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A BRIEK COURSE IN 
 
 ADVANCED ALGEBRA 
 
 Being Course Two in Mathematics in the 
 University of Wisconsin: 
 
 CONTAINING 
 
 I. Imaginaries. 
 
 II. Discussion of the Rational Integral 
 Function of x. 
 
 III. Solution of Numerical Equations of 
 
 Higher Degree. 
 
 IV. Graphic Representation of Equations. 
 V. Determinants. 
 
 C. A. VAN VELZER and CHAS. S. SLIGHTER. 
 
 EV.-VNSTON, ILL.: 
 
 UNIVERSITY PRESS. 
 
-.i/X^ 
 
 ADVANCED ALGEBRA. 
 
 CHAPTER I. 
 
 IMAGINARIES. 
 
 I. Let us notice what is meant by Algebraic Quantity, and 
 how the notion of it may originate. The primitive conception of 
 number is used when we enumerate the marbles in a box, and say: 
 o, I, 2, 3, 4, etc. Our simple scale \s, arithmetical quantity , and 
 it runs down to a definite nothing, or zero, and stops. But let us 
 attempt to apply this scale in the measurement of other things. 
 Suppose we are estimating time. Where is the zero from which 
 all time is to be measured in one direction, or sense? There is no 
 such zero, as in the case of the marbles; for you can conceive of 
 no event so far past but what other events preceded it. We are 
 forced to select a standard event, and measure the time of other 
 events with reference to the lapse bifore or afte7' that. The zero 
 used is an arbitrary' one, and there is quantity in reference to it in 
 two opposite senses, future and past, or, as is said in algebra, 
 positive and negative. We are likewise obliged to recognize 
 
 quantity as extending both ways from a zero in the attempt to 
 measure many other things; in locating points along an east and 
 west line, no point is so far west but what other points are west 
 of it, hence could not be looted in the arithmetical scale; the 
 same in measuring force, which may be attractive or repulsive; 
 or motion, which maj- be toward ox from; etc. Thus our notion of 
 algebraic quantity as we name this kind of number, has arisen. 
 
 Because of the peculiar analogy between our notion of time 
 •and of algebraic qunntity, algebra has been called the science of 
 
2 ADVANCED ALGEBRA. 
 
 pure time. All quantities are measured exactly as a past and 
 future, or, graphically, along a line, in both directions from a 
 zero point. But, with algebraic numbers, we never get out of the 
 line. This kind of quantity, although more general than arith- 
 metical number, is really quite restricted. We observe, at once, 
 that there is an opportunity of enlarging our conception of quan- 
 tity if we can only get out of our line, or one way spj'ead as .some 
 say, and explore the region without. We may expect, then, an 
 extension of our notion of quantity which will enable us to con- 
 sider, along with the points of our Hue, those which lie without. 
 
 2. It is interesting to note that any number may be regarded 
 as the symbol of an operation; and that thereby some original con- 
 ceptions may be conveniently extended so as to contain a meaning 
 of greater range. Thus lo may be regarded, not only as feji, but 
 as denoting the operation of taking unity ten times ; to express 
 this, we may write lo i. In the same way, 5 denotes the opera- 
 tion of taking unity five times. Now we notice that an exponent 
 will have an extended significance. It means to repeat the opera- 
 tion designated; i- e., the operation designated is to be performed, 
 and performed again on the result and performed again on this 
 result ; and so on, equal to the number of times told by the expo- 
 nent. Thus 10'' means to perform the operation of repeating 
 unity ten times (told in 10) and then to perform the operation of 
 repeating this result ten times, ?. e., loUo-i). Also lo'^ means 
 io[io(ioi)]. Then, of course, the exponent zero can only mean 
 that the operation on unity denoted by the quantity is not to be 
 performed at all; i. e., unity is to be left unchanged; thus 10" or 
 io"-i = i. An expression hke v/ 4 , looked upon as a symbol of an 
 operation denotes an operation such that if the operation were to be 
 performed twice the result would be the same as is indicated by the 
 expression under the radical itself; /. e.,^'~^ stands for the opera- 
 tion which must be repeated to be equivalent to quadrupling, 
 which is indicated by the expre.ssion uader the radical (4); hence 
 y/ 4 represents the operation oi doubling. The expre.ssion — 1 as 
 
IMAGINARIES. 3 
 
 a symbol of operation is a reverser, denoting the operation of 
 changing from one sense to the opposite sense. 
 
 3. We remember that such expressions as 3 + \/^I^, <:+\/^, 
 \/^, etc., were forced upon our notice in the solution of quad- 
 ratic equations. It is customary to call such Imaginaries, be- 
 cause of the presence in them of a term like\/^ which, evidently, 
 does not correspond to any algebraic quantity whatever. But, re- 
 membering the restricted nature of algebraic quantity, it is possible 
 that such expressions are unreal only in an algebraic sense; that 
 if the restriction can be removed by an extension of our concep- 
 tion beyond a mere linear or past-and-future notion of quantity, 
 the expression may, perhaps, become as real to us as algebraic 
 numbers now are. 
 
 While v/^ is not an algebraic number, yet we may give to 
 it any definition as a symbol of operation which is consistent with 
 this one condition: (v/^)'= — i; that is, two of the opera^ons 
 must be a reversal.. So, as — i may be defined as such a symbol 
 which operates to turn a straight b 
 
 line through iSo°, in a similar way ^ 
 
 we make this 1 7 
 
 4. Definition. Thisexpres- "^ 
 
 sion \/^ is defined as that symbol (- (^zz'ya O 
 which denotes the operation of turn- ^^^ 
 
 ing a straight line through an angle 
 
 A 
 
 iT 
 
 of 90° in the positive direction. It | 7 
 
 is customary in mathematics to con- "^ 
 
 si Jer rotation opposite to that of the 
 hands of a watch as positive rotation. 
 
 In the figure, let a beany line. Then a operated on by^^, 
 /. <?., -v/^-a is a turned ?<!/>, or positively, through 90°. Now, of 
 course, >/— i can operate on y/~i a just as well as on a. Then 
 ■v/— i(-v/— i^) is v/— I turned positively through 90°; 
 v^— |[n/— i(v/— I «";] is ^~i{y/^.a) turned through 90°; etc. 
 
 As we are at liberty to consider two turns of 90° as the same 
 as one turn of 180°, .•. v/_,(\/^^) = (— 1)«. Also nn= 
 
4 ADVANCED ALGEBRA. 
 
 (—i)OB,.-. On=— iV—ra), hut s/^i{- a )= on, .-.— (^"i-a) 
 
 ^■<y—i( — a). Thus the student may show many like relations. 
 
 5. In order that imaginaries may be conveniently used in 
 algebraic work the following useful conventions governing the or- 
 dinary algebraic operations with them are adopted: 
 
 a. The Associative Lazv, by which the result of a repeated 
 operation is to be regarded the same irrespective of the mode of 
 grouping the elements, so long as their order is not disarranged. 
 That is, {c+s/'^vd)-\-{e-\-^^/) means the same as 
 
 c+iV^^id+o + V^if. Also. {^—x-a){b) = {s/'^i){ab). 
 
 b. That s/^^i a is to mean the same as v/^^ and vice versa. 
 
 c. The Commutative Law, by which the result of several 
 operations is to be regarded the same irrespective of the order in 
 which they are performed. That is, ^v/ZTf means the same as 
 y/^{ a ; s/^^v/I^ means v/^\/— i « ; c-\-ds/^^\-\-e-\-f\/^^\^ 
 c-\-e-\-ds/^\-\-fs/^^\. While it seems more natural to write 
 ■v/^ « than as/—\, because of the definition of \/—\, yet it is 
 the universal practice to write the latter. Of course the equation 
 iOv/^ = -s/^-io, or better iov/^i = n/^ 10 i, means that the 
 resjilt 0/ performing the operation of turning unity through go° 
 and performing upon the 7'esult the operation of taking it ten 
 times is the same as the result of performing the operation of taking 
 luiity ten times and pcrformiyjg upon this result the operation of 
 turning thro2cgh go° . 
 
 d. The Distributive I^aw in the case of an operation per- 
 formed upon a7i aggregate, by which the resul*; of such operation 
 
 upon the aggregate of several terms is to be regarded the same as 
 the aggregate of the results of the operation upon each term sep- 
 arately. That is, (a-\-b)s/^ means the same as a\/^+b\/^. 
 
