continue the operation with these polynomials, in the same manner as with the preceding. III. Continue this series of operations until a remainder is ob- tiiincd which will exactly divide the preceding divisor: this last divisor will be the greatest common divisor ; but if a remainder is obtained which is independent of the principql letter, and which will not divide the co-efiicients of each of the proposed polynomials, it shows that the proposed polynomials are prime with respect to each other, or that they have not a common factor. EXAMPLES. 1. Find the greatest common divisor of the polynomials a3 _ a-^jj _|_ 3ai2 _ 3^3^ and a^ _ ^^b + Ah"^. First Operation. Second Operation. a3 _ (M + 3ab^ - 3^3 4^2j~— a62_3i3 |a2 _ 5ab + 4Z»2 a +46 ~ a2 — 5ab -f 4F — Aab + 4*2 la — b a- U 0. 1st rem. 19aZ/2 _ 19^3 or, 19b^{a-b). Hence, a — Z> is the greatest common divisor. We begin by dividing the polynomial of the highest degree by that of the lowest ; the quotient is, as we see in the above table, a + 4b, and the remainder Idab"^ — I9b^. But, 19a/y2 — 19Z>3 = 19^2 (a _ i). Now, the factor 19^2^ -^yill divide this remainder without dividing a2 — 5ab + 4^2 ; hence, the factor must be suppressed, and the question is re- duced to finding the greatest common di^-isor between a"^ — 5ab + 462 and a — b. Dividing the first of these two polynomials by the second, there is an exact quotient, a — 4b ; hence, a — b is the greatest com- mon divisor ol liie two given polynomials. To verify this, let each be divided by a — b. ci-i.\i\ X ■ GREATEST COMMON DIVISOR. 303 2. Find the greatest common divisor of the polynomials 3a^ — oaW + 2ai* and 2a* — "iaW + M. We first suppress a, which is a factor of each term of the first polynomial : we then have 3a^ — baW + 2^-* II 2a* — 3aW + ¥. We now find that the first term of the dividend will not con i.iin the first term of the divisor. We therefore multiply the divi- dend by 2, which merely introduces into the dividend a factor not common to the divisor, and hence does not affect the common divi- sor sought. We then have 6a* — 10a2Z»2 -i- Ah* jl2a* — 3a2/;2 + b* 6a* - 9a2<^2 _|_ 3^4 I 3 — aW + b* We find after division, the remainder — a^h"^ + ¥, which we put under the form — b"^ (a^ — b"^). We then suppress — i^, and divide 2a* — 3a2Z/2 -j- b* 2a* — 2a2i2 Z*2 2a2 — ^2 — a2^2 _|_ bi — aW + i*. Hence, a2 — b"^ is the greatest common divisor. 3. Let it be required to find the greatest common divisor be- tween the two polynomials — 3^3 + 3ai2 _ an + a^, and Ah'^ — 5ab + a2. rirst Operation. — V2b^ + 12a52 — Aa^b + 4a3 ist rem. 2d rem. or. — 3ai2 — a2^ _j_ 4^3 — 120^2 _ Aa%+ 16a3 1 4P — 5ab + a2 — 3^, — 3a — 19a2^ + 19a3 19«2(_ b + a). Second Operation. 4^2 _ ^nh + «2 — ah -\- a2 ' 0^ — i 4- a Ah + a llcnce, — b ■\- a, or a — i, is the greatest common divisor. 304 ELEMENTS OF ALGEBRA. [CHAP. X In tlie first operation we meet with a difficulty in dividing tlie two polynomials, because the first term of the dividend is not exactly divisible by the first term of the divisor. But if we ob- serve that the co-efficient 4, is not a factor of all the terms of the polynomial 4i2 — 5ab + a^, and therefore, by the first principle, that 4 cannot form a part w. the greatest common divisor, we can, without affecting this com mon divisor, introduce this factor into the dividend. This gives — 12^*3 + 12ai2 _ 4^2^ ^ 4^3^ and then the division of the terms is possible. Effecting this division, the quotient is — 3h, and the remainder is — 3ai2 _ a^ + 4«3. As the exponent of b in this remainder is still equal to that of b in the divisor, the division may be continued, by multiplying this remainder by 4, in order to render the division of the first term possible. This done, the remainder becomes — 12a52 _ 4a^ + iGa^, which divided by 4b^ — 5ab + «^, gives the quotient — 3a, which should be separated from the first by a comma, having no con- nexion with it. The remainder after this division is — I9a^ + 19a3. Placing this last remainder under the form 19a^ (— b -\- a), and suppressing the factor IQa^, as lorming no part of the com- mon divisor, the question is reduced to finding the greatest com- mon divisor between 4J2 _ 5ab + a2 and — b -\- a. Dividing the first of these polynomials by the second, we obtain an exact quotient, — 4& + a : hence, — Z> + a, or a — b, is the greatest common divisor sought. 288. In the above example, as in all those in which the ex- ponent of the principal letter is greater by unity in the dividend than in the divisor, we can abridge the operation by first multi- plying every term of the dividend l)y the square of the co-elh- CHAP. X.] GREATEST COMMON DIVISOR. 305 cient of the first term of the divisor. We can easily see that by this means, the first partial quotient obtained will contain the first power of this co-efficient. Multiplying the divisor by the quotient, and making the reductions with the dividend thus prepared, the result will still contain the co-efficient as a factor, and the division can be continued until a remainder is obtained of a lower degree than the divisor, with reference to the principal letter. Take the same example as before, viz., — 3P -f 3aP — a^b -\- a^ and 45^ — bah + d^, and multiply the di\'idend by the square of 4 r= 16 ; and we have First Operation 120^/2 _ 4^2^ _|_ 16- lained which contains — led — c2, as a factor of its two co- 310 ELEMENTS OF ALGEBRA. [CHAP. X. efTicients ; for 2Jc2 -{- c^ =z — c (— 2cd — c-). This factor being suppressed, the remainder is reduced to a — c, which Avill exactly divide a^ — c^. Hence, a — c is the required greatest common divisor. 293. There is a remarkable case, in which the greatest com- mon divisor may be obtained more easily than by the general method ; it is, when one of the two polynomials contains a letter which is not contained in the other. In this case, it is evident, that the greatest common divisor is independent of this letter. Hence, by arranging the polynomial which contains it, with reference to this letter, the required com- mon divisor will be the same as that which exists between the co-ejffi- cients of the different powers of the principal letter and the second polynomial. By this method we are led, it is true, to determine the greatest common divisor between three or more polynomials. But they will be more simple than the proposed polynomials. It often hap- pens, that some of the co-efficients of the arranged polynomial are monomials, or, that we can discover by simple inspection that they are prime with each other ; and, in this case, we are cer- tain that the proposed polynomials are prime with each other. Thus, in the example 1, treated by the first method, after having suppressed the common factor a — c, which gives the results, (^2 — c^ and 2ad — c^, we know immediately that these two polynomials are prime with each other ; for, since the letter a is contained in the second and not in the first, it follows from what has just been said, that the common divisor must divide the co-efficients 2d and — c^, which is evidently impossible ; hence, they are prime with respect to each other. 2. Let it be required to find the greatest common divisor of the two polynomials, 2bcq + 30/np + 185c + 5mpq and 4adq — A2fg + 2Aad — Ifgq, by the last principle. We observe, in the first place, that the two polynomials do no contain any common monomial factor. CHAP. X..] GREATEST COM.MOV DIVISOR. 311 Since q is common to tlie two polynomials, we can arrange them with reference to this letter, and follow the ordinary rale. But h is found in the first polynomial and not in the second. If then, we arrange the first with reference to b, which gives {3cq -\- 1 8c) b + 30mp -f- 5mpq, the required greatest common divisor will be the same as that whici. exists between the second polynomial and the two co-efficients 3cq + ISr and 30/np -j- 5mpq. Now, the first of these co-eflicients can be put under the form 3c {q -f- 6), and the other becomes 5mp (§' + 6) ; hence ^ + 6 is a common factor of these co-efiicients. It will therefore be suffi- cient to ascertain whether 7 + 6, which is a prime divisor, is a factor of the second polynomial. Arranging this polynomial Avith reference to q, it becomes {4ad --7fg)q-42fg+2iad; as the second part, 2iad — A2fg = 6 [Aad — Ifg), it follows that this polynomial is divisible by 5' + 6, and gives the quotienl Add — Ifg. Therefore, j -f 6 is the greatest common divisor of the proposed polynomials. Remark. — It may be ascertained that q-\-6 is an exact divisor of the polynomial {Aad-lfg)q + 2Aad-A2fg, by a method derived from the property proved in Art. 278. Make 7 + 6 = 0, or 7 = — 6, in this polynomial ; it becomes {4ad -7fg) X - 6 4- 24ad - 42fg = ; that is, — 6 substituted for q reduces the polynomial to ; hence q -\- Q is a divisor of this polynomial. This method may be advantageously employed in nearly all the applications of the process. It consists in this, viz., after obtain- ing a remainder of the first degree with reference to a, when a is the principal letter, make this remainder equal to 0, and dcdncf the value of a from this equation. If this value, substituted in the remainder of the 2d divr ■ destroys it, then the remainder of the 1st degree, simplified A ^ 292, is a common divisor. If the remainder of the 2d de>:r;'r 312 ELEMENTS OF ALGEBRA. [CHAP. X. does not reduce to by this substitution, we may conclude that tliere is no common divisor depending upon the principal letter. Farther, having obtained a remainder of the 2d degree, with reference to a, it is not necessary to continue the operation any farther. For, Decompose this "polynomial into two factors of the \st degree. »vhich is done by placing it equal to 0, and resolving the result- jquation of the 2d degree. When each of the values of a thus obtained, substituted in the remainder of the 3d degree, destroys it, it is a proof that the re- mainder of the 2d degree, simplified, is a common divisor ; when only one of the values destroys the remainder of the 3d degree, the common divisor is the factor of the 1st degree with respect to a, which corresponds to this value. Finally, when neither of these values destroys the remainder of the 3d degree, we may conclude that there is not a common divi- sor depending upon the letter a. It is here supposed that the two factors of the 1st degree with reference to a, are rational, otherwise it would be more simple to perform the division of the remainder of the 3d degree by that cf the second, and when this last division cannot be performed ex- actly, we may be certain that there is no rational common divisor, for if there was one, it could only be of the 1st degree with re- spect to a, and should be found in the remainder of the 2d degree, which is contrary to the hypothesis. 3. Find the greatest common divisor of the two polynomials 6x5 — 4,T* — lla;3 — 3a;2 _ 3a; _ 1 and 4x4 _|_ 2^3 — IQx^ + 3,r — 5. Ans. 2x'^ — 4x'^ -\- x — 1 4. Find the greatest common divisor of the polynomials 20x6 _ i2oc5 -(_ iG-ci _ 153;3 _|_ ]4j,2 _ 15^, ^ 4, and 15x4— 9^3 + 47a;2 — 21a; + 28. Ans. 5x" — Sa? + 4. 5. Find the greatest common divisor of the two polynomials 5a^^" + 2a^3 + ca^ — 3a-b^ + hca aiid aS 4- 5a^d — aW + ba%d. Ans. a^ _f_ a5. CHAP. X.] TRANSFORMATION OF EQUATIONS. 313 Tran.fjormadoii cf Equations. The transformation of an equation consists in changing its form without affecting the equality of its members. The object of a transformation, is to change an equation from a given form, to another form that is more easily resolved. First Transformation. To make the Denominators disappear from an Equation. 29 i. If we have an equation of the form x"' + Px'"-! + Qx'"-2 _(_ ... r.c + Z7= 0, and make a? ^ -4- ; k we shall have, after substituting this value for x, and multiplying every term by k^, y"" + P%'"-i + Qk-y^-^ -f P^3y'"-3+ . . . + Tk'^-^y + f/A"' — 0, an equation in which the co-efficieuts of y are equal to those of the given equation, multiplied respectively by yt°, A\ k"^, k^, k*, &c. This transformation is principally used to make the denominators disappear from an equation, when the co-efficient of the first term is unity. As an example, take the equation of the 4th degree, a „ c e s If we make x = -~, k y being a new unknown quantity and k an indeterminate quantity, we have ak ^ ck"^ „ ek^ gk* Now, there may be two cases — 1st. Where the denominators b, d, f A, are prime with each other. In this hypothesis, as k is altogether arbitrary, take k = hdfh, the product of the denominators, the equation will then become r/ + adfh . y3 + ch'^dfW . y^ + eb^dfW . y + gb^d^fP = 0, in which the co-efficients are entire, and that of its first term unity. 