 We agree by the above that any operation upon imaginaries 
 or by imaginaries is to be regarded as associative, commutative 
 or distributive, if it would be so regarded when the imaginaries 
 are replaced by ordinary quantities, and not otherzuise. 
 
 6. W^e will now interpret the powers of s/^ by means of 
 the new significance of an exponent and by the above convention -^. 
 

 
 
 w 
 
 lAGINARIES. 
 
 (^/= 
 
 ir 
 
 or is/- 
 
 =7)" I 
 
 = -ri 
 
 
 (s/- 
 
 i)^ 
 
 or (^/I: 
 
 T)'-i 
 
 = s/~i 
 
 = 2 
 
 (V= 
 
 T)'-' 
 
 
 
 =— I 
 
 =={' 
 
 (^/= 
 
 7)-' 
 
 = Cv/- 
 
 i)-^ v/= 
 
 T — ^/- 
 
 -,=t' 
 
 (s/- 
 
 7)^ 
 
 = (n/= 
 
 T)'-'(n/= 
 
 T)-^ = + i 
 
 = i' 
 
 (x/- 
 
 T)-"' 
 
 = (n/= 
 
 0* V= 
 
 7 =n/=T 
 
 = l' 
 
 (v/= 
 
 I)" 
 
 = (v/= 
 
 ~lf(K/- 
 
 I)'=— I 
 
 = f' 
 
 (v/= 
 
 0' 
 
 = (n/= 
 
 T/n/- 
 
 ~ /~ 
 
 ' —i' 
 
 (v/= 
 
 'iT 
 
 = {%/= 
 
 DXn/- 
 
 i/ = + i 
 
 = f 
 
 etc. 
 
 
 etc. 
 
 
 etc. 
 
 etc. 
 
 Art. 
 
 15- 
 
 Whence it is seen that all even powers of \/^r are either+ 1 
 or — I and all odd powers either \/^i or — V^. Reconcile this 
 with the figure in Art. 4. 
 
 7. Theorem. Any polynomial of the form 
 
 can be put in the typical fonii, a + d^/^. 
 
 The symbols c,,, r,, c,, etc., stand for any coefficients, and n is 
 a whole number. 
 
 (i.) As a particular case take 
 
 c,(s/~iy+c^(s/-iy+cXs/~if-^r.js/-iy+cy-i+c,= 
 
 which, (by Art. 5, c) equals c^ — r.j+c+r,,^/-^ — c.^ZTx+c^sZ—i, 
 which (by Art. 5, d) equals {c^ — '^:i+fO + (''o — ^2+0^/— i- which is 
 of the form a-\-b^--\. 
 
 (2.) In any case the odd powers of v/^ will be either v/^ 
 or — V^; the even powers will be +1 or — i, which are real. 
 Hence the result may be expressed as an aggregate of real quan- 
 tities and an aggregate of quantities operated on by-v/^, which 
 is an expression of the form a-\-bs/^, by Art, 5, c and d. 
 
 Q. E. D. 
 
ADVANCED ALGEBRA. 
 
 8. Any imaginary, in the typical form, may be taken as the 
 representation of the position of a point in a plane. For, suppose 
 c+^v/HT is the imaginary. Let O be the zero point and OX the 
 positive direction. Lay off OA=+c, and at A erect d>/^. Then 
 c-\-dy/—\ defines the position of the point Pwith reference to O. 
 Then OA-\-AP, or OP, is a geometrical representation of 
 c-\-d'y—\. In the same manner let c—d^^, — c—dy/~\, and 
 —c-\-d>/—\ be con.structed. 
 
 9. The meaning of some of our conventions may now be il- 
 lustrated. Let us construct c-\-dy/—i-\-c-{-fs/—i- 
 
 e c 
 
 1 
 
 
 
 F 
 I H 
 
 G 
 B 
 
 T 
 
 e 
 
 P 
 
 c / 
 
 1 T 
 
 El 
 
 i 
 
 rf JD X 
 
 
 
 ^ 
 
 
 c 
 
 i 
 
IMAGIXARIES. 7 
 
 The first two terms, c-\-dy/—i, give OA + AB, locating B. 
 The next two terms, c+J\/—i, give BC+CP, locating P. Hence 
 the entire expression locates the point P with reference to O. 
 Now if the original expression be changed in any manner allowed 
 by our conventions, the result is only a different path to the same 
 point. Thus: 
 
 c+c+dV-i+fV-i is the path OA, AD, DC, CP. 
 {c+e) + [d-\-f)K/—i is the path OA, AD, DP. 
 c^d^-x+c+tV-i is the path OE, EH, HC, CP. 
 e+d^'^i+/s/~+c is the path OE, EH, HE, EP, etc. 
 The student should consider other cases. 
 
 Query. Arc there any methods of locating P with the same 
 four elements, which the figure does not illustrate? 
 
 10. Definition. Two imaginaries are said to be Conju- 
 gate when they differ only in the sign of the term containing 
 ^_j. Such are c-\-ds/—\ and c — d^—i. 
 
 11. Theorem. Conjugate imaginaries have a real sum 
 and a real product. 
 
 For, r+ ds/—\ + c —dy/^i 
 
 = c+c+d^:^i—d^-i. Art. 5, c. 
 
 = 2c+{d-d)^~i = 2c _ Art. 5, d. 
 
 And {c-\-ds/^i){c— dy/^), by distributing the second paren- 
 thesis through the first, (Art. 5, d) equals 
 
 c{c - dV~i ) + dV^i ( c - d>/—i ) 
 =c--cd^~i_ + dy/~i c-ds/^^d^^i Art. 5, d. 
 =r-\-d'+eds/^~cds/^. Art. 5, c. 
 
 =r + d'+{ed—cd)s/—i=r^d' Art. 5,^/. 
 
 Q. E. D. 
 
 12. Theorem. 7 he sum, product, or quotient of tii'o imagi- 
 naries is, in general, an imaginary of the typical form , 
 
 I,et the two imaginaries be c-\-d^y~\ and e-\-f\/—\. 
 a. Their sum is {c-\-ds/~\)-\-{e^f^/~\) 
 
 = { r+ e) + ( ds/—i +/\/~i ) Art. 5 , r and a . 
 = (^-f^)-f («'+/) v/-i Art. 5, d. 
 
 which may be written a-\-b\/^\. 
 
S ADVANCED ALGEBRA. 
 
 b. The product is {c+ds/—Y){e+fV—i) 
 
 = ce + c/^—i + ed^—i + df^—i ^~i Art. 5 , d and c. 
 
 = (ce—d/) + (c/-\-ed)^—i Art. 5, a and d. 
 
 which may be written a-^-d-,/—!. 
 
 c. Their quotient is 
 
 c->rd^-i _ (r+^v/-i)(g-A/-i) 
 ^+/"x/-i {e+jV-x){e-fV-i) 
 By the preceding, the numerator is of the form a'-\-b'\/—\. 
 By Art. 11, the denominator equals e' -{-/'. Then the quotient 
 
 equals 
 
 ^'"^^'^-^ ^' +^A Art. 5,^. 
 
 which is of the form a-\-b\/—i. Q. E. D. 
 
 13. ThEuREI.i. If an imaginary is equal to zero, the imagi- 
 nary and real part are separately equal to zero. 
 
 Suppose <a; + i^v/— 1=0 
 then a= — bs/—\. 
 Now it is absurd tor a real quantity to equal an imaginary, 
 except they each be zero. 
 
 Therefore ^-=0 and <^=o. Q. E. D. 
 
 14. Corollary. If two iviaginaries be equal, then the real 
 and the imaginary pa7^ts must be respectively equal. 
 
 If f+rt'v/— i=^+A/^i, then {c—e)^-{d-f)V-\=o and 
 c—e=o and (f— y"=o. 
 
 15. For the purpose of abbreviation, it is customarj^ to rep- 
 resent the symbol V—i by i. Then the typical imaginary is 
 a -f bi. But the student must keep clearly in mind the meaning 
 of i, and also the meanings of the powers of z. See Art. 6. 
 
CHAPTER II. 
 
 The Rational Integral Function of x. 
 
 1. Definition. A. Variable is a quantity which changes 
 vaUie. Variables are represented by the last letters of the 
 alphabet. 
 
 2. Definition. A Constant is a quantity which has a fixed 
 value. Constants are represented by the first letters of the alpha- 
 bet and by numbers. 
 
 3. Definition. A Function of a quantity is a name applied 
 
 to any mathematical expression in which the quantity appears. 
 
 I — ,r 
 
 2ax; x-y'x'; . ; s/a-.v'. 
 
 These are all functions of x. In the same manner we speak of 
 functions of several quantities. The second expression above 
 may be called a function of .r andj'. Obviously, a function of a 
 quantity might be otherwise defined as any expression which de- 
 pends upon the quantity for its value. 
 
 4. Definition. A R.ational Function of a quantity is one 
 in which the quantity is not involved in a radical, or affected with 
 an irreducible fractional exponent. 
 
 5. Definition. An Integral Function of a quantitj^ is one 
 in which the quantity does not appear in the denominator of a 
 fraction, or is not affected with a negative exponent. 
 
 6. A function may be both rational and integral 
 in which case it is called a Rational Integral Function 
 This is the function we propose to consider, but, while limit 
 ing ourselves to the consideration of this class of func 
 tions, it is possible that some light may be incidentally 
 thrown upon others. At least we notice that such a function as 
 7"-|-5v/v — 5, is, in some respects, the same as x^+5.r — 5. 
 
 7. Such expressions 3.5, fiinction of x, function of a, function 
 of x-\-h, etc., are abreviated into F{x), F{a), F(x-{-h), or <f{x), 
 f(a), (f{x-\-h) or a similar expression. It must be kept well in 
 
lo ADVANCED ALGEBRA. 
 
 mind that /% 9', etc., are not coefficients. The expression /(:r) 
 will be exclusively used for a rational integral function of .i", with 
 real coefficients. 
 
 If fix) and /(a), or F (yX) and F{a), occur together in the 
 same discussion, f{a) stands for what /(.r) becomes when a is 
 put for .r. Thus, if /(.v) is x'' — 7.r'+2x-|-4, then /(«■) is 
 d — 7a-'+2fl' + 4. In the same way, /(.i-+/>) stands for what 
 f{x) becomes, when {x-\-h) is put for x. 
 
 8. It is evident that 
 a^x"-j-a^x"~^-\-a.^x"~'^-\-a.^x''~^-\- . . . -\-a^^_.^j(r-\-a^^_^x^a^^ 
 
 stands for any rational integral function of -v, where ra is a whole 
 number, and the symbols, a^, a^, «^, a.^, etc., stand for any coeffi- 
 cients, positive or negative, integral or fractional, commensurable 
 or incommensurable. We confine ourselves, for our purposes, to 
 ?ral coefficients. 
 
 If we suppose/(x) to be divided through by the coefficient of 
 the highest power of .v, then the following will represent any /(a): 
 
 .V"+/.,.v"-'+A-i:"-^+A-V"-'+ . . . -f-A-r^"""'"'+A-rl"'^'+A- 
 
 If none of the above coefficients are zero, the function is said 
 to be complete. The term, a,,, or A^. is called the absolute term. 
 
 9. Definition. The Degree of /(.v) is the exponent of the 
 highest power of x. In general, the degree of a function is 
 spoken of in the same way as the degree of an equation, and is 
 determined in like manner. 
 
 10. Definition. An}^ real quantit}^ or any imaginary, 
 which substituted for .i" in /(.a-) makes/(.t-) vanish, we will call a 
 Root of/(.v}. 
 
 1 1 . Definition. A Rational Integral Equation containing 
 one unknown is one Vv'hich can be placed in the form /(.r)=o; 
 that is, in the form 
 
 a^"-\-a^x''^^-\-a.pc"~''--\-a.p:"'''^-\- . . . -\-a^^_^x-\-a^=o (i) 
 A rational integral equation may also be represented by 
 
 .r"+/),.v—+A.-^"~"+A^'""H • • • ^-/'„-r^'+A=o (2) 
 
 since equation (i; is unchanged if we divide through by the co- 
 efficient of .r". When the equation /(-f)=o is written out in 
 either of above forms it is commonly spoken of as the General 
 Equation of the «th degree. 
 
RATIONAL INTEGRAL FUNCTION OF .r. ii 
 
 Matty equations which are not in above form, that is. are not 
 rational and integral, may take that form through the ordinary- 
 processes of transformation, such as: 
 
 I x' - 2 
 
 A" +2 \/ 20— 3-1- 
 y/io — 3-r= X* — 4 
 20 — 3.v=.i-' — 8-V-+ 16 
 .1-^ — 8.v'4-3-v— 4=0 
 which is a rational integral equation of the fourth degree. Some 
 equations, however, cannot be reduced to a rational integral equa- 
 tion of finite degree by an}' process of transformation; such are 
 
 I I 
 
 log x-\-x'-^a; x-\-c^'=a; {x-\-a)'^-\-{x-^b)"'^=( ; etc. 
 
 12. Dep'inition. a Root of an equation is any value of the 
 unknown which satisfies the equation. 
 
 13. When the final result in any investigation in applied or 
 pure mathematics takes the form of a rational integral equation, 
 the problem of finding a value of .v naturalh' arises, and it is one 
 of the objects of the study of the subject of the Theory of Equa- 
 tions to determine such. In order, then, to throw all the 
 light possible on the.se equations of higher degree, we will begin 
 by taking up the left member of such equation, considered as the 
 function of a variable, and endeavor to reach a fair conception of 
 the nature and properties of such functions in general. This 
 method of procedure in itself might be a justification for speaking 
 both of the roots of an equation and the roots of a function; but it 
 is further observed that these expressions are reall}- the same, since 
 the root of/ (a-), being defined as the value of .r which makes/(.r) 
 \-anish, must also be the value which satisfies the equation, 
 
 /(.r)=o. Nevctheless, when speaking of a root of/(.t-), the 
 thoughts associated with the expression in our mind are different 
 from the considerations uppermost when we speak of a root of 
 /(x)=o; the first is the more convenient expression, when we are 
 considering any function of x without reference to its history or 
 application; the latter expres.sion is u.sed when we are considering 
 an equation of which /(.i) is the left member; we have in mind, 
 when using /"(a), .v as a variable, unrestricted in value; but in 
 
12 ADVANCED ALGEBRA. 
 
 /(.v)=o, .V appears under restrictions or conditions and is fixed in 
 value, although unknown. 
 
 14. Theorkm. The difference between f{x) and /{a) is divis- 
 ible by -V — '/.. 
 
 We are to prove ^ =a quotient without a remainder. 
 
 X 'I- 
 
 Now/(a') is rt„.v" + ^?,.r"~' + <7.,.f"~-+ . . . +a^^_.,x--\-a^^_^.\-+a^^ Art. 8. 
 and/('/) is a^^'i." + a ^'/■"~^ + a ,"."~- + . . . +a^^_./r-\-a^^_^"-\-a^^ Art. 7. 
 _ j\x)— /{'>:) ^ 
 
 X — '/ 
 
 ^n( -^'" - "■") + a, (x—- «"-') + alx"--' - o"-') + + a„^lx'-o') + a„_, (x-a) 
 
 X 'I. 
 
 equals some quotient without a remainder, since difference of like 
 powers is divisible bv the difference of the quantities them.selves. 
 
 Q. E. D. 
 
 15. Examples. Divide each of these functions b}^ .v — '/. 
 until the quotient does not contain .r, and notice the remainder: 
 
 1. .V-— 4A-+7. 3. 4-1"'— 3-^'' +5-1'— I- 
 
 2. x'' — .r-'+2. 4. bx' — ex'- — dx-^b. 
 
 16. Theorem. IVhenf^x) is divided by x—i, the remain- 
 der, after the quotient does not contain .v, is equal to f {»■). 
 
 Now, from Art. \x, -^^-''-'^^''^=.some qotuient 
 x — « 
 
 .-. ■'^l:^=some quotient +^^-^-^ 
 x — "- X — '/ 
 
 .•. /(a) is the remainder. Q. E. D. 
 
 17. Theorem. Any fix) is divisible by x minus a root. 
 Let the root be '/. 
 
 Now from Art. 1 4,-^-^-^^"-^^"^= some quotient. But /"('/) is 
 
 X- a 
 
 zero; (Art. \o.). 
 
 -'-^= some quotient. n t? n 
 
 18. Theore.m. Conversely, If any fi^x) is divisible by x 
 7nimis »■, <>■ is root. 
 
 Then/f.i") is (.V — «)Q, which will vanish when .r=«, .'• <i- is 
 a root. Q. E. D. 
 