314 ELEMENTS OF ALGEBRA. [CHAP. X. We can determine the values of x correspondiog to those of 3, from the equation, _ y * ~ bdfk' 2d. When the denominators contain common factors, we shall evidently render the co-efficients entire, by making k equal to the smallest multiple of all the denominators. But we can simplify still more, by giving to k such a value that A\ k-, P, . . . shall contain the prime factors which compose b, d, f, h, raised to pow- ers at least equal to those which are found in the denom.inators. Thus, the equation 5^5^ 7 13 „ •x* x^ A a;- X =: 0, 6 12 150 9000 becomes bk , 5^2 ^ 7A-3 13A* after making x = -^, and reducing the terms. First, if we make k = 9000, which is a multiple of all Ine other denominators, it is clear, that the co-efficients become whole numbers. But if we decompose 6, 12, 150, and 9000, into their factors, we find 6 = 2x3, 12 = 22x3, 150 = 2x3x52, 9000 = 23x32x53; and by simply making i = 2 X 3 X 5, the product of the different simple factors, we obtain F = 22 X 32 X 52, F = 23 X 33 X 53, /t* = 2* X 34 X 5% whence we see that the values of k, F, k^, k*, contain the prime factors of 2, 3, 5, raised to powers at least equal to those which enter into 6, 12, 150, and 9000. Hence, the hypothesis A = 2 X 3 X 5, is sufficient to make the denominators disappear. Substituting th) value, the equation becomes ^ 5.2.3.5 3 5.22.32.52 ^ 7.23.33.53 13.2*.3'^.54 _ ^ 2^3^ ^ "* 22;3~~ ^ 2.3T52" ^ 23r3'2:53" ~ ^ CHAP. X.] TRANSFORMATION OF EQUATIONS. 315 which reduces to yt - 5.5y3 + 5.3.5y _ 7.22.32.53/ — 13.2.32.5 = ; or y* — 25y3 + 375y2 _ I260y — 1170 = 0. Hence, we perceive the necessity of taking k as small a number as po.ssible : otherwise, we should obtain a transformed equation, having its co-efficients very great, as may be seen by reducinc the transformed (equation resulting from the supposition k = 9000. Hence we see, that any equation may be transformed into another equation, of which the roots shall be a multiple or sub-multiple of those of the given equation. EXAMPLES. 1. oc^ :r 3^2 H — -X = 0. 3 36 72 y Making x = ^-, and we have o 6 y3_ 14y2-(- lly — 75 = 0. 5 13 , 21 32 ^ 43 1 2. x^ — -— a:' + -r a'3 x^ x = 0. 12 ^40 225 600 800 Making x — — -^ — = -^, and we have 2-^.3.0 60 ys _ 55y _^ I890y3 - 30720y2 - 928800y + 972000 = 0. Second Transformation. To make the second Term disappear from an Equation. 295. The difficuUy of resolving an equation generally diminishes with the number of terms involving the unknown quantity. Thus the equation a?2 = g, gives immediately, x =z zt. y q, while the complete equation x"^ -{- 2px -^ q z=i 0, requires preparation before it can be resolved. Now, any given equation can always be transformed into another equation, in which the second term shall be wanting. For, let there be the general equation -jm ^ p^m 1 _j. Qy.m-7. ^ _ _ ^ Tx + U = 0. Suppose X = u + x', 316 ELEMENTS OF ALGEBRA. [CHAP. X II being unknown, and x' an indeterminate quantity. By substitu- ting u -\- a/ for X, we obtain {71 + xJ^+P^u 4- ^)"'~^ + Q (" + x'Y'-^ . . . -\-T{u-{-x')-{-U = 0. Developing by the binomial formula, and arranging according to the decreasing powers of u. we have -f mx + P + m m — 1 + (m - 1) Pa/ + Q + + a?"» + Pa;""-i + Qx^-^-z + . . . + Tx" + f^ ). = Since a;' is entirely arbitrary, we may dispose of it in such a way that we shall have P mx' 4- P = ; whence, ,t' z= . m Substituting this value of x' in the last equation, we shall obtain an equation of the form, in which the second term is wanting. If this equation were resolved, we could obtain any value of x corresponding to that of u, from the equation X ■= u -\- x , or X = u . 7n Whence, in order to make the second term of an equation dis- appear, Substitute for the unknown quantity a new unknown quantity, uni- ted with the co-ej^cient of the second term, taken with a contrary sign, and divided by the exponent of the degree of the equation. Let us apply the preceding rule to the equation ■T^ + 2px = q. [f we make x = u — p, we have (m — p)'-* -{- 2p [u — p) = q ; CHAP. X.] TRANSFORIIATIOX OF EQUATIONS. 317 and by performing the multiplications and reducing, u^ —f = q, whicti gives u =z ± -^ q -\- p^ -^ and consequently, a;= — p ± -^ q -{- p"^. 296. Instead of making the second term disappear, it may l)e required to find an equation which, shall be deprived of its third fourth, or any other term. This is done, by making the co-efR- cient of u corresponding to that term equal to 0. For example, to make the third term disappear, we make, in the above-transformed equation TTl 1 m x"^ + (»? - 1) -P^r^ + Q = 0, from which we obtain two values for x^, which substituted in the transformed equation reduce it to the form ^m _j_ P'um-\ ^ R^uni-3 _ ^ T'u + U' = 0. Beyond the third term it will be necessary to resolve an equa- tion of a degree superior to the second, to obtain the value of a/ • and to cause the last term to disappear, it will be necessary to resolve the equation a/m + Par"»-i . . . + Tx" + U — 0, which is what the given equation becomes when a:' is substituted for x. It may happen that the value m ' which makes the second term disappear, causes also the disap- pearance of the third or some other term. For example, in order that the third term may disappear at the same time with the second, it is necessary that the value of x' which results from the equation m shall also satisfy the equation 771 — 1 m x"^ -\-{m-l)Px' -{- Q = 0. P Now. if in this last equation, we replace a/ by , Ave have m m — l P2 P2 m- . — -(/rt-l) — + Q = 0, or (m - 1) P^ -2mQ = 0; 2 Trr m 318 ELEMENTS OF ALGEBRA. and consequently, if [CHAP. X. P2 = 2mQ m — 1 the disappearance of the second term will also involve that of the the third. Formation of derived Polynomials. 297. The relation which has been used in the two preceding articles, indicates that the roots of the transformed equation are equal to those of the given equation, increased or diminished by a certain quan- tity. Sometimes this quantity is introduced into the calculus, as an indeterminate quantity, the value of which is afterward determined by requiring it to satisfy a given condition ; sometimes it is a par- ticular number, of a given value, which expresses a constant dif- ference between the roots of a primitive equation and those of another equation which we wish to form. In short, the transformation, which consists in substituting u -\- x for X, in a given equation, is of very frequent use in the theory of equations. There is a very simple method of obtaining, in prac tice, the transformation which results from this substitution. To show this, let us substitute for x, u + x' in the equation then, by developing, and arranging the terms according to the ascending powers of ti, we have + P a;'"*-! + (m — 1 ) Px'^-^ + + . . . ^Tx" +T + U u-{- m- m — 1 1.2 ,771 — 2 „ , + (m — 1)— -— Pa:""-3 + (m-2)^^Qy--* + . .. u^-\- . . .u" >=0 If we observe how the co-efficients of the different powers of u are composed, we shall see that the co-'sfficient of vP, is what the Cl'.AP. X.J FORMATION OF DERIVED POLYNOMIALS. 319 tirst member of the given equation becomes vrhen x' is substitut*>d in place of a; ; we shall denote this expression by X' The co-efficient of u^ is formed from the preceding term X', by multiplying each term of X^ by the exponent of 3/ in that term, and then diminishing this exponent by unity; we shall denote '.bis co-efficient by Y\ The co-efficient of ti^ is formed from Y\ by multiplying each term of Y' by the exponent of x' in that term, dividing the prod- uct by 2, and then diminishing each exponent by unity. Repre- senting this co-efficient by -;-, we see that Z' is formed from Y^, in the same manner that Y^ is formed from X' . In general, the co-efficient of any power of u, in the above- transformed equation, may be found from the preceding co-efficient in the following manner : viz., By taking each term of that co-efficient in succession, multiplying it by the exponent of x\ dividing by the number which marks the place of the co-ejicient, and diminishing the exponent of x' by unitij. The law by which the co-efficients -7/ v/ 1.2' 1.2.3' are derived from each other, is evidently the same as that which governs the formation of the terms of the binomial formula (Art, 203). The expressions, T, Z\ Y', TP . . . . are called derived polynomials of X\ because each is derived from the one which precedes it, by the same law as that by Avhich Y^ is deduced from X\ Hence, generally, A derived polynomial is one which is deduced from a given poly- ni mial, according to a fxed and known law. Recollect that X^ is what the given polynomial becomes when t is substituted for x. Y' is called the first-derived polynomial ; Z' is called the second-derived polynomial V' is called the third-derived polynomial &c., &c 320 ELEMENTS OF ALGEBRA. [CHAP. X. We should also remember if we mal;e u = 0, we shall have, a/ = X, whence X' will become the given polynomial, from which the derived polynomials will then be obtained. 298. Let us now apply the above principles in the following EXAMPLES. 1. Let it be required to find the derived polynomials from ;1. equation 3a;* + 6x3 — 3a;2 + 2a; + 1 = = X Now, u being zero, and x' = x, we have from the law of forming the derived polynomials, X= X' = 3a:* + 6a;3 — 3a;2 + 2x + 1 ; F = 12x3+ 18x2 — 6x ^2; Z' = 36x2 + 36:r _ 6 ; V = 72x + 36 ; W = 72. It should be remarked that the exponent of x in the terms 1, 2 — 6, 36, and 72, is equal to ; hence, each of those terms dis- appears in the following derived polynomial. 2. Let it be required to cause the second term to disappear in the equation x^ — 12x3 + 17x2 _ 9a, _(_ 7 == 0. 12 Make (Art. 295), x=:m-j = ?<4-3; whence, x^ = 3. The transformed equation will be of the form Z' V ^2^2x3 and the operation is reduced to finding the values of the co-efficients Z' V X, T, 2 ' 2.3' Now, it follows from the preceding law for derived polynomials, that X = (3)*-12. (3)3+17. (3)2-9. (3)1+7, or X^=-110; T =4.(3)3-36.(3)2+34.(3)1-9, or - - F =-123: ^ =6.(3)2-36.(3)1 + 17, or - . - - |^=-37: — =4.(3)1-12 = 0. 2.3 '^ ^ 2.3 CHAP. X.] FORMATION OF DERIVED POLYNOMIALS. 3Vi' Therefore the transformed equation becomes u^ — 37«2 _ 123« — 110 = 0. 3. Transform the equation 4x^ — 5a:2 + 7x — 9 = into another equation, the roots of which shall exceed those of the given equation by unity. Make, X = u — 1 ; whence x^ = — 1 ; and the transformed equation will be of the form ^1.2 '1.2.3 Hence, we have K' = 4. (-1)3- 5. (-1)2 + 7. (-1)1— 9, or X' z=-25j y^ =12. (-1)2-10. (-1)1 + 7 - - - - F = + 29; Z' Z' ^-^^■(-^y-^ Y=-i^^ 2.3 2.3 ^ Therefore, the transformed equation becomes 4w3 _ 17«2 + 29 w — 25 = 0. 4. What is the transformed equation, if the second term be made «o disappear in the equation a,5 _ lOx* + 7a;3 + 4a; — 9 = ? Ans. u^ — 33«3 _ 118»2 _ I52u — 73 = 0. 5. What is the transformed equation, if the second term be made 4} disappear in the equation 3a:3 + 15x2 + 25j: — 3 = ? 152 Ans. 3«3 — — = 0. 9 6. Transform the equation 3a;* — Ux^ + 7a-2 — 8a; — 9 = ■jito another, the roots of which shall be less than the roots of the ifiven equation by -;— . Ans. 3u-* — 9u^ — 4«2 u = 9 9 21 /)'" -f P (,r + 7/)^"-! + . . . = (x + w — g) (x + w — Z-) . . . ; or, changing the order of x and u, in the second member, and re- garding X — (I, X — b, . . . each as a single quantity, (x-I-M)'"+P(,r + (/)"'-i . . . =:[u+x — a) {u + x — b) . . . (m + x — /). Now, by performing the operations indicated in the two members, we shall, by the preceding Article, obtain for the first member, X + Ya + — «^ + ... w^ ; X being the first member of the proposed equation, and Y, Z . . . the derived polynomials of this member. With respect to the second member, it follows from Art. 284, 1st. That the part involving m°, or the last term, is equal to the product (x — a)[x — b) . . . {x — /) of the factors of the proposed equation. 2d. The co-efficient of u is equal to the sum of the products of these m factors, taken m — 1 and m — 1 . 