RATIONAL INTEGRAL FUNCTION OF nc 
 
 19. A short mctJiod of dividing- miy f ix) by x — a. 
 
 Suppose the /(.r) to be a^y''-{-a^x^ + a.,x''-\-a.^x^ + a^x+a.^ 
 
 ,, 1 ^ a,x'+a,x^-\-a.,x''-\-a.,x' + a,x+a- 
 
 IN owlet -" — — ' — — — ■' — ^—^ — s 
 
 X — a 
 
 Remainder 
 
 X — a 
 
 where A, B, C, etc.. are undeterraiiud coefficients. Put R {or Re- 
 nt i-indcr. 
 
 Then a^y ' + c? ,.v' + a.,x' -f- a.^x"' + a ^x-\-a.^ 
 
 = (.i-— a) I Ax'-\-Bx'^Cx'-VDx+E^^ -\ 
 
 = Ax'+B—oA ).v' + ( C—"B)x''^ D~iC)x'+{E—'iD ) + R—'-E. 
 
 Equating coefficients, 
 
 A =«„ I f ^~ ^n I 
 
 B—aA = a^ I B=aA-{-a^ 
 
 C—aB=a.. \ ^, . C = aB+a, 
 
 r^ r- ■ \ Therefore, { j^ r~ \ 
 D — aC^a., f I D='i.C-\rCi., 
 
 E—aD=a\ I E=aD+a^ 
 
 R—aE=a. I I R=aE+a.^ 
 
 Arranging the right hand column of equations horizontally: 
 
 Coefficients in dividend, <:7„ +^7, -\-a., -\-a.^ -\-a^ +<^5 
 
 -\-aA +aB +aC +"D ^-aE 
 
 Coefficients in quotient, A B C D E R 
 
 The following is seen to be the law of the coefficients, A, B, 
 C, etc., in the quotient: The first coefficient in the quotient is the 
 same as the first coefficient in the dividend. The second coeffi- 
 cient in the quotient is the second coefficient in the dividend plus 
 a times A, the one just found. The third coefficient in the quo- 
 tient is the third coefficient in the dividend plus a times B, the 
 one just found. And ^;n' coefficient in the quotient is the corre- 
 sponding coefficient in the dividend plus « times the preceding one 
 in the quotient. 
 
 The process of finding the coefficients in the quotient, and the 
 remainder, is more apparent in a particular case: 
 
14 ADVANCED ALGEBRA. 
 
 Find the quotient of 7.1-^+8.1-' — 6a--' — 15.V' — 8.r+4 by x — 2. 
 Coefficients in dividend, 7 +8 — 6 — 15 — 8 +4 (2 
 
 14 44 76 122 228 
 Coefficients in quotient, 7 22 38 61 114 232 
 
 2 T,2 
 
 Henc2 the quotientis, 7.i-*+22.t''+38-v- + 6i.i-+ 1 1^-\ — f — 
 
 Find the quotient of .v^ — 81 by x — 3: 
 Coefficients in dividend, 1000 — 81 (3 
 
 3 9^ 27 81 
 
 Coefficient in quotient, i 3 9 27 o 
 
 Quotient, _r'+3A- + 9.v-f 27, 710 remainder. 
 
 This method of obtaining the quotient is called synthetic divi- 
 sion. It will evidently apply when the dividend is a function of 
 any degree. 
 
 Query : What change will there be if the divisor is x-\- '/ ? 
 
 20. Examples: 
 
 1. Divide x*— 5a-'+i2a-+4-1'— 8 by .r— 2. 
 
 2. Divide x^-\- 1 i.v-+36.r+ 15 by A-+5. 
 
 3. Prove 3 is a root of x' — 6A-+ii.r — 6. Art. 17. 
 /. Prove 6 is a root of x'— i2A''+47.r"' — 72.1-+36. 
 
 5. Prove — 6 is a root of -v' — 4.1-' — 29.^-+ i56.r — 180. 
 
 6. Find value of x'-\- i6.r — 4.1- -j- 1 when x^z. Art 16. 
 
 7. Find value of .i''+9.i- — 19X— 76 when .1 = 5. 
 
 21. Theorem, a.ssumed. Every f{x) has at least one root. 
 This means that there is some real quantity, or some imagi- 
 nary, which put for x in 
 
 a^'-\-a^x"~^-^a.^x^'-''-k- . . . -f rt,,_,.r+i7„_,.r-f «, 
 makes the aggregate value o. The theorem here assumed is one 
 whose demon.stration is too difficult to be given in this course. It 
 may be found in Todhunter's and other treatises on Theory cf 
 Equations. 
 
 22. Theorem. Imaginary roots enter in pairs. 
 
 Suppose r+'^^ is an imaginary root of /(.1-). We' will prove 
 that /'—''/ is also a root. 
 
 We know /(.v) to be divisible by x—y — "'/ (Art 17). If it 
 is also divisible by x— /-+''/, then ;- — 'V is a root (Art. 18) Let 
 us divide /(a-) by (.v— /—'>/) Cr—r+'V), or [(.v— ;')"+''']■ Let the 
 
RATIONAL INTEGRAL FUNCTION OF JC- i5 
 
 quotient l»e O and stop when the remainder is of the form S.v-\- T. 
 Then /(.r) is Qlix-rf + '^'I^Sx^ T. 
 
 Put .1=/' + ''/ 
 
 Then, by hypothesis, f{j-\-oi)=o. 
 
 Sr-\-S<H-\- T=-o. 
 .'. 5"'5z=o and 5=0. .. 7^=0 
 since the real and imaginary parts are separately equal to zero. 
 
 f(x) is divisible by (x—-i' — ^i){x — y + i^i) without remainder. 
 y — in is a root. 
 
 Q. E. D. 
 
 23. An exactly similar proof shows that surd roots, of the 
 form r+\/'', enter in pairs, if all the coefficients of/(-v) are rational. 
 
 24. CoROLivARY. Every fi^x) is -divisible by a rational inte- 
 iiral fitndio)! of the first or second degree, since it has one real or 
 two imaginary roots. 
 
 25. Theorem. Any f(yX) of the nth degree has n roots and 
 no more. 
 
 a„-v" + «j.v"~' + a„.v"~^H- . . , -\-a^^_.^x'--\-a^^_^x-\-a^^ has one root, real 
 or imaginary, (Art. 21). Call it '/,. Then/(x) is exactly divisible 
 by .V — «j (Art. 17). The quotient is of the form 
 
 a^x"-^-\-Bx"-'-+Cx"-''^ . . . +Hx'+Kx+M. 
 Then/(.v) is {x—a^)(a^y''-'-^Bx"--'+Cx"-' + . . .-^ Hx'+Kx+M). 
 Since the second factor is a rational integral function of .r, it has 
 at least one root. Call it '/,. Then, 
 /■(.v)is (.V— /,)(.v— ■A,)(«„-i-'-^'+^,.i-'-'+t>"-*+ . . . +R\x-^M^) 
 
 Again, the third factor must have a root, suppose a.^, whence 
 fix) is (.V— r/,)(A-— aJ(.r-aJ(a„.v"-^'+.9,.r"-'+ C,x"-'-\- -^K^x+M.^ 
 and so on. Whence 
 
 /(.v) IS (.V— ^/,)(.v-«,)(.v— «,)(-V--«J . . . .{x—".)a^^. 
 It is obvious, therefore, that /'(.v) will vanish when x has any 
 ©f the values '-(j, 'a„ "■.^, etc., and for no others. Q. E. D. 
 
 In the above, if any of the supposed roots are imaginary, divi- 
 sion by X — a would render some of the coefficients of the quotient, 
 B, C etc., imaginary. This could be avoided, however; for having 
 one imaginary root in hand, we know that there is another, and 
 could divide by a quadratic factor, as in Art. 22. All the 
 
i6 ADVANCED ALGEBRA. 
 
 coefficients in the divisor being real in this case, those of the 
 quotient would be real also. 
 
 26. Corollary. Ajiv /(x) may be represented by 
 {x~-<i^){,x—i.X^—'^.,){-^-—<i,) {x—'j.Ja^ 
 
 27. Corollary. Every /{x) of an odd degree has, at least, 
 one real root. Since it has an odd number of roots altogether, and 
 an even number of imaginary roots. 
 
 28. Examples. Form the/(A-) whose roots are: 
 /• 3. 2, 1. _ 2. 3, —2, I. 
 
 3- 3, 2, W-^^ 2—3s/~i. 4. o, 3, v/ 2 , —>/ 2- 
 
 29. Theorem. Multiplying or dividing J\x) by a constant 
 does not affect the roots. 
 
 f{x) is (.V— aj(.v— «,)Gv— '/J U-—'\.K 
 
 Pffx) is p{x—a^){x—a^){x—L.^ (-V— '^.)«o 
 
 /^-^is ' (.V— a.jC.r— 'Aj(-v-'.,j (-V— '^)«o 
 
 All of these, obviously, vanish for the same value of .v. 
 
 Q. E. D. 
 
 30. Corollary. Changing the signs of all the terms o/f{x) 
 does not affect the roots. This is multiplying by — i. 
 
 31. Theorem. In any fix), the coefficient of the highest 
 power being unity, the coefficients of the other powers are functions of 
 the negatives of the loots. 
 
 Since we know -v"+/i-r"~'+/),,i-"-'+A-^"""'++A-2-'^'"'+/'.-r'^'+A 
 can be represented by (jr — '\){x — 'A,)(-V — /,) .... (-v — '-cj, it is 
 evident that the values of A. A' A- etc., can be had in terms of the 
 roots by forming the product of the binomial factors, as in the 
 demonstration of the binomial theorem. Such product is: 
 
 -^-i—'WV'—'W'^'—'W, ■ ■ ■ — '^.-/^,-.'^,)-^"+ • • +(^'^'^■/^:. • • «J- 
 
 />,, or thefrst parenthesis, is the sum of the negatives of the roots. 
 
 A, or the second parenthesis, is the sum of the products of the neg. 
 atives of the root taken t7i'o at a time. 
 
 A, or the third parenthesis, is the sum of the products of the nega- 
 tives of the roots taken three at a time ; and so on. 
 
 A, or the nth parenthesis, is the product of the negatives of all the 
 roots. Q- E. D. 
 
RATIONAL INTEGRAL FUNCTION OF x. i; 
 
 32. Corollary. T/ic roofs 0/ /(x), the coefficient of the 
 highest power being unity, are all factors of the absolute term. 
 
 2,2,- Theorem. If two numbers be substituted for x in fix), 
 giving results with contrary signs, a)i odd )iumber of real roots lie 
 betzveen the values substituted. 
 
 Let 'ij, o._^, a,^, , . . . 't-y be all the real roots of /(.i"), arranged 
 in the order of their magnitude, '/., being the greate.st and '/,• the 
 least. Suppose /"(.v) to be divided by each one of the real roots 
 and let ^{x) be the quotient, which, of course, is divisible by x 
 niinus each of the imaginary roots. Then 
 
 /I1-) is (.v-/.,)(.r— /.J(.v-'/.3) .... {x~u.,)c{x). 
 Now, from what is known about imaginary roots (x\rt. 22) <{'yx) 
 must be of the form 
 
 [Ci - dY^e'\{x—ff^g'\ etc. 
 and, since it is the product of the sum oi squares, must be positive, 
 for all real values of .r. 
 
 Now suppose -v to have a value greater than <i.^. Then all 
 the factors are plus and hence f {x) is plus. Let .r decrease grad- 
 ually and become less than '/, but greater than 'a^; then the first 
 factor is — and all the rest +, hence /"U") is — . Now let x be- 
 come less than «., but greater than '/,; then two factors are — and 
 the rest +, hence /\.v) is +. Again, let .v vary until it is be- 
 tween '/., and '/.^; three factors are — and the remainder -f, hence 
 f{x) is +. And so on. 
 
 Whence it is evident that / (.v) changes signs every time we 
 1 ass over a real root. Hence, if two numbers should be put for 
 -V in /"(.v), giving results wilh contrary signs, there must be an 
 odd nu:n!jer of real roots between the two values substituted. 
 
 Q. E. D. 
 34- Examples. Show that 
 
 /. .r'— 6.r— 13 has a root between 3 and 4. 
 2. .1' — i2.x'-f 12.V — 3 has a root between 2 and 3. 
 35. Theorem. Any fLx), the coeff dent of the highest pozcer 
 of X being unity, and }io)ie of the rest irreducible fractions, cannot 
 have a root 7vJ/icIi is an irreducible fraction . 
 
 Letsuch f{x) be .r" + /;;,.r""'-)-;;/.,.r"~-+. . . + w _.x-\-i)i , where 
 
,S ADVANCED ALGEBRA. 
 
 w,, w,,, w., etc., are not irreducible fractions. If this /(.v) can 
 
 vanish when A' is an irreducible fraction, let that fraction be ,. 
 
 Then / ^ is r„ + w,7„:i^ + ?«._, ,„_,,+ . . . +/''/„^, ,^-)-w,,. 
 
 If the value of this is zero, it will be zero when multiplied by ,5""'. 
 
 But \_^+n!^u"~^ + )}i J "■"'-. . . + w„_i,^"~''^ + ,^"~'w„ cannot be zero; for 
 
 '/" 'J- 
 
 ^j is an irreducible fraction, since , is; and the rest of the function 
 
 does not contain a fraction, since none among «, r^, in^, iii.,, di.., etc., 
 are such; and a fraction must be combined with a fraction to make 
 
 zero. Therefore it is impossible for ,^ to be a root. 
 
 QE.D. 
 36. Theorem. Any J\x) cati be trausfonned into a v'(j') 
 whose coefficient of the highest power is unity and ?ione of the other 
 coefficients irreducibk fractions. 
 
 Suppo.se /(-v) in the form (Art. 8), 
 
 .v"-f-/',-v"-'+A-^-"-'+ . . +/'„-,,-r+A-r^'+A- 
 If any of the coefficients, />,, p.,, />,,< ^^c, are fractions, let their 
 common denominator be q. Put .v ==' , then f{.x) becomes 
 
 q" 9" 9" ' 9 
 
 Multiply through by q'\ 
 
 y"-VqP,y"~'^fP._y'~''^ ■ ■ ■ +/'~'A-,j'+/A 
 
 where none of the coefficients can be irreducible fractions, since 
 each is multiplied by q, their common denominator. 
 
 Q.E. D. 
 
 37. Query. What relation exists above between the roots 
 of/(.r) and <s(y) ? 
 
 38. Examples. Transform these functions into others, so 
 that the coefficient of the highest power will be unit}' and no coef- 
 ficients irreducible fractions: 
 
RATIONAL INTEGRAL FUNCTION OF X. 19 
 
 s , 14 . 28 8 
 
 9 27 27 
 
 ^- 3^-* — 40-v'+ 1 2,ox' — 1 20.V+ 27 ■ 
 , II ., , I 
 
 ^- 8 i.r'— 1 8.1-^-36.1-+ 8. 
 39. Theorem. A?iy /{x) can be transformed into a <s{x) 
 -whose roots differ from those qff(x) by any assigned quantity. 
 
 Take/Gv) as x-'+p^x"~'p.X'''+ . . . +/„-,-r+A- 
 Suppose .V to bej'+//. Then/(.r) is /(_>'+//), or 
 
 {y+h)"+p^{v+hy'-'+p.Av+h)"-'+p.^y+h)"-'+ . . . +A- 
 Expanding the terms by binomial theorem, f{x+h)^= 
 ,,..+ „/„.^-.+ "J»T_il ,.,,-.+ n(n-2}ip),,y..,^ 
 
 P,r-'+ p'^-n^r-'+p!"-^^' >.y-+ .... 
 
 p.,v"-"--{-pIn — 2)hv"-''+ .... 
 Aj'"-''+ • • • • 
 Collecting in terms of powers of r, we have/( ]'+//) = 
 
 _,''■ + (;?//+/., )!■"-'+ I "^"^^^^'' +P,in—i)h+p.^^yv"--' 
 
 I I-2-3 1-2 ^- ' ' ]- 
 
 which may be called f ( )')• More terms can evidently be obtained 
 by expanding further by binomial theorem. Since x^=y-\-h, y is 
 always // less than .v, so the roots of t-'(j') must be h less than 
 those of /"(.t). By originally taking .\-=i'+//, the roots of <f{y) 
 would be // greater than those of /(.v). Q- E D. 
 