3d. The co-efficient of ifi is equal to the sum of the products of these m factors, taken m — 2 and m — 2 ; and so on. Moreover, since the two members of the last equation are iden- tical, the co-efficients of the same' powers are equal (240 \ Hence, X = (x _ o) (x - h) (x _ c) . . . (x — /), which was already known. Hence also, Y, or the first-derived polynomial, is equal to the sum of the products of the m factors of the first degree in the proposed equation, taken m — 1 and m — 1 ; or equal to the sum of all the quotients that can be obtained by CHAP. X.] EQUAL ROOTS. 2ZS dividing X by each of the m factors of the frst degree in the pro- posed equation ; that is, 2i Also, ■ — , that is, the second-derived poHTnonii il, divided by 2, is equal to the sum of the products of the m factors of the pro- posed equation taken m — 2 and m — 2, or equal to the sum of the quotients that can be obtained by dividing X. by each of the factors of the second degree ; that is, Z X X X + 2 [x — a){x~l) [x ~ a){x — c) ' ' ' [x — k)[x — I)' and so on. Of equal Roots. 300. An equation is said to contain equal roots, when its first member contains equal factors. When this is the case, the de- rived polynomial, which is the sum of the products of the m fac- tors taken m — I and ?/j — 1, contains a factor in its diflerent parts, which is two or more times a factor of the proposed equa- tion (Art. 299). Hence, there must be a common divisor between the frst member nf the proposed equation, and its frst-derived polynomial. It remains to ascertain the relation between this common divi- sor and the equal factors. 301. Having given an equation, it is required to discover ivhether it has equal roots, and to determine these roots if possible. Let us make X = x'" + Px^-^ + Qx'"-^ -{-... + Tx + U z= 0, and suppose that the second member contains n factors equal to X — a, n' factors equal to x — b, n" factors equal to x — c . . .> and also, the simple factors x — p, x — q, x — r . . . ; we shall then have, A"= (:r — o)" {x — bY{x — cy . . .{x -p)(x — q){x — r) (1). We have seen that Y, or the derived pohmomial of X, is (he !fU7n of the quotients obtained by dividing X by each of the m fac- tvrs vf the frst degree in the proposed equation (Art. 299). 324 ELEMENTS OF ALGEBRA. [CHAP. X Now, since X contains n factors equal to x — a, Ave shall have n partial quotients equal to ; and the same reasoning ap- * X — a nlies to each of the repeated factors, x — b, x — c. . . . M-ore over, we can form but one quotient for each simple factor, which is of the form, _X_ _X_ X X — p' X — g X — r Therefore, the first-derived polynomial is of the form, nX nX i}"X X X X , _. Y= + - + + . . . + — + + + . • . (2). X — a X — b X — c X — y; x — q x — r Dy examining the form of the value of A' in equation (1), it is plain that {x - aY-\ {x - hY'-\ {x - cY"-^ . . . are factors common to all the terms of the polynomial ; hence the product {x — ff)"-! X (x — i)"'-! X {x — c)"'^-^ . . . is a common divisor of Y. Moreover, it is evident that it will also divide X: it is therefore a common didsor of A and Y\ and it is their greatest common divisor. For, the prime factors of X are x — a, x — h, x — c . . ., and X — p, X — q, X — /•...; now, x — p, x — q, x — r, cannot di- vide y, since some one of them will be wanting in some of the parts of y, while it will be a factor of all the other parts Hence, the greatest common divisor of A' and Y is D = (x - «)"-' (.r - //)"'-' {x - r)"'^-i . . . ; that is, The grpcilest coniiiKin divisor is composed of the product oj those factors which enter two or more times in the given equation, each raised to a poiver less by unity than in the primitive equation. 302. From the above we deduce the following method for find- ing the equal ronis. To discover whether an equation A'= contains any equal roots, furm Y or the derived polyno?niul of X ; then seek fur the greatest common divisor hetivccn X and y; if oixi caimot be obtained, the equation has no equal roots, or equal faciors. CHAP. X.] EQUAL ROOTS. 325 If we find a common divisor D, and it is of the first degree, or of the form x — h, make a" — h ■= 0, whence x zrz h. We then conclude, that the equation has two roots equal to h, and has but one species of rqual roots, from which it may be freed by dividing X by (.r - hy.' If D is of the second degree with reference to x, resolve the equation D ^ 0. There may be two cases ; the two roots will be equal, or they will be unequal. 1st. When we find D :=: (x — h)-, the equation has three roots equal to h, and has but one species of equal roots, from which it can be freed by dividing X by (x — //)•*. 2d. When D is of the form (,r — //) (r — h'), the proposed equation has tico roots equal to h, and tiro equal to h', from which it may be freed by dividing A' by (x — hy [x — h'Y, or by D"^. Suppose now that D is of any degree whatever ; it is necessary, in order to know the species of equal roots, and the number of roots of each species, to resolve cumptetely the equation D = 0. Then, every simple root of D ivill be twice a root of the given equation ; every double root of D will be three times a root of the given equation ; and so on. As to the simple roots of X = 0, we begin by freeing this equation of the equal factors contained in it, and the resulting equation, X' = 0, will make known the simple roots. EXAMPLES. 1. Determine whether the equation 2x* — 12x3 + 19^.2 _ 6x + 9 = contains equal roots. We have for the first-derived polynomial (Art. 297), 8x3 _ 36^2 _^ 28x — 6. Now, seeking for the greatest common dinsor of these poly- nomials, we find Z) = X — 3 = 0, whence x = 3 ; hence, the given equation has two roots equal to 3. 326 ELEMENTS OF ALGEBRA. [CHAP. X Dividing its first member by [x — 3)^, we obtain 2a;2 +1=0; whence a' = zh — y — 2. The equation, therefore, is completely resolved, and its roots are 3, 3, +i-/3^ and _— /-2. 2. For a second example, take a;5 _ 2a;* + 3x^ — 7a;2 -f 8a; — 3 = 0. The first-derived polynomial is 5a;* — 8x^ + 9a;2 — 14jc -f 8 ; and the common divdsor, 3^2 — 2a; + 1 = (a; — ] )2 : hence, the proposed equation has three roots equal to 1. Dividing its first member by (x _ 1)3 = a;3 — 3a;2 -\-3x — \, the quotient is -1±V^11 x^ -{- X -\- 3 = ; whence x = — ; thus, the equation is completely resolved. 3. For a third example, take the equation a;^ + 5^6 + 6a;5 — 6a;* — 15a;3 — 3a;2 + 8a; -f 4 = 0. The first-derived polynomial is 7x6 ^ 30^5 -f. 30a;* — 24x3 _ 45^,2 _ 6a; + 8 ; and the common divisor is X* + 3x3 _|_ a;2 _ 3a, _ 2. The equation X* + 3x3 + x2 — 3x — 2 = cannot be resolved directly, but by applying the method of equal roots to it, that is, by seeking for a common divisor between its first member and its derived polynomial 4x3 + 9x2 _|. 2a; — 3, we find a common divisor, x + 1 ; which proves that the square ©f X + 1 is a factor of x* + 3x3 ^ ^2 _ ^x — 2, and the cube of X -|- 1, a factor of the first member of the given equation. CHAP. X.] ELIMINATION. 327 Dividing a:* 4- 3x3 4- a;2 — 3a; — 2 by (3, _f. 1)2 _ j,2 _|. 2a; + 1, we have x^ + x — 2, which being placed equal to zero, gives the two roots a: = 1, x = — 2, or the two factors, x — 1 aiul X + 2. Hence we have X* + 3x3 + x2 — 3x — 2 = (x + 1)2 (.^ _ 1) (a, _^ 2). Therefore, the first member of the proposed equation is equal to (x + 1)3 (x — 1)2 (x + 2)2; that is, the proposed equation has three roots equal to — 1, two equal to +1, and two equal to — 2. 4. What are the equal factors of the equation x' — 7x6 ^ 10x5 4- 22x* — 43x3 — 35x2 ^ 43^ + 36 = 0. Ans. (x — 2)2 (x — 3)2 (x + 1 )3 = 0. 5. What are the equal factors in the equation x^ — 3x6 ^ 9a;5 _ ] 9j;4 _^ 27x3 _ 33_r2 _|_ 27x — 9 =: 0. Ans. (x — 1)3 (x2 + 3)2 = 0. Elimination. 303. To eliminate between two equations of any degree what- ever, involving two unknown quantities, is to obtain, by a series of operations, performed on these equations, a single equatiun which contains but one of the unknown quantities, and which gives all the values of this unknown quantity that will, taken in connexion with the corresponding values of the other unknown quantity, satisfy at the same time both the given equations. This new equation, which is a function of one of the unknown quantities, is called the fnal equation, and the values of the un- known quantity found from it, are called compatible values. Elimination by Means of Lideterminate Multijjlif^rs 304. Let there be the equations a X -\- b y — c =0, a'x -\- b'y — c' = 0. If we multiply the first by m, and subtract the second from tlu' product, we have {ma — a') X + (mb — b'^ y — 7«c 4" c' = . . . (1 ). 328 ELEMENTS OF ALGEBRA. [CHAP. X. Now, since the value of m is entirely arbitrary, we may give it such a value as to render the co-efficient of* x zero, which gives ma — a' z= 0, whence m := + a and {mb — b')7j— mc + c' = (2). Substituting in equation (2) the value of m, and we have a a'c — ac' acf — a'c " a' a'h — ab' aV — a'b — .0 — a Had we chosen to attribute to m such a value as to render the co-efficient of y zero in equation (1), we should have had b' mb — h' = 0, whence ot = -— b and [ma — a') x — mc -{- c' ■= (3). Substituting in equation (3) the value of m, we obtain Z _ ' b'" " I'c- be' ~ y , ab' — a'b —r- -a — a b The above values for x and y are the same as those deter- mined in Art. 97. The principle explained above is applicable to three or more equations, involving a like number of unknown quantities. 305. Of all the known methods of elimination, however, the method of the common divisor is, in general, the best ; it is this method which we are going to develop. Let f(x, j/)=0 = A, and / (x, y) = = 5, be any two equations whatever, in which / and /'' denote any func- tions of X and y. Suppose the final equation involving y obtained, and let us try to discover some property of the roots of this equation, wliich may serve to determine it. L t 1/ — a he one of the values of y which will satisfy both the given equations. This is called a compatible value of y. It is plain, that, since this value of y, in connexion with a certain value of x, will satisfy both ('(piations, that if it be substituted in thom, there will result two CHAP. X.] ELI.MIXATION. 329 equations involving x alone, which will admil of at least one com- mon va/iif! of X ; and to this common value there will correspond ft common divisor involving x (Art. 279). This common divisor will be of the first, or of a higher degree with respect to x, ac- cording as the particular value of y = a corresponds lo one or more values of x. Reciprocally, every value of y which, substituted in the two equa- tions, gives a common divisor involving x, is necessarily a compati- ble value, because it then evidently satisfies the two equations at the same time with the value or values of x found from this com- mon divisor when put equal to 0. 306. We will remark, that, before tlie substitution, the first mem- bers of the equittiims cannot, in general, have a common divisor which is a I'uriction of one or both of the unknown quantities. For, let us suppose for a moment that the equations ^ = 0, B = 0, are of the form A' X Dz^O, B' X D =:0. D being a function of x and y. Making separately D = 0, we obtain a single equation involving two unknown quantities, which can be satisfied with an infinite number of systems of values. Moreover, every system which ren- ders D equal to 0, would at the same time cause A'D, B'D to vanish, and would consequently satisfy the equations A = Q and B = 0. Thus, the hypothesis of a common divisor of the two polyno- mials A and B, containing x and y, would bring with it as a con- sequence that the proposed equations were indeterminate. There- foie, if there exists a common divisor, involving x anJl y, of the two polynomials A and B, the proposed equations will be indeter- minate, that is, they may be satisfied by an infinite number of systems of values of x and y. Then there woidd be no data to determine a final eouation in y, since the number of values of y is infinite. If the two polynomials A and B were of the form A' X D, B' X D, D being a function of x only, we might conceive the equation Z) -— resolved with reference to x, which would give one or 330 ELEMENTS OF ALGEBRA. [CHAP. X more values for this unknown. Each of these values subslituteJ in the equations A' X D^O and B' x D =^ 0, would verify them, without regard to the value of y, since D must e nothing, in consequence of the substitution of the value of x Therefore, in this case, the proposed equations would admit of a finite number of values for x, but of an infinite number of values for y, and then there could not exist a final equation in y. Hence, when the equations A = 0, J5 = 0, are determinate, that is, when they admit only of a limited nnmhet of systems of values for a; and y, their first members cannot have for a common divisor a function of these unknown quantities, un- less a particular substitution has been made for one of theso quantities. 