 40. HoRNERS' Method of computing the coefficients of 
 c-( )'), in above. 
 
 Suppose f{x) is <i:„.v'-f-rz,.r"~' + fl!._,.v"~- + . . -{- a ^^_.,x' -\- a ^^_^x + a ^^. 
 Put .1 = 1'+/^ and suppose/ (-1") then becomes 
 
 A,y" + A^y"-'+A.j'"--'+. . .+.-i,,_,_,j'"---'+.^„_,j'+.^,„ 
 where A^^,A^,A.^. etc., are undetermined coefficients. Now, 
 y=x — h. Therefore, 
 a^y-"-{-a^x"~^-\-a.^x"~--j- . . . +«„-.,- v' + «„_,.v+(r,, 
 
 =A^iv--hr+A,(x—hy-'+A.sx-hr-'+ ..." 
 
 + A „_,(.v— //)■- -f A„_, (x—h ) + ^„ . 
 
20 ADVANCED ALGEBRA. 
 
 Hence, we see that .-J,,, the last undetermined coefficient, is 
 the remainder when /(a-) is divided by x — /;. The quotient is 
 
 whence it is apparent that ^„_i, the next to the last undetermined 
 coefficient, is the remainder when this last expression is divided 
 by x—h; and the same way for the other A' s. ilencc, the unde- 
 termined coefficients, beginning at the last, are the successive 
 remainders, as/(-r) is continuously divided by x — h. 
 
 As an application of this, suppose it was required to reduce 
 the roots of.i' — 5.1" — 5-r +25 by 2. We would divide successive- 
 ly by x — 2 by synthetic division as follows: 
 
 I c^ ^ +25 2 oi"' abreviated thus: 
 
 2 — 6 — 22 I 
 
 I 
 
 3 
 
 — II 
 
 3 
 
 
 2 
 
 — 2 
 
 II 
 
 X 
 
 — I 
 
 -13 
 
 '-^:; 
 
 
 2 
 
 II 
 
 
 ■13 
 
 I I A., 
 
 II II 
 
 Result: .r-f .r— 13.1 +3. 
 
 41. Examples. Transform each J\x) into a cr( rj whose 
 roots shall diffi^r as assigned. 
 
 /. -X"*-|-4.v" — 2.1"'' — 12.V+9, roots to be 3 greater. 
 
 2. x'' — 9.v''+23.f — 15, roots to be i less. 
 
 3. -v^ — 6.1'— 13, roots to be 3 less. 
 
 42. Theorem. Any f{x) can be transformed into a e( i-), 
 witl: the yiext highest po'wer of v zvanting. 
 
 In Article 39 put .v=i — ^' ,- i e., Ii=— . Then the co- 
 ;/ n 
 
 efficient of the next highest power of r becomes ( — «— -f /',)oro. 
 
 Q. E. D. 
 
 43. Examples. Transform each into cr { y) with the next 
 highest power wanting. 
 
 /. .r'— -9.r+23.r — 15. 
 
RATIONAL INTEGRAL FUNCTION OF x. 21 
 
 J. 6.x-'' — I ijr'^+ 6x — I . 
 
 /. Transform ax^'+dx' + cx+d, where a, d, c, d, are whole 
 numbers, into a function of the form 2^-\-h-i-f/2, where / and m are 
 whole numbers. 
 
 44. Theorem. // the signs of all terms containing the odd 
 pozvers, or the even pozvers, of x be changed, f{x) is transformed into 
 f(-x) or-f(-x), the roots of either of the latter being the negatives of 
 the roots off(x). 
 
 = {x-a^){x-a^){x-a^ . . . {x-u.J (l). 
 
 Putting — .r for x throughout the equation, we shall have/(jr) 
 with the signs of the odd powers changed, equal to the product of 
 n factors of the form ( — x — '/}, or 
 
 fi _ ^-w ^ •^'" -/'r^'"~'+A v""'-/',.i-"-'+ . .-/>„_,.r+A' if '^ '"^^ e^*^^ X 
 '. --i""+/>i-v"~'-/),.r"~--f/',a"~'- . --/^-r^'+A' if ^^ i^ °^^ ^ 
 = (-i)"(.v+'/,)(.v+'/..;(x+'g . . ."U-+'/). (2). 
 
 Multiplying this by — i , we will have f{x) with the signs of the 
 even powers changed, or 
 
 -/■(- 
 
 _ ( - x"+p^x"- ^-p.,x" '+pyX" '•'-.. +p„-r'^'-p„ \in is even ( 
 ( x" - p^x"~^ +Pt'^'"'''- Pt^"~''' + • • -'rpn-v^ ~J'n if '^ is odd i 
 = (— i)"^'Cr+r/,)(x+'/.,)(x+'/,) . . . Cv+'.J (3). 
 
 Now, by changing the odd powers in fix) we must obtain 
 what is contained in brackets in (2), which equals f{ — x); and 
 from the latter part of (2), it is evident the roots are the negatives 
 of the roots of (i ) or/(x). 
 
 By changing the even powers in/'(x) we must obtain what is 
 in brackets in (3), which equals -f(-x); and from the latter mem- 
 ber of the equation, it appears that the roots are the negatives of 
 those of (i) orf(x). Q. E. D. 
 
 45. Theorem. fff(x) has r roots equal to '/., the derivative 
 has r — i roots equal to a, the second derivative has r — 2 roots equal 
 to <>., and so on. 
 
 Suppose f{x) has r roots equal to «, then 
 
 /(^) = (x-a)'-^(^) 
 
22 ADVANCED ALGEBRA. 
 
 where <fiX) stands for the quotient of/(.v) by {x — '/)'' 
 
 Since Z>.r/(r) contains .r-'/. as a factor r- i times, it has r— i 
 
 roots equal to '/•. 
 
 Call the above derivative /(.r). Then, if <f,(x) represents 
 the quotient as above, 
 
 /^(x)=(x—ay-W,(x) 
 
 Now the derivative of this will have r- 2 roots equal to '/; and 
 so on. Q- E. D. 
 
 46. Corollary. {/'/{■^') has r eqtial roots, it and its first 
 derivative tvill have a co})imo7i divisor of the form, {x — ».)'"'^ 
 
 47. Examples. Test each for equal roots by finding the 
 highest common divisor of/(-v) and its derivative. 
 
 /. .;t--'-6.r^+ii.r-6. 
 
 2. X* '- 5. V '■ + 1 2x' + 4-;v — 8 . 
 
 J. 2.r'-7.r'+4-^-|-4- 
 
 4 .v'+4-r' - 2.1-' - 1 2.r+9. 
 
 48. Theorem. No f{x) can have more positive roots than it 
 has changes of signs from -\- to — and — to -h- 
 
 Suppo.se we have any polynomial which, when arranged ac- 
 cording to the descending powers of .r, has the signs -|- and - 
 occurring in the order given in Af below. Let this polynomial be 
 multiplied by .r-'/, corresponding to the introduction of a positive 
 root '/.. Representing only the signs in the multiplication, it is as 
 follows: 
 M. +' +■' -\-' —' —' -' +' -f' -" +'" -f" 
 
 a. 
 b. 
 
 
 _i 
 
 2 
 
 ^3 
 
 + * 
 
 +^ 
 
 
 7 
 
 9 
 
 
 +;; 
 
 u 
 
 P. 
 
 4 
 
 f 
 
 f« 
 
 +' 
 
 r 
 
 — ' 
 
 +'" 
 
 r 
 
 12 
 
 We have written the signs of some of the terms f because a 
 + term is combined to a - term and it is unknown which is the 
 greater. The original polynomial will be spoken of as J/, the 
 first and second partial products as a and b, and the 
 final product as P. Figures have been attached to the 
 signs so that they may be .spoken of by number. It will 
 be noticed that while we use the particular order of 
 
RATIONAL INTEGRAL FUNCTION OF x. 23 
 
 signs given in J/, yet the demonstration we give is perfectly gen- 
 eral, — applicable whatever the order of signs in the polynomial. 
 
 ( I ). We will first sho.v that the product has at least as many 
 changes of sign as the multiplicand. 
 
 The si.i;ns in a are the same as those in ;J/, and the signs of b 
 are those of J/ reversed and put one place to the right. 
 
 Consic.er an}- two consecutive signs of J/, the {k — i )th and 
 X'th. They are either alike or different. 
 