307. From this it is easy to deduce a process for obtaining the final equation involving y. Since the characteristic property of every compatible value of y is. that being substituted in the first members of the two equa tions, it gives them a common divisor involving a-, which they had not before, it follows, that if to the t^vo proposed polynomials, ar- ranged with reference to .r, we apph'' the process for finding the greatest common divisor, \vq shall generally not find one. But, by continuing the operation properly, we shall arrive at a remainder independent of x, but which is a function of y, and which, placed equal to 0, will give the required final equation. For, every value of y found from this equation, reduces to nothing the last remain- der of the operation for finding the common divisor ; it is, then, such, thai substituted in the preceding remainder, it will render this remainder a common divisor of the first members A and B. Therefore, each of the roots of the equation thus formed, is a com- patible value of y. 308. Admitting that the final equation may be completely re- solved, which would give all the compatible values, it would after- ward be necessary to obtain the corresponding values of x. Now, it is evident that it would be suBinent for this, to substitute the different values of y in the remainder preceding the last, put the polynomial involving x which results from it, eijual to 0, and find CHAP. X.] ELIMINATION. 331 from it the values of x ; for these polynomials are nothing more than the divisors involving a:, which become common to A and B. But as the final equation is generally of a degree superior to the second, we cannot here explain the methods of finding the values of y. Indeed, our design was principally to show that, two equations of any degree being given, we can, without supposing the resolution of any equation, arrive at another equation, containi?}g only one of the unknown quantities which enter into the proposed equations. EXAMPLES. 1. Having given the equations A — x^ -{■ xy + y"^ — \ = 0, B = x^ + y3=0, to find the final equation in y. First Operation. x^ + y^ x^ -+■ yx"^ -\- [y"^ — ]) X x"^ -{- xy -\- y X — y = Q — yx^ — (y2 — l^ X -{- y^ — yx'^ — y'^x — y-^ -{- y R = X -\- 2y3 — y = 1st remainder. Second Operation. a;2 + yx + y2 _ 1 JU4. 2y3 - x2 + (2y3 — y)x a: - (2y3 — 2y) — (2y3 — 2y) X + y^ _ 1 — (2y3 — 2y) X — 4y^ + 6y* — 2y2 R^ = 4y'5 — 6y^ + 3y2 — 1 = 2d remainder Hence, the final equation in y, is 4y6 _ Qyi 4- 3y2 — 1 = 0. If it were required to find the final equation in x, we observe that X and y enter in the same manner into the original equations ; hence, x may be changed into y and y into x, without destroying the equality of the members. Therefore, 4x6 _ Q^i + 3x2 — 1 = is the final equation in x. 332 ELEJIENTS OF ALGEBRA. [CHAP. X. 2. Find the final equation in y, from the equations A = x^ — Si/x"^ + (3y2 — y + 1 ) ^ — y^ -\- rp- — 2y = 0, B zzz x^ — 2i/x + y2 — y ;_ 0. First Operation. ^3 _ 3y^2 _|. (3^2 _ y + 1) ^ _ y3 ^ y2 _ 2y II a:2 — 2jy + / _ y X" — 2yx'^ -\- (y2 — y) j; — yx2 + (2y2 + 1) a; _ y3 _|. y2 _ 2y — ya.'2 + 2y^a? — y^ -{■ y^ v = Q X — 2y = R. Second Operation. x"^ — 2xy + y" — y x^ — 23:y 2y Hence, r x^Q' f — y =Q is the final equation in y. This equation gives y = 1 and y = 0. Placing the preceding remainder equal to zero (Art. 308), and substituting therein the values of y = 1 and y = 0, we find for the corresponding values of x, X — 2 and x = 0; from which the given equations may be entirely reserved. CHAF. XI. 1 RESCiLUTIOX OF NUMERICAL EtiUATIONS. 333 CHAPTER XL RESOLUTION OF NUMERICAL EQUATIONS INVOLVING ONE OR MORE UNKNOWN QUANTITIES. 309. The principles established in the preceding chapter, are applicable to all equations, whether their co-efficients are numeri- cal or algebraic, and these principles are the elements which are to be employed in the resolution of equations of the higher de- grees. It has been already remarked, that analysts have hitherto been able to resolve the general equations only of the third and fourth degrees. The general formulas which haA'e been obtained for the resolution of algebraic equations of the higher degrees, are so complicated and inconvenient, even when they can be applied, that the problem of the resolution of algebraic equations, of any degree Avhatever, may be regarded as more curious than useful. Therefore, analysts have principally directed their researches to the resolution of numerical equations, that is, to those which arise from the algebraic translation of a problem in which the given quantities are particular numbers. Methods have been found, by means of which, the roots of a numerical equation of any given de- gree, may always be determined. It is proposed to develop these methods in this chapter. To render the reasoning general, we will represent the proposed equation by X =1 x"" + Px""-! + Qj;'"-2 + . . . =0. in which P, Q . . . denote particular numbers which are real, and either positive or negative. 334 ELEMENTS OF ALGEBRA. [CHAP. XT First Principle. 310. If we substitute for x a number a, and denote by A what X becomes under this supposition ; and again substitute a -\- u for X, and denote the new polynomial by A' : then, u may he taken so small, thai the difference between A'' and A shall he less than any assignable quantity. If now, we denote hy B, C, D, what the co-efEcients Z V Y, — , (Art. 297), become, when we make a? = a, we shall have for the polynomial X, under the supposition that a + « is substituted for x, equal to A -\- u{B -\- Cu + Du"^ + . . . u'"-^) = A\ Now, the quantity M (5 + Cw 4- Du^ 4- . . ■ w"'~0 is the difference between A'' and A J s-'^i it is required to show that this difference may be rendered less than any assignable quan- tity, by attributing a value sufficiently small to Ji. Let us take the most unfavorable case that can occur, viz., let us suppose that every co-efficient is positive, and that each is equal to the largest, which we will designate by K. Then, Zm (1 + ?i 4- w^ + • • • W"'^) = u{B -hCu+ . . . + M^-i) ; and in any other case, Ku (1 4- w + m2 + . . . u'»-i) > w (^ 4- C« -f . . . w-"-'). But we have, by Art. 61, (1 u'"\ ) ; 1 — u / Ku ^^ . ^ Ku and (1 — u^) < 1 — u 1 — u when « < 1. This being premised, if we wish the difference between A' and A to be less than any number N, let us make u such, that Kn ^ AT I.- 1 -1 ^ ^ . — or <^ iV which requires that, u z= or < _ , , 1 — w A + A CHAP. XI.] RESOLUTION OF NUMERICAL EQUATIONS. 335 and anv value of u which will fulfil this last condition, will satisfy the inequality iTzi (1 + u + u2 + . . . u'"-i) < N, and consequently, render « (5 -(- Cu + Brfl + . . . ti'"-^) < iV; in which the inequality is greater even than in the expression above. Second Principle. 311. If two numbers p and q, substituted in succession in the place of X in a numerical equation, give two results affected with contrary signs, the proposed equation contains a real root, compre- hended between these two numbers. Let us suppose that p, when substituted for x in the equation X =zO, gives + R, and that q substituted in the equation X=0, gives — R'. Let us now suppose x to vary between the values of p and q by so small a quantity, that the difference between any two cor- responding consecutive values of X shall be less than any assign- able quantity ; in which case, we say that X is subject to the law of co7itinuity, or that it passes through all the intermediate values between R and — R\ Now, a quantity which is constantly finite, and subject to the law of continuity, cannot change its sign from positive to nega- tive, or from negative to positive, without passing through zero : hence, there is at least one number between p and q which will Siitisfy the equation X = 0, Knd consequently, one root of the equation lies between these numbers. 312. We have shown in the last article, that if two numbers be substituted, in succession, for the unknown quantity in any equation, and give results affected with contrary signs, that there will be at least one real root comprehended between them. We are not, '^rv/e ?r, to conclude that there may not be more than one ; nor 336 ELEMENTS OF ALGEBRA. [CHAP. XI. that the substitution, in succession, of two numbers which include roots of the equation, will necessarily give results afl'ected with contrary signs. Third Prhiciple. 313. When an uneven number of the real roots of an equation is comprehended between two numbers, the results obtained l and q. These two quantities Y' and Y" , are afTected with the same sign ; for, if they were not, by the second principle there would be at least one real root comprised between p and q, which is contrary to the hypothesis. To determine the signs of the above results more easily, divide the first by the second, and we obtain ( p-«)(p- Z> )(p-c)... X F (y_a)(y-Z.)(y-c)... X r-' which can be written thus. p — a p — b p — c X r- X ~ q — a q — b q — c y/, CHAP. XI.] LIMITS OF REAL ROOTS. ii'l7 Now, since the roots a, b, c, . . . are comprised between p av.d q, we have p a, h, c, d . . ., and 9 a, h, c, d . . .; whence we deduce p — a, p — b, p — c, . . 0, ajnd q — a, q — b, q — c, . . . 0. Hence, since p — a and q — a are affected with contrary signs, as well as p — b and q — b, p — c and q — c . . ., the partial quotients p — a p — b p — c q — a q — o -, Sic, Y' are all negative. Moreover, -^7-^ is essentially positive, since Y and Y" are affected with the same sign ; therefore, the product p — a p — b p — c Y' X IT- X X . . . -jT-,, , q — a q — q — c 1" will be negative, when the number of roots, a, b, c . . .. compre- hended between p and q, is uneven, and positive when the number is even. Consequently, the two results {p-a){p-b){p-c) . . . X F, and (? — o) (?—*)(? — '^) •• • X F'^ will have contrary or the same signs, according as the numbei of roots comprised between p and q is uneven or even. Limits of Real Roots. 314. The different methods for resolving numerical equations, consist, generally, in substituting particular numbers in the pio- posed equation, in order to discover if these numbers verify it, or whether there are roots comprised between them. But by reflect- 22 338 ELEMENTS OF ALGEBRA. [CHAP. XI ing a little on the composition of the first member of the general equation X'n -f Pa;"'-! + Qa;'"-2 ...-}- Tx -\- U = 0, we become sensible, that there are certain numbers, above ^^hicb it would be useless to substitute, because all numbers above a certain limit, would give positive results. 315. Let it now be required to resolve the following question: To determine a number, ivhich suhstitaled in place of x will ren- der the first term x"* greater than the arithmetical sum of all the other terms. Suppose all the terms of the equation to be negative, except the first, so that ^m _ p^m-l _ Qx'n-2 _ _ ^ fx — U = 0. It is required to find a number for x which will render a;'" > P:>;"'-i + Qa;"'-^ +...-{- Tx + U. Let k denote the greatest co-efficient, and substitute it in place of the co-efficients ; the inequality will then become y,m y Ji^m-l _^ ^^m-2 ^_ . . _ _j_ ^jj ^ ^_ It is evident that every number substituted for x which will satisfy this condition, will for a stronger reason, satisfy the pre- ceding. Now, dividing this inequality by x"*, it becomes k k k k k k Making x = k, the second member becomes — - = 1 plus a k series of positive fractions. The number k will therefore not sat- isfy the inequality; but by supposing a; = /t + 1, we obtain for the second member the series of fractions K fC fi K rC Y^ '^ (M^Tp "^ {k + 1)3 + • ■ • + (^ + I )"•-! + {k+ i)-"' which, considered in an inverse order, is ar. increasing geometri- k cal progression, the first term of which is — , the ratio ^ ^ ' (A + 1 )'» k k -{- 1, and the last term — ; hence, the expression for tl e k -f- i sum of all the terms is (Art. 192), CHAP. XI.] ORDINARV LIMIT OF POSITIVE ROOTS. 339 A__ k r+T ■ ^ + ^^ "~ (TTTp 1 which is evidently less than unity. Now, any number > (^ + 1), put in place of x, will render the k k sum of the fractions 1 + • • • still less. Therefore, a; x^ The greatest co-efficient plus unity, or any greater number, being substituted for x, icill render the jirst term x"* greater than the arith- metical sum of all the other terms. 316. Every number which exceeds the greatest of the posi- tive roots of an equation, is called a superior limit of the positive roots. From this definition, it follows, that this limit is susceptible of an infinite number of values. For, w^hen a number is found to ex- ceed the greatest positive root, every number greater than this, is also a superior limit. But since the largest of the positive roots will, when substituted for X, merely reduce the first member to zero, it follows, that we shall be sure of obtaining a superior limit of the positive roots by finding a number, which, substituted in place of x renders the frst member positive, and ivhich at the same time is such, that every greater number will also give a positive result. Hence, the greatest co-rfficient of x plus unity, is a superior limit of the positive roots. Ordinary Limit of the Positive Roots. 317. The limit of the positive roots obtained in the last article, is commonly much too great, because, in general, the equation contains several positive terms. We will, therefore, seek for a limit suitable for all equations. Let a"*""" denote that power of x, corresponding to the first neg- ative term which follows x^, and let us consider the most unfavor- able case, viz., that in which all the succeeding terms are negative and affected with the greatest of the negative co-efficients in the equation. Let S denote tliis co-efiicient. What conditions will render j,m y Sa-"'-" + 5'a;'"-"-' + . . . Sx + S^. 340 ELEMENTS OF ALGEBRA. [CHAP. XI. Dividing both members of this inequality by a?'", we have 5 -S -S , S S Now, by supposing X = \/S -\- 1, or for simplicity, making "y/ S = S' which gives, >S = *S''", and a? = .S' 4- 1 or yS + 1, will, when substituted for x, render the sum of the fractions S S X" a'''+i "T- • • • • still smaller, since the numerators remaining the same, the de nominators will increase. Hence, y *S + 1, and any greater number, will render the first term a;'" greater than the arithmetical sum of all the negative terms of the equation, and will consequently give a positive result for the first member. Therefore, Unity increased by that root of the greatest negative co-efficient whose index is the number of terms which precede the first negative term, is a superior limit of the positive roots of the equation. If the co-efficient of a term is 0, the term must still he counted. Make « = 1, in which case the first negative term is the sec- ond term of the equation ; the limit becomes V^ + 1 = S + 1 ; I'iiat. is, the greatest negative co-efficient plus unity. CHAP. XI.] SMALLEST LIMIT IN ENTIRE NUMBERS. 341 Let ra r= 2; then, the limit is y 2, will render the poly- nomial of the first degree positive. But 2, substituted in th(5 polynomial of the second degree, gives a negative result ; and 3, or any number > 3, gives a positive result. Now 3 and 4, substituted in succession in the polynomial of the third degree, give negative results ; but 5, and any greater number, gives a positive result. Lastly, 5 substituted in X, gives a negative result, and so does 6 ; for the first three terms, x* — 5x^ — 6x^, are equivalent to the expression x^(x — 5) — 6^2, which reduces to when x — 6 ; but X = 7 evidently gives a positive result. Hence 7, which here CHAP. XI.] SUPERIOR LI.MIT OF NEGATIVE ROOTS. 343 Stands for x\ is a superior liinit of the positive roots of the giinn equation. Since it has been shown that 6 gives a negative re- sult, it follows that there is at least one real root between 6 and 7. 2. Applying this method to the equation x^ — 3x* — 8x3 _ 25x^ + 4a: — 39 = 0, i,he superior limit is found to be G. 3. We find 7 to be the superior limit of the positive roots of the equation x5 _ 5x* — 13x3 _(_ i7a;2 _ 69 = 0. This method is seldom used, except in finding incommensurable roots. Superior Limit of negative Roots. — Inferior Limit of posi- tive and negative Roots. 319. Having found the superior limit of the positive roots, it only remains to find the inferior limit, and the superior and infe- rior limits of the negative roots. Let, L = superior limit of positive roots. L' = inferior limit of positive roots. U^ = superior limit (that is, numerically) of negative roots t 1/''' =: inferior limit of negative roots. 1st. If in anv equation X = 0, we make x = — , we have a V derived equation 1"= 0. We know from the relation x = — , thai y the greatest positive value of y will correspond to the smallest of X ; hence, designating the superior limit of the positive roots of the equation 1" = by L, we shall have — = L\ the inferior j-j limit of the positive roots of the given equation. 2d. If in the equation X=0, we make a: = — y, which gives the transformed equation Y^i 0, it is clear that the positive roots of this new equation, taken with the sign — , will give the nega- tive roots of the given equation ; therefore, determining, by thf known methods, the superior limit L of the positive roots of ili'' equation y=:0, we shall have — i ^ i^', the superior limit (nu- merically) of the negative roots of the proposed equation. 344 ELEMENTS OF ALGEBRA. [CHAP. XI. 3d. Finally, if we replace x, in the given equation, by , y and find the superior limit L of the transformed equation Y = then, L'^' = =- will be the inferior limit (numerically) of the negative roots of the given equation. Consequences deduced from the preceding Principles. First. 320. Every equation in which there are no variations in the signs, that is, in which all the terms are positive, must have all of its real roots negative; for, every positive number substituted for x, will render the first member essentially positive. Second. 321. Every complete equation, having its terms alternately posi- tive and negative, must have its real roots all positive ; for, every negative number substituted for x in the proposed equation, would render all the terms positive, if the equation was of an even de- gree, and all of them negative, if it were of an odd degree. Hence. their sum could not be equal to zero in either case. This principle is also true for every incomplete equation, in which there results, hy substituting — y for x, an equation liaving afl its terms affected with the same sign. Third. 322. Every equation of an odd degree, the co-efficients of which are real, has at least one real root affected with a sign contrary to that of its last term. For, let ^m ^ p^m-\ -{.... Tx ± U = 0, be the proposed equation ; and first consider the case in which the last term is negative. By making a; = 0, the first member becomes — U. But by giving a value to a: equal to the greatest co-efficient plus unity, or {K -\- 1)) the first term x^ will become greater than the arith- metical sum of all the others (Art. 315), the result of this sub- stitution will therefore be positive ; hence, there is at least one CHAP. Xr.J CONSEQUENCES OF PRECEDING PRINCIPLES. 345 real root comprehended between and K -{- \, Avhich root is posi live, and consequently alFected with a sign contrary to that of tlit- last term (31]). Suppose now, that the last term is positive. Making a; = 0, we obtain + U for the result ; but by putting — (iiL + 1) in place of x, we shall obtain a negative result, since the first term becomes negative by this substitution ; hence, the equation has at least one real root comprehended between and — {K -\- 1), which is negative, or affected with a sign contrary to that of the last term. Fourth. 323. Every equation of an even degree, which involves only real co-efficients and of wJiich the last term is negative, has at least two real roots, one positive and the other negative. For, let — Z7 be the last term ; making a? == 0, there results — U. Now substitute either K+l, or — {K -\- \), K being the greatest co-efEcient in the equation. As in is an even number, the first term x^ will remain positive ; besides, by these substitutions, it becomes greater than the sum of all the others ; therefore, the results obtained by these substitutions are both positive, or afl^ected with a sign contrary to that given by the hypothesis x=0; hence, the equation has at least two real roots, one positive, and compre- hended between and K -\-\, the other negative, and compre- hended between and — (if + 1), (311). Fifth. 324. If an equation, involving only real co-efficients, contains im- aginary roots, the number of such roots must be even. For, conceive that the first member has been divided by all the simple factors corresponding to the real roots ; the co-efiicients of the quotient will be real (Art. 278) ; and the quotient must also be of an even degree ; for, if it was uneven, by placing it equal to zero, we should obtain an equation that would contain at least one real root (322) ; hence, the imaginary roots must enter by pairs. Remark. — 325. There is a property of the above polynomial quotient M'^hich belongs exclusively to equations containing only imaginary roots; viz., every such equation ahoays remains ■posi- tive for any real value substituted for x. 346 ELEMENTS OF ALGEBRA. [CHAP. XI For, by substituting for x, K-\- I, the greatest co-efficient plus unity, we could always obtain a positive result ; hence, if the polynomial could become negative, it would follow that when placed equal to zero, there would be at least one real root com- prehended between K -\- I and the number which would give a negative result (Art. 311). It also follows, that the last term of this polynomial must be positive, otherwise a; = would give a negative result. Sixth. 326. When the last term of an equation is positive, the number of its real positive roots is even ; and when it is negative, the w.m- her of such roots is uneven. For, first suppose that the last term is -f- U, or positive. Since by making a; = 0, there will result + U, and by making x=K-i- 1, the result will also be positive, it follows that and K + 1 gi^'^ two results affected with the same sign, and consequently (Art. 313), the number of real roots, if any, comprehended between them, is even. When the last term is — U, then and K -{- 1, give two re- sults affected with contrary sigTis, and consequently comprehend either a single root, or an odd number of them. The reciprocal of this proposition is evident. Descartes'' Rule. 327. An equation of any degree rvhatever, cannot have a greater numl^er of positive roots tha?i there are variations in the signs of its terms, nor a greater number of negative roots than there are per- manences of these signs. A variation is a change of sign in passing along the terms, and a permanence is when two consecutive terms have the same sign. In the equation x — a = 0, there is one variation, and one posi- tive root, X =z a. x\nd in the equation x -\- b = 0, there is one permanence, and one negative root, a? = — b. If these equations be multiplied together, there will result ar; equation of the second degree. a' — a + b x — ab !