 First suppose them aliker'^ then the /'th sign of a is the san-e 
 as the X'th sign of J/, while immediately under this is the (y^— i)th 
 sign of b, which is of the opposite kind, so that the /tth sign 
 of the product /^ is .-^ Second suppose the {k — i)th and X'th 
 signs of yl/ different ;t then, as before, the A'th sign of <a; is the same 
 as the /'th sign of J/, but the sign immediately under this is 
 the [k^i )th sign of b, which is the {k — i)th sign of J/ reversed, and 
 hence is the same as the /tth sign of a. Therefore the k'&i sign of 
 P\s the same as the /^th sign of M, unambiguous and unchanged. 
 Passing along the signs of J/ and /"from left to right, it is evi- 
 dent that \\\^ first sign of P is the same as the first sign of M and 
 that the following signs of /^are all f, but as soon as the sign in M 
 changes from + to — or — to -f , the second sign of this change 
 appears in P in the same position as in J/. Therefore, in this 
 portion of M and P, f.ie signs begin alike and end alike, and as 
 there is one change of sign in M there must be one or more (in- 
 deed, some odd number) changes in P, and, as this holds good 
 every time the sign of M changes, therefore there are certainly as 
 many changes of sign in /'as m M. 
 
 (2) We will show next that there cannot be the same num- 
 ber of changes of signs in P as in M. 
 
 If tlie first sign of a polynomial is like the last, the number of 
 changes of signs between the first sign and the last must be even ; 
 and it the first sign differs from the last sign, then there are an 
 odd number of changes of signs within it. Now the last sign of 
 P is necessarily different from the last sign of M. Therefore if 
 the first and last signs of M are alike, the first and last signs of 
 P are unlike; and if the first and last signs of J/ are unlike, the 
 first and last signs of P are alike. Hence, if there is an even 
 
 * In this case k would be either 2, 3, 5, 6, 8, or 11, if illustration on opposite page is used. 
 t To illustrate this reasoning, k may be taken as either 4, 7, 9 or 10 in work on op. page. 
 
24 ADVANCED ALGEBRA. 
 
 number of changes in M, there is an odd number in P, and if 
 an odd nnvcvh^x in M, then an even number in P. Hence /'cannot 
 have the same number of changes as M. 
 
 Now we have shown: (i.) that P has at least as many, 
 changes as J/; (2.) that P cannot have the same number as J/. 
 Therefore P has more changes of signs than M. That is, the re- 
 sult after muhiplying by x — «, or introducing a positive root, 
 contains more changes of signs than the original polynomial. 
 Since at least one additional change is brought in for each positive 
 root which may be introduced, no f{x) can have more positive 
 roots than it has changes of signs from -f to — and — to +. 
 
 Q. E. D. 
 
 49. Corollary. Nof{x) can have more negative roots than 
 there are changes of sigtis from -^ to ~ and — to -j- , after the signs 
 of all the odd or even powers have been changed. See Art. 44. 
 
 The above theorem and corollary con.stitute what is known 
 as Descartes'' Rule of Signs. 
 
 50. Examples. 
 
 /. Show that x" — i has one positive root and no other real 
 root. 
 
 This function, being the difference of like powers, is divis- 
 ible by X — I ; whence + 1 is a root. There is onl}- one change of 
 sign, hence it can have no more than one positive root, hence 
 none other than + i . Changing the signs of all the odd or even 
 powers of x, there are no changes of signs, hence no negative 
 roots. 
 
 2. Show that A"'' — I has two real roots o\\\y , one + and one — . 
 
 3. Show that x^-\-a has no real roots. 
 
 4. Discuss the roots x" — a when n is odd and also when n 
 is even. 
 
 J". Discuss the roots of .i""4-a, when 71 is either odd or even. 
 
CHAPTER III. 
 
 SOLUTION OF NUMERICAL EQUATIONS. 
 
 1. As was noted at the beginning of the last chapter, one of 
 the objects of the study of the Theory of Equations is to establish 
 methods for solving any rational integral equation containing one 
 unknown. When, by any mathematical process, we obtain the 
 values of x from an equation, the result is, " by the adoption 
 of a t^rm primarily applicable to the mode or process of its dis- 
 covery, called the solution of the equation." Thus 2 and 3, the 
 values of . I from .r'— 5,r + 6=o, may be spoken of as ihe sohiiion of 
 this quadratic, as well as the process of completing the square, 
 etc , by which they are obtained. This double use of the word 
 solution, as designating either a result or the process by which the 
 result is obtained will hardly lead to any confusion. 
 
 2. Definition. A Solution ol an equation is any function 
 of the known quantities which represents the values of x. 
 
 The solutions of the equations of the first degree, x-\-a=o and 
 
 x'^-\-ax-\-b=o, are already familiar to us; they are, respectively, 
 
 J — azt^/a''' — 4<!>. 
 
 j»r= — a, and x= ^ 
 
 2 
 
 Similar solutions for the equations of the third and fourth 
 degree, x'-\-ax''-{-bx+c=o and x'-\-ax'-\-bx''-ircx+d=o, have been 
 obtained, but they are scarcely of value in the solution of numer- 
 ical equations. These are called Algebraic Solutions, since they 
 involve only the ordinary algebraic operations, such as addition, 
 subtraction, multiplication, division and involution and evolution 
 to commensurable powers and roots. Abel, a Norwegian mathe- 
 matician, has proved that the algebraic solution of the general 
 equation of fifth and higher degrees can not be obtained; that is, 
 the result can not be expressed by means of the ordinary algebraic 
 symbols of operation alone. The function of the known quantities 
 which expresses the values of .i", is not within the range of alge- 
 braic analvsis. 
 
26 ADVANCED ALGEBRA. 
 
 While it is true that an algebraic solution of the general equa- 
 tions of high degree is impossible, j^et our knowledge of mathe- 
 matical processes is sufficient to enable us to find the real roots of 
 any numerical equation whatsoever. The means adopted for 
 accomplishing this leads to a solution satisfactory in all respects, 
 giving the values of the roots exactly, if commensurable, or to 
 any desired degree of approximation, if incommensurable. It 
 will be found that the mode of procedure depends upon the 
 properties of f{x) established in the last chapter. We shall 
 begin with an illustration of tlie general method by 
 particular example and .shall afterwards summarize the process in 
 a general statement of advice for any case. 
 
 3. Solution of the equation 
 
 x'—\yix'—/^^—iy2x--\-^,y2x-\- =0. (I). 
 
 a. Put.r=V2j', which transforms it into the following equa- 
 tion (II. Art. 36), which has no root that is an irreducible frac- 
 tion: (II. Art. 35). 
 
 v'C j')=y— 3/— i6y-f 281-+ 72j'+ 32=0. ( 2). 
 
 The real roots of this must be either integers or incommensur- 
 able fractions. 
 
 b. In seeking for integral roots, we need only search among 
 the factors of the absolute term, (II. Art. 32) which are: =ti, ±2, 
 ±4, ±8, ±16, ±32. We may now test by dividing (2)h\y 
 minus each. (II. Art. 18). 
 
 I -3 —16 -(-28 +72 +32 (+1 I —3 —16 +28 +72+ 32 ( + 2 
 _j_i _ 2 — 18+10+82 +2 — 2 — 36 — 16+112 
 
 r —2 —18 +10 +82 + 114 I —I —18 —8+56+144 
 
 + 1 not a root. +2 not a root. 
 
 I —3 —16 +28 +72 +32 ( + 3 I —3 —16 +28 +72 +32 (+4 
 
 + 3 o —48 —60 +36 +4+4 — 48 — 80 —32 
 
 r o — 16 — 20+12+78 I +1 — 12 — 20 — 8 o 
 
 + 3 not a root. +4 w a root. 
 
 This last quotient gives us an equation of a lower degree 
 containing the remaining roots: that is 
 
 y +y — I iy- — 20) — 8 = o. (3) . 
 
SOLUTION OF NUMERICAL EQUATIONS. 27 
 
 We need now tr}' only factors of 8, of which we have already 
 tried +1, +2. +4. We will try the small negative numbers be- 
 fore trying +8. Dividing byj'+i, corresponding to a root— i, 
 we have: 
 
 I -f I —12 —20 +8 (—1 
 — I 0+12 — 8 
 
 I o — 12 — 8 o — I is a root. 
 
 The equation containing the remaining roots is 
 
 J''— 12) — 8 = 0. (4). 
 
 Using the untried factors of 8, we find no more integer roots. 
 Hence the equation must have three incommensurable roots, or 
 one incommensurable and two imaginary. 
 
 c. It is now well to test for equal roots, for if two or three of 
 these incommensurable roots are equal, we can find them very 
 readily. But y''—\2y — 8 and the derivative 3^ — 12 have no 
 common di^dsor, hence there are no equal roots. (II. Art. 46) 
 
 d. We next attempt to locate the incommensurable roots, 
 by assigning difierent values to j' and calculating the correspond- 
 ing values of f(j'j. (II. Art. 33). If we put any value, a, ior y 
 in f(ji'), we can compute the value of c{a) by the short method of 
 division; for f(a) is the remainder when ^iy) is divided hy y — -a. 
 