-• CHAP. XI.] DESCARTES* RULE. 347 It' a is less than b, the equation will be of the first form (Art. 144) ; and if a > i the equation will be of the second form ; that is, a < 6 gives a;- + 2j;a; — y = 0, and a '^ b " x"^ — 2px — ^ = 0. In the first case, there is one permanence, and one variation, and in the second, one variation and one permanence. Since in either form, one root is positive and one negative, it follows that there are as many positive roots as there are variations, and as many negative roots as there are permanences. The proposition will evidently be demonstrated in a general manner, if it be shown that the multiplication of the first member by a factor x — a, corresponding to a positive root, introduces at least one variation, and that the multiplication by a factor x -\- a, corresponding to a negative root, introduces at least one permanence. Take the equation ^m _|_ Ax'"-'^ db Bx"'-'^ rb Cx'"-^ ± . . . ±: Tx ± U = 0, in which the signs succeed each other in any manner whatever. By multiplying by x — a, we have 3^+'^ dzAlx'^dzB — a I q= Aa -i± C a:'"-2 ± . . . ±U =1= Ta 1=0. The co-efficients which form the first horizontal line of this product, are those of the given equation, taken with the same signs; and the co-efficients of the second line are formed from those of the first, by multiplying by a, changing the sigTis, and advancing each one place to the right. Now, so long as each co-efficient of the upper line is greater than the corresponding one in the lower, it will determine the sign of the total co-efficient ; hence, in this case there will be, from the first term to that preceding the last, inclusively, the same variations and the same permanences as in the proposed equation ; but the last term qp Ua having a sign contrary to that which im- mediately precedes it, there must be one more variation than in the proposed equation. When a co-efficient in the lower line is aff'ected with a sign contrary to the one corresponding to it in the upper, and is also greater than this last, there is a change from a permanence of sign to a variation ; for the sign of the term in which this happens. 348 ELEMENTS OF ALGEBRA [CHAP. XI. being the same as that of the inferior co-efRcient, must be con- trary to that of the preceding term, which has been supposed to be the same as that of its superior co-efficient. Hence, each time we descend from the upper to the lower line, in order to determine the sign, there is a variation which is not found in the proposed equation ; and if, after passing into the lower line, we continue in it throughout, we shall find for the remaining terms the same variations and the same permanences as in the given equation, since the co-efficients of this line are all affected with signs contrary to those of the primitive co-efficients. This sup- position would therefore give us one variation for each positive root. But if we ascend from the lower to the upper line, there may be either a variation or a permanence. But even by sup- posing that this passage produces permanences in all cases, since the last term ^ Ua forms a part of the lower line, it will be ne- cessary to go once more from the upper line to the lower, than from the lower to the upper. Hence, the new equation must have at least one more variatio7i than the jjroposed ; and it will be the same for each positive root introduced into it. It may be demonstrated, in an analogous manner, that the mul- tiplication by a factor x -{- a, corresponding to a negative root, would introduce one permanence more. Hence, in any equation, the number of positive roots cannot be greater than the number of VARIATIONS of sigus, uor the number of negative roots greater than the number of permanences. Consequence. 328. When the roots of an equation are all real, the number of vositive roots is equal to the 7iumber of variations, and the number of negative roots to the number of permanences. For, let m denote the degree of the equation, n the number of variations of the signs, p the number of permanences ; we shall have w = ra + p. Moreover, let n^ denote the number of positive roots, and p' the number of negative roots, we shall have m = 71'' + p' ; whence n -\- p ^ n' -\- p\ or, n — n' =. p' — p. Now, we have just seen that n' camiot be > n, nor can it be ess, since p' cannot be > p ; therefore we must have n' = n, and p' = p. CHAP. XI.] COMMENSURABLE ROOTS OF EQUATIONS. 319 Remark. — 329. When an equation wants some of its terras, we can often discover the presence of imaginary roots, by means of the above rule. For example, take the equation x^ -\- px -{- q ^= 0, p and g being essentially positive ; introducing the term which is wanting, by affecting it with the co-efRcient rt : it becomes x^ ± . x'^ -{- px -\- q = 0. By considering only the superior sign, we should only obtain permanences, whereas the inferior sign gives two variations. This proves that the equation has some imaginary roots ; for, if they were all three real, it would be necessary by virtue of the supe- rior sign, that they should be all negative, and, by virtue of the inferior sign, that two of them should be positive and one nega- tive, which are contradictor^/ results. We can conclude nothing from an equation of the form x^ — pa; + ? = ; for, introducing the term dz . x^, it becomes x^ ± .x^ — px -\- q = 0, which contains one permanence and two variations, whether we take the superior or inferior sign. Therefore, this equation may have its three roots real, viz., two positive and one negative ; ori two of its roots may be imaginary and one negative, since its last term is positive (Art. 326). 0/ the commensurable Roots of Numerical Equations. 330. Every equation in which the co-efficients are whole num- bers, that of the first term being unity, will have whole numbers only for its commensurable roots. For, let there be the equation ^m J^ p^m~l ^ Q^m-2 ^ _ _ ^ Tx + U =: ; in which P, Q . . . T, U, are whole numbers, and suppose that it were possible for one root to be a commensurable fraction — Substituting this fraction for x, the equation becomes fjm fjm — 1 «m— 2 fj ^,m ^ ^ Im-X ^ "< Jm-2 -f • • • "T -l ^ T 350 ELEMENTS OF ALGEBRA. [CHA^. XI. whence, multiplying both members by b'"~'^, and transposing, — = — Pa'^-^ — Qa'^-^ — ... - Tab""-^ — Ub"'-K b But the second member of this equation is composed of a series of entire numbers, while the first is essentially fractional, for a and b being prime with each other, c"* and b will also be prime with each other (Art. 118), and hence this equality cannot exist; for, an irreducible fraction cannot be equal to a whole number. Therefore, it is impossible for any commensurable fraction to satisfy the equation. Now, it has been shown (Art. 294), that an equation containing rational, but fractional co-efficients, can be transformed into another in which the co-efficients are whole num- bers, that of the first term being unity. Hence the research of the commensurable roots, either entire or fractional, can always be re- duced to that of the entire roots. 331. This being the case, take the general equation and let a denote any entire number, positive or negative, which will verify it. Since a is a root, wc shall have the equation am + p„'"-i 4- . . . + Ra^ Jr Sa^ + Ta+ U =0 . . .{\). Replace a by all the entire positive and negative numbers be- tween 1 and the limit + L, and between — 1 and — U^ : those which verify the above equality will be roots of the equation. But these trials being long and troublesome, we will deduce from equa- tion (1), other conditions equivalent to this, and more easily verified. Transposing all the terms except the last, and dividing by a, equation (1) becomes — = - a-"-! - Pa'"-2 — . . , — Ra"- - Sa—T . . . (2). a ^ ' Now, the second member of this equation is an entire number ; hence — must be an entire number ; therefore, the entire roots of a the equation are comprised among the divisors of the last term. Transposing — T" in equation (2), dividing by a, ,and making \- T = T'', we have, a T — = - a^-2 - Pa""-^ . . . — Ra- S . . . (3). CHAP. XI.] COMMENSURABLE ROOTS OF EQUATIONS. 351 T The second member of this equation being entire, — , that is. a the quotient of the division of - + r by a, a is an entire number. Transposing the term — S and div iding by a, Ave have, by supposing T' a a = — a"'~ •3 _ Par-i — . . . -R .(4). The second member of this equation being entire, a , that is, the quotient of the division of T \- S by a a IS an entire number. By continuing to transpose the terms of the second member into the first, we shall, after m — 1 transformations, obtain an equa- tion of the form 9L^-a-P. a Then, transposing the term — P. di^dding by a, and making Q' P' P' ~ + P=P', we have — =— 1, or ^1=0 a a a This equation, Avhich is only a transformation of equation (1), is the last condition which it is requisite for the entire number a to satisfy, in order that it may be known to be a root of the equation. 332. From the preceding conditions we conclude that, when an entire number a, positive or negative, is a root of the given equa- tion, the quotient of the last term, divided hy a, is an entire number. Adding to this quotient the co-efficient of x\ the quotient of this sum, divided by a, must also be entire. Adding the co-efficient of a?^ to this last quotient, and again divi- ding by a, the new quotient must also be entire; and so on. 352 ELEMENTS OF ALGEBRA. [CHAP. XI Finally, adding the co-efRcient of the second term, that is, of x'^~^, to the preceding quotient, the quotient of this sum divided by a, must be equal to — \ ; hence, the result of the addition of unity which is the co-eficient of x^, to the preceding quotient, must be equal to 0. Every number which will satisfy these conditions will be a root. and those which do not satisfy them should be rejected. All the entire roofs may be determined at the same time, as follows : After having determined all the divisors of the last term, write those which are comprehended between the limits + L and — L^'' upon the same horizontal line; then underneath these divisors write the quotients of (he last term by each of them. Add the co-eff.cient of x^ to each of these quotients, and write the sums underneath the quotients which correspond to them. Then divide these sums by each of the divisors, and write the quotients underneath the corresponding sums, taking care to reject the fractional quotients and the divisors which produce them; and so on. When there are terms wanting in the proposed equation, their co-efficients, which are to be regarded as equal to 0, must be taken into consideration. EXAMPLES. 1. What are the entire roots of the equation oci — x'^ — 13a;2 + 16a: — 48 = ? The superior limit of the positive roots of this equation (Art 317), is 13+ 1 = 14. The co-efficient 48 need not be considered, since the last two terms can be put under the form 16 (a; — 3); hence, when « > 3, this part is essentially positive. The superior limit of the negative roots (Art. 319), is — (1 + V48), or - 8. Therefore, the divisors of the last term which may give roots, are 1, 2, 3, 4, 6, 8, 12; moreover, neither + 1, nor — 1, will satisfy the equation, because the co-efficient — 48 is itself greater than the sum of all the others : we should therefore try only the positive divisors from 2 to 12, and the negative divisors from — 2 to — 6 inclusively. CHAP. XI.] COMMENSURABLE ROOTS OF EQUATIONS. 353 By observing the rule given above, Ave have 12, 8, 6, 4, 3, 2, — 2, - 3, - 4, - 6 - 4, — 6, -8 -12, — 16, —24, + 24, + 16, + 12, + 8 + 12, + 10, + 8. + 4, 0,-8 + 40, + 32, + 28, + 24 + 1, •■ + 1, 0,-4 — 20, - 7, — 4 -12, .. -12, — 13, — 17 -33, -20, — 17 - 1, - 3, + 5, 3, , - 4, , - 1, .., + 4, - 1, •• The frst line contains the divisors, the second contains the quotients arising from the division of the last term — 48, by each of the divisors. The third line contains these quotients augmented by the co-elTicient +16, and the fourth the quotients of these sums by each of the divisors ; this second condition excludes the divisors +8, +6, and — 3. The fifth is the preceding line of quotients, augmented by the co-efficient — 13, and the sixth is the quotients of these sums by each of the divisors ; this third condition . excludes the divisors 3, 2, — 2, and — 6. Finally, the seventh is the third line of quotients, augmented by the co-efficient — 1, and the eighth is the quotients of these suras by each of the divisors. The divisors + 4 and — 4 are the only ones which give — 1 ; hence, + 4 and — 4 are the only entire roots of the equation. In fact, if we divide a;* — a:3 — 13a'2 + \Q,x — 48, by the product (a? — A) (x -\- 4), or x"^ — 16, the quotient will be at^ — a; + 3, which placed equal to zero, gives ^ = -1^1/317; 2 2^ ' therefore, the four roots are 4 — 4 1 1 /- - 4 V- 2 2^ 11 and 2 2 ^ 2. What are the entire roots of the equation a;4 _ 5a;3 4- 25x — 21 =0? 23 354 ELEMENTS OF ALGEBRA. [CHAP. XT. 3. What are the entire roots of the equation 15a:5 _ 19^4 _|_ 6a;3 _i_ 15^,2 ^jg^ + 6 = 0? 4. What are the entire roots of the equation 9x^ + 30a;5 + 22x4 _f_ lOx^ + 17x2 — 20a; + 4 = 0. Sturms' Theorem. 333. The object of this theorem is to explain a method of de- termining the number and places of the real roots of equations in- volving but one unknown quantity. Let X=0....(1), represent an equation containing the single unknown quantity x\ X being a polynomial of the m"' degree with respect to x, the co-efficients of which are all real. If this equation should have equal roots, they may be found and divided out as in Art. 302, and the following reasoning be applied to the equation which would result. We will therefore suppose X = to have no equal roots. 334. Let us denote the first-derived polynomial of X by X,, and then apply to X and X^ a process similar to that for finding their greatest common divisor, differing only in this respect, that instead of using the successive remainders as at first obtained, we change their signs, and take care also, in preparing for the division, neither to introduce nor reject any factor except a positive one. If we denote the seA'-eral remainders, in order, after their signs have been changed, by X^, X^ . . . X^, which are read X second, X third, &c., and denote the corresponding quotients by Qj, Qj . . Qr-i> we may then form the equations Z=X,Q,-X, ....(2), X^ = -A2Q2 — x^ Xn-\ — X^Q^ A, n+1 Ar_<] (3). iSmce by hypothesis, JT = has no equal roots, no common di- visor can exist between X and X^ (Art. 300). The last remainder — Xj., will therefore he different from zero, and independent of a;. CHAP, XI.] STURMS' THEOREM. 355 335. Now, let us suppose that a number p has been substituted for X in each of the expressions X, J^j, X^ . . . X^_i ; and that the signs of the results, together with the sign of X^, are arranged ill a line one after the other : also that another number q, greater than p, has been substituted for x, and the signs of the resuUa arranged in like manner. Then will the number of variations in the signs of the first ar- rangement, diminished by the number of variations in those of the second, denote the exact number of real roots comprised between p and q. 336. The demonstration of this truth mainly depends upon the four following properties of the expressions X, X^ . . . X„, &c. I. Let a be a root of the equation X = 0. If we substitute a -{- u for X, and designate by A what X becomes, and denote the derived polynomials by A\ A^^, A''\ &c.; we shall have (Art. 299), A + A'U + — - 7^2 + «"•. But since by hypothesis, a is a root of the equation X = 0, we have A = 0, and hence the above expression becomes A'^ A"' « (A^ + — « 4- 2-3- "'••••+ «"'-0 ; in wliich A' is not zero, since the equation j^ = is supposed not to contain equal roots. Now we say, that u can be jnade so small, that the sign of the quantity within the parenthesis shall be the same as that of its first term. We attain this object, by finding for u a value which shall ren- der, numerically. A' > 2 " + 2.3 ti2 + &C / ■ >> V /A'' 4- ^/// 7/. 4- &c that is, a condition which will always be fulfilled (Art. 310), when A' ^ . M = or < — , K being the greatest co-efFicient of u. K+A! ° II. If any number be substituted for x in these expressions, it is impossible that any two consecutive ones can become zero at the same time. 356 ELEMENTS OF ALGEBRA. [CHAP. Xi For, let X„_„ X^, -^n+u be any three consecutive expressions, Then among equations (3), we shall lind Xn-i — X^Qn — -^n+l • • • • (4), from which it appears that, if X„_i and X^ should both become for a value of x, X,^J^^ would be for the same value ; and since the equation which follows (4) must be we shall have X„^2 = for the same value, and so on until we should find X^ = 0, which cannot be ; hence, X,j_i and X„ can- not both become for the same value of x. III. By an examination of equation (4), Ave see that if X„ be- comes for a value of x, A''„_i and A'',,^, must have contrary signs ; that is, if any one of the expressions is reduced to by the substitution of a value for x, the preceding and following ones will have contrary signs for the same value. IV. Let us substitute a -\- u for x in the expressions X and X^, and designate by U and ZJ, what they respectively become under this supposition. Then (Art. 297), we have U = A -\- A'u + A" — ^ &c. \. ; - ■ . . (5), U, = A,-{-A\u-\-A'\^+ &c. in which A, A' , A", &,c., are the results obtained by the sunsti tution of a for x, in X and its derived polynomials ; and Ai A\ &c., are similar results derived from X^. If now, a be a root of the proposed equation X = 0, then A = 0, and since A' and A^ are each derived from A""!, by the substitution of a for a-, we have A' = A^, and equations (5) become U =A'u + A-^+Slc.} ^„. 2 S (6). Uy= A'+ A'^w + &c. ) Now, the arbitrary quantity u may be taken so small that when added to a, it will but insensibly increase it, and when subtracted from a, it will but insensibly diminish it; in which cases, the signs of the values of \J and V^ will depend upon the signs of their first terms ; that is, they will be alike w^hen u is positive or when c + " is substituted for x, and unlike when u is negative or whsn CHAP. XI.] STURMS' THEOREM. 357 a — u is substituted for x. Hence, if a number insensihly less than one of the real roots of X =: be substituted for x in X and X„ the results will have contrary signs, and if a number insensibly greater than this root be substituted, the results will have the same sign. 337. Now, let any number as k, algebraically less, that is, nearer equal to — oo, than any of the real roots of the several equations X=0, X, = . . . X^_, = 0, be substituted for x ia them, and the signs of the several results arranged in order ; then, let x be increased by insensible degrees, until it becomes equal to h the least of all the roots of the equa- tions. As there is no root of either of the equations between k and h, none of the signs can change while x is less than h (Art. 311), and the number of variations and permanences in the several sets of results, will remain the same as in those obtained by the first substitution. When X becomes equal to h, one or viore of the expressions X, X,, &c., will reduce to 0. Suppose X^ becomes 0. Then, as by the second and third properties above explained, neither -X„_, nor Xnj^y can become at the same time, but must have contrary signs, it follows that in passing from one to the other (omitting Xn = 0), there will be 07ie and only one variation • and since their signs have not changed, one must be the same as, and the other contrary to, that of X„, both before and after it becomes ; hence, in passing over the three, either just before X^^ becomes or just after, there is one and only one variation. Therefore, the reduc- tion of X„ to neither increases nor diminishes the number of variations ; and this will evidently be the case, although several of the expressions X^, X^, &c., should become at the same time. If X =^ h should reduce X to 0, then h is the least real root of the proposed equation, which root we denote by a ; and since by the fourth property, just before x becomes equal to a, the signs of X and Xi are contrary, giving a variation, and just after pass- ing it (before x becomes equal to a root of X, = 0), the signs are the same, giving a permanence instead, it follows that in passing this root a variation is lost. In the same way, increasing x by insensible degrees from x ^ a -\- u until we reach the root of AT = next in order, it is plain that no variation will be lost or gained in passing any of the roots of the other equations, but that 358 ELEMENTS OF ALGEBRA. [CHAP. XI in passing this root, for the same reason as before, another varia- tion will be lost, and so on for each real root between k and the number last substituted, as g, a variation will be lost until x has been increased beyond the greatest real root, when no more can he hst or gained. Hence, the excess of the number of variations ob- tained by the substitution of k over those obtained by the substi- tution of g, will be equal to the number of real roots comprised between k and g. It is evident that the same course of reasoning will apply when we commence with any number p, whether less than all the roots or not, and gradually increase x until it equals any other number q. The fact enunciated in Art. 335 is therefore established. 338. In seeking the number of roots comprised between p and q, should either p or q reduce any of the expressions Xi, X^, &c., to 0, the result will not be affected by their omission, since the number of variations will be the same. Should p reduce X to 0, then p is a root, but not one of those sought ; and as the substitution oi p -\- u will give X and X, tlio same sign, the number of variations to be counted will not be affected by the omission of X = 0. Should q reduce X to 0, then q is also a root ; and as the substitution of 5' — u will give X and X, contrary signs, one varia- tion must be counted in passing from X to X^. 339. If in the application of the preceding principles, we ob- serve that any one of the expressions X,, X2 . . . &c., X,^ for in- stance, will preserve the same sign for all values of x in passing from p to q, inclusive, it will be unnecessary to use the succeed- ing expressions, or even to deduce them. For, as X^ preserves the same sign during the successive substitutions, it is plain that the same number of variations will be lost among the expressions X, Xi, &c. . . . ending with X„ as among all including X^. When- ever then, in the course of the division, it is found that by placing any of the remainders equal to 0, an equation is obtained with imaginary roots only (Art. 325), it will be useless to obtain any of the succeeding remainders. This principle will be found very useful in the solution of numerical examples. 340. As all the real roots of the proposed equation are neces- sarily included between — oo and -f oo, we may, by ascertaining CHAP. XI.] STURMS' THEOREM. 359 the number of variations lost by the substitution of these, in suc- cession, in the expressions X, X^ . . . X„, . . &c., readily determine the total number of such roots. It should be observed, that it will be only necessary to make these substitutions in the first terms of each of the expressions, as in this case the sign of the term will determine that of the entire expression (Art. 315). 341. Having thus obtained the total number of real roots, we may ascertain their places by substituting for x, in succession, the values 0, 1, 2, 3, &c., until we find an entire number which gives the same number of variations as +03. This will be the smallest superior limit of the positive roots in entire numbers. Then substitute 0, — 1, — 2, &c., until a negative number is obtained which gives the same number of variations as — oo This will be, numerically, the smallest superior limit of the negative roots in entire numbers. Now, by commencing with this limit and observing the number of variations lost in passing from each num- ber to the next in order, we shall discover how many roots are included between each two of the consecutive numbers used, and thus, of course, know the entire part of each root. The decimal part may then be sought by some of the known methods of ap- proximation. EXAMPLES. 1. Let 8x3 — 6a; — 1 = = Z. The first-derived polynomial (Art. 297), is 24*2 _ Q^ and since we may omit the positive factor 6, without affecting the sign, we may write 4a:2 - 1 =: X,. Dividing X by X,, we obtain for the first remainder, — 4x — 1. Changing its sign, we have 4* + 1 = X^. Multiplying X^ by the positive number 4, and then dividing by X2, we obtain the second remainder — 3 ; and by changing its sign + 3 = Z3. The expressions to be used are then X=8x^ ~6x- 1, Z, = 4a:2 - 1, X^ = 4x +1, X3 = + 3 360 ELEMENTS OF ALGEBRA. TOHAP. XI. Sul)stituting — 00 and then + oo, we obtain the two following arrangements of signs : — 4" — + 3 variations, + + + +•• .0 There are then three real roots. If now, in the same expressions we substitute and + 1, and then and — 1, for x, we shall obtain the three following arrangements : For X = -{- 1 + + + + variations, x= \- + 1 x= -I - -\- - i- 3 As a; = + 1 gives the same number of variations as + °o, and X = —1 gives the same as — oo, +1 and — 1 are the smallest limits in entire numbers. In passing from — 1 to 0, two variations are lost, and in passing from to + ], one variation is lost; hence, there are two negative roots between — 1 and 0, and one positive root between and + 1. 2. Let 2x* — 13x2 _|_ lOa: -19 = 0. If we deduce X, X,, and Xg, Ave have the three expressions X = 2x* — 13x2 + lOx - 19, X, = 4x3 _ 13^ + 5^ Xg = 13x2 _ 15-^ ^ 38. If we place X^ = 0, we shall find that both of the roots of the resulting equation are imaginary ; hence, X^ will be positive for all values of x (Art. 325). It is then useless to seek for X, and X4. By the substitution of — 00 and + co in X, X,, and Xj, we obtain for the first, two variations, and for the second none ; hence, there are two real and two imaginary roots in the proposed equation. 3. Let x3 — 5x2 + 8x — 1 = 0. 4. X* — x3 — 3x2 + x2 — a: - 3 = 0. 5. x5 _ 2a;3 + 1 = 0. Discuss each of the above equations. CHAP. XI.] RESOLUTION OF CUBIC EQUATIONS. • 361 Young's Method of resolcing Cubic Equations. 342. Every numerical equation of the third degree may be re- duced to the form x^ -{- Ix^ + ex - N (1), in which b, c, and N, are known numbers. Since this equation will have at least one real root (Art. 277), let us find, either by Sturms' theorem or by trial, two consecutive numbers, either integral or decimal, which being substituted for ar, will give results with different signs. We then know that one of the values of x will lie between them (Art. 311), and consequently, that the smallest number will be the first figure of the required root. Let us designate this figure by r. Now, if we neglect the re- maining figures of the root, and regard r as the approximate value of X, we shall have N r^ -\- hr^ -{- cr ^:z N ; whence, r = . r^ -\- or -\- c Having found r, denote the remaining part of the root by y ; then X ^= r -\- y. Substituting this value of x in the given equation, we have ^3 __ y.3 _[_ 3j-2y _|_ 3r?/2 _[_ y3 \ bx"^ = br^ -f 2bry + hf > = iV ex z= cr -\- cy ) and by adding and arranging with reference to y, y3 + (3r + b)y'^ + (3r2 -\. 2br + c) y + (r^ + Z-r2 + cr) = N. But we may simplify the form of this equation, by making Z/' = 3r + b, c' = 3r2 + 2Z/r + c, N' = N — {r^ + b?-^- + cr) ; which will give if + b^^ + c^y = N' (2). The form of this equation is entirely similar to that of the given equation ; and if we denote by s the first figure of y and make tne same supposition as before, w^e shall have wlience, s = . *2 ^ b's -i- ) + 3r2 + 2ir + c ' fl + b"t + c'' "" f [« + 3 (r + 5) + i] + 3^2 + 2b's + c" ' y2 _^ 5///y _|_ g/// „ [y 4. 3 (r _f. 5 ^ f ) _}. Jj _[_ 3^2 4. 2b" t 4- C' &c., &c., &c CHAP. XI.] RESOLUTION OF CUBIC EQUATIONS. 36.S The value of r being found, and c a known number, tlie de- nominator 7-(r4-5)H-c will be known. This forms the first di- visor, and dividing N by it, the first figure of the quotient will be r, as before found. Multiplying the divisor by r and subtracting the product from N, we obtain N' , the second dividend. It will be seen that the three right-hand terms in each denomin- ator, are formed from the preceding figure of the root. These make trial divisors for each figure of the root after the first. Having found N\ we form its trial divisor and then see how often it is contained in N^, which gives s. We then form the complete divisor which we multiply by s, and subtract the result from N'', which gives N^''. We then form its trial divisor, find the figure t, after which we find the complete divisor for t, and then multiply it by the quotient figure t and subtract the result from iNP'', giving N'"'''; and similarly for all the following figures of the root. 344. By examining the table of Example I, on the next page, and comparing it with the formulas, we see, that if under any divi- sor we write the square of the figure of the root which the divisor determines, and then add it to the two numbers directly above, their sum will be the trial divisor for the next figure of the root. Hence, we have the following RULE. I. Write doicn c, the co-effcient of x, and on the same line, to the right, place the known number N, and set in the quotient the Jirst figure of the root found by trial. n. To this figure of the root add b, the co-ejfficient of x-, and then multiply the sum hy the figure of the root, and add the product to c, and the sum will be the first divisor, which is then to be multiplied by tlie quotient figure and the product subtracted from N. HI. Under the first divisor write the square of the first figure of the root, and then add it to the last two sums, and the result will be tJie trial divisor for the next figure of the root. IV. Having found the next figure of the root, add to it three times the figures of the root already found, and also the co-efficient t : then multiply the sum by the figure of the root and add the prod- uct to the trial divisor, and the sum will be the entire divisor, which must then be multiplied by the figure of the root, and the product subtracted from the last dividend. The process for determining other figures of the root is entirely similar. 364 ELEMENTS OF ALGEBRA. [CHAP. XI. + 05 + IN + lO l>1 r3 3 w CO G II a II ■^ ]l ^ u 'S ^ ^ CO + ■5^ + II is II •v. II ^ II + + 5: + > fe: %> fe; •*o fe; ^ <: II II II II II II in to II CO 00 CJ r- ^ 00 rsf CX) CO CO XI ^ ^ CO C5 o rH Tf o C5 CO o ^H 'Tf 05 CO lO ct CO CO o CO C9 T-H T— 1 o o o w CN( =3 a .9 & • jV CO = a H + ^ + oo = q CO O 1 CO in in fN> r* -.-« Tf Tt< (>J CO in '^ Tf o CT5 CO in CO CO CD Of) o 00 o 00 f^ en t> J> CO CO o o w o CO o in ^ ^ o w o rs> in CO CO CO CO 00 + u (N + (N ^ II + lO V + !;. V II V. >-l II . + + + + 1 ^ c^ ^ " + + + + > + ^ + + ^ ^ + CO + + + + + CO + + to + CO + .2 n3 TS r^ K CO ^ CHAP. XI.] RESOLUTION OF CUBIC EQUATIONS. 36') Remark. — The operations in tlie example of the table, are all performed according to the directions of the rule ; but more deci- mal places have been used in the dividends and divisors, in the latter part of the work, than were necessary. Had we admitted but three places of decimals in the dividends, and rejected all other places to the right, as fast as they occurred, we should still have 'lad the root equally true to at least four places of decimals. But since the figures of the root are decimals, it follows that if the num- ber of decimal places in the dividend does not exceed three, the decimal places in the corresponding divisor should not exceed two ; and for every succeeding figure found in the root, one place may be struck ofl" from the right of the divisor. After finding a certain number of figures of the root, it will occur that the numbers to be added to the divisors will fall among the rejected figures, after which the remaining figures of the root will be found by simple division. It should be observed, however, that when places are rejected from the divisor, that whatever would have been carried had the complete multii)lication been performed, is still to be carried to increase the last figure retained ; and when- ever the left-hand figure of those rejected, either in the dividend or divisor, exceeds 4, the last figure retained is to be increased by 1. The following is the last example, wrought on the principle of admitting but three places of decimals into the dividends. The rejected figures, both in the dividends and divisors, are placed a little to the right. 6 20 26 4 60 5.76 55 .76 .16 61 .68 30 44 62 844 004 62 3 1 892 76325 62 .4 75 9 12. 4257 4- 52 23 9 22 .304 1 .596 1 .240 088 .356 312 .312 827625 .044 484375 043 001 366 ELEMENTS OF ALGEBRA. [CHAP. SI. 2. Find one root of the equation x^ + x- = 500. This equation is the same as x^ + x- + Oa? = 500 ; hence 6 = 1 c = 0, and N = 500. The first figure of the root found by trial, is 7. 56 56 49 Tel 13 .56 174 .56 36 188 .48 .2381 188 .7181 .0001 188 .9563 . 1669 189, . 123|2 189, .290 .005 500 [7.61727975, &C., = X 392 108 104 .736 3Ught the lence ends, imals 3 .264 J-, C C 't3 CJ 1 .887181 <« 2 S "^ 2 ] .376819 ■^ O q;} O 1 . 323862 C^ -.^ a (J . 052957 . 037859 exam cimal ay ah s fron 3 pla . OlSODa .013251 ^ cd s C-, >o .<^ 'E, S .001847 bD '^ 03 Kr .001704 ■a r^ ^ - .000143 ^•1^ ^'^ .000133 Rem y ret ividei ejectii fler h .000010 . 000009 _o --d >-, c; " o . 1|8|9.12|9|5 3. F'ind one root of the equation x^ — 17a:- + 54a: = 350. Ans. a- = 14 . 954, &c. 4. Find one root of the equation x^ + 2x'^ + 3x =: 13089030. Ans. X = 235. 5. Find one root of the equation x^ 4- Sx^ — 23x = 70. Ans. X = 5 . 1345, &c. Remark. — In the preceding solutions only one root has been obtained, yet the others may be found with equal facility, by find- ing by trial the first figure in each and then proceeding by the rule already given. There is, however, a shorter inelhod for de- termining the remaining roots. Subtract the root found, taken with a contrary sign, from the co- efficient of the second term of the given equation, and denote the remainder by a. Divide the absolute term by the root found, and denote the quotient by b ; then will the roots of the equation a;2 -f- CT -f & = be the two remaining roots of the giA^en equatiop CHAP. XI.] RESOLUTIOX OF TIIGHE.'J KQUATION3. 367 Method of resolving Higher Equations. 345. The general method of resolving cubic equations, has been explained in Art. 342. We shall now add from Young's Algebra, the method of resolving equations of a higher degree. It has not been thought best to give the general investigation, but merely to add, for the solution of an equation of any degree, the follow- ing general RULE. I. Transpose the absolute term to the second memher of the equation. Then, beginning with the co-efficient of the first term, arrange the co- efficients of the first member in a row, placing the absolute term to the right. II. Having found the first figure of the root, multiply it by the first co-efficient and add the product to the second co-efficient ; then multiply the sum by the same figure of the root and add the product to the third cn-effiicient ; and so on to the last co-efficient : the last sum will be the first divisor, which multiply by the figure of the root and subtract the product from the absolute term : the result will be the second dividend. III. Perform the same operations on the first co-efficient and the set of swns found, as was performed unth the co-efficients, and the last sum xvill be the trial divisor for the second figure of the root. Then perform the same ojjerations on the first co-efficient and the second set of sums, only stop in the column of the last co-efficient but one. Repeat the same op- eration on the first co-efficient and the last set of sums, but stop in the next left-hand column, and so on until you stop in the column of the sec- ond co-efficient. IV. Then find from, the trial divisor the second figure of the root, taking care that it be not too large. Take this second figure, and per- form unth it on the first co-efficient and the last set of sinus the same operations as were performed on the co-efficients with the first figure of Oie root; and the sum in the last column will be the second divisor, which multiply by the second figure of the root and subtract the product from the second dividend. V. The next trial divisor, the next figure of the root, and the true divisor, arc found by the principles already explained, and the placet^ of figures in the root may be carried as far as necessary. 368 ELEMENTS OF ALGEBRA. [CHAP. XI EXAMPLES. 1. Find the root of the equation x* — 3x^ + 75a? = 10000. Operatio7i. -3 75 10000 [9.8860027, &c., , = y 9 81 702 6993 9 78 777 3007 9 162 240 2160 2677. ,5616 18 2937 329 .4384 9 243 483 409. 952 306 . 1662 27 3346. 952 23 .2722 9 ,8 29. 512. 44 44 434, .016 23 .2616 36. 3780. 968 106 ,8 30, ,08 46 . 110 78 28 37. ,6 542 . 52 3827. , 07|8 . 8 30, ,72 46 .36 27 T 38. .4 573 . ,24 3873, .44 .8 3 . , 14 3 .50 3|9, .|2 576 .3|8 3876 . 9|4 3 579 . 1 • 1^ 3 .5 3880 .4 3 5|8|3 Remark. — The work, in the example, has been contracted by omitting or cutting off decimal places, as in the operations for tho cube root, and in equations higher than the third degree, the con- tractions may be begun after the first decimal place of the root is found. By using one period of four decimal places, the root has been found to eight places of figures. With another period of Your places, that is, by beginning the contractions later, we should have found four additional places, or ic = 9 . 88600270094. THE END. QA. Davies - Elements of algebra: including iSh Sturms* theorem. D28E CHEUJf ENGINEERING AND MATHEMATICAL >o ( A