 (II. Art. 16). 
 ) ''' -^- oy' — 1 2j' — 8 = 0. 
 
 I +0 — 12 — 8 ( + 2 r +0 —12 —8 (-f3 
 
 + 2 -f4 —16 
 
 + 3 +9 —9 
 
 I -f2 —8 —2^=R=^{2) 
 
 I +3 —3—17 = ^=^(3) 
 
 I +0 — 12 — 8 ( + 4 
 
 I +0 —12 - 8 (-2 
 
 + 4 +16 +16 
 
 — 2 -f 4 +16 
 
 I +4 + 4 + 8 = A'=cr(4) 
 
 I —2 — 8 +8=A'=f( — 2) 
 
 etc. , etc. , 
 
 etc. , etc. 
 
28 
 
 ADVANCED ALGEBRA 
 
 In like manner we find the values of <f{r) when 
 values and tabulate them as in the margin. The 
 first column contains the values assigned to )' and 
 the second contains the corresponding values of <f(r)- 
 It is then noticed that there is a root between +3 and 
 + 4 and between o and — i and between — 3 and — 4. 
 (II. Art. 33). 
 
 e. The roots thus located can be determined to 
 any degree of accuracy in the manner following. Since 
 equation (4) has a root between +3 and +4, the first 
 figure of the root is 3. Then transform the equation, 
 by Horner's method, into one whose roots are 3 less, 
 is as follows : i + o — 12 — 8 ( 3 
 
 + 3 +9 —9 
 -17 
 
 has 
 
 other 
 
 V 
 
 V'(J') 
 
 —4 
 
 — 21 
 
 —3 
 
 + I 
 
 — 2 
 
 + s 
 
 — 1 
 
 4- 3 
 
 
 
 — 8 
 
 + 1 
 
 — 19 
 
 + 2 
 
 -24 
 
 + 3 
 
 — 17 
 
 + 4 
 
 + 8 
 
 The 
 
 work 
 
 + 3 
 
 + 6 
 
 + 3 
 
 + I.S 
 
 I I +9 
 The resulting equation is 
 This 
 
 j'''-|-9_r'+ 151/ — 17=0. ('5"). 
 must have a root between some of these values: .0, .1, 
 .2, .3, .4, .5, .6, .7, .8, .9, i.o. By trial it is found to lie between 
 .7 and .8, thus: 
 
 I +9.0 4-15.00 —17.000 
 4- .7 + 6.79 +15-253 
 
 I 4-9-7 +21.79 - 1.747 
 
 The first figure of this root of (5 
 second ft ^s; lire of a root of {^). 
 
 Now depress the roots of ("5) by 
 work is as follows: 
 
 I +9.0 +15.00 — 17.000 { 
 
 + • " +6.79 +1 5-253 
 I +9.7 +21.79 I —1.747 
 
 I +9.0 +15.00 —17.000 (.8 
 + .8 + 7.84 +18.372 
 
 r +9.8 +22.84 + 1.272 
 
 therefore. 7, which is the 
 
 Ijv Horner's method. The 
 
 1 + 10.4 
 
 + 29.07 
 
 + 11. 1 
 
SOLUTION OF NUMERICAL EQUATIONS. 29 
 
 The resulting equation is 
 
 j'"'+i i.ij'"'+29.o7i — 1.747=0 (6). 
 
 This must have a root between some of these values: .00, .01, 
 .02, .03, .04, .05, .06, .07, .08, .09. .10. By trial it is found to 
 lie between .05, and .06, thus: 
 
 I -fiiio +29.0700 — 1.747000 (.05 
 + •05 +-5575 +1-481375 
 
 I +11. 15 -(-29.6275 — .265625 
 
 I +11. 10 -(-29.0700 — 1.747000 (.06 
 + .06 -(-.6696 +1.784376 
 
 I +11. 16 +29.7396 + .037376 
 
 Hence 5 is the first figure of a root of (6j, which is the second 
 figure of a root of (5), which is the third figure of a root of 
 (4); whence this root of ('4), correct to three figures, is 3.75. 
 
 We now depress the roots of (6), by .05. The work is as 
 follows: 
 
 1 +11. 10 +29.0700 — 1.747000 (.05 
 •05 -5575 I -481375 
 
 1 +11. 15 +29.6275 
 
 ^5 -5600 
 
 I +11.20 I +30.1875 
 •05 I 
 
 ^65625 
 
 I I +11-25 
 
 The resulting equation is, 
 
 j''+ii.25_i''+3o.i875_r — . 265625. (7). 
 
 By trial this equation is found to have a root between .008 and .009. 
 Whence 8 is the first figure of a root of (7 ), the second figure 
 of a root of (6), the third figure of a root of (5), or the fourth 
 figure of a root of (4), whence a root of (4) complete to four 
 figures is 3.758. It is evident that the root can be determined 
 to 2ix\y degree of accuracy by a continuation of this process. 
 
30 ADVANCED ALGEBRA. 
 
 In like manner we might have found the other incommen- 
 surable roots of (4), either the one between ^3 and — 4, or o and 
 — I. For, an incommensurable negative root can be found if the 
 negative roots be transfoimed into positive ones before applying 
 Harnett's Method. This can be done by II. Art. 44. 
 
 We can obtain the approximate values of the other two roots 
 from the quadratic resulting from removing the approximate root 
 from (7 ). 
 
 I +11.250 +30.187500 — .265625000 (.008 
 .008 .090064 .242220512 
 
 I +11.258 +30.277564 
 
 J'-' +11.258J' +30.277564=0 (8). 
 
 y- +11.258J' +31.695641 = 1.418075 
 whence )'= — 4.439 and — 6.S19. 
 
 But, remembering that these roots are 3.75 less than those of (4), 
 we really have, as the roots of (4), 
 
 j'= — 689 and — 3 069- 
 
 /. Finally, collecting all the values of r found, and remember- 
 ing that .v= Y'zy, we obtain these results as the solution of 
 
 y? — lY-zX^— 4.1'' — 3 1 2 x" + 4 V2 .r + I = o 
 
 J' = — 4, or — I, or +3.758+, or — .689, or — 3. 069 
 
 x= — 2, or — 1-2, or +1.879+, or — .345, or — 1.535. 
 
 4- The Principle of Trial Divisors. The process given 
 above for finding the successive figures of a root would be found 
 to be ver>^ tedious, since each successive figure is determined from 
 among several digits by actual trial. But it wnll be found that 
 after one or two figures of the root are obtained as above, that a 
 suggestion of the next figure can generally be obtained in a ver>' 
 
SOLliTlON OF NUMKRICAL EQUATIONS. 31 
 
 simple wa}-. To illustrate this, consider equation (7) in the 
 last article: 
 
 k''+ 1 1. 25)'' 4- 30. 1875J' — .265625=0. 
 
 We know that v is some number of thousandths phis some- 
 thing. We are determining the figures of the root one at a time, 
 and at present merely desire the number of thousandths. By prop- 
 er transformations in the equation it is evident that 
 
 ^^ -265625 
 
 -f- 1 1.251'+ 30. 1875 
 
 Now, since 1' is known to be a fraction, and less than one hun- 
 dredth, the most valuable lermin the denominator of the fraction is 
 30.1875; for, the higher the powers of y the less account they 
 are when )' is a fi action. Hence 
 
 !■= ■" ^:^ ^ nearlv= .000879-!- (8). 
 
 30.1875 
 
 Whence it is quite certain that 8 is the first figure of j', or 
 the fourth figure of a root of (4). Therefore, in any case, when 
 two or three figures of a root have been obtained, a suggestion of 
 the next figure can be had by dividing the absolute term of the 
 depre.s.sed equation by the coefiicient of j/. 
 
 This is known as the principle of trial divisors. Of course as 
 we find more figures of the root and continually depress the roots 
 of the equation, the smaller r becomes and the more surely can 
 we rely upon the .suggested value. Thus two figures of the ap- 
 proximate value of 1' in (8) are really correct, as will be .seen. 
 
 We now give a more elaborate arrangement of the work of 
 Horner's method as used in the last article. The lines that ex- 
 tend across the page are the divisions between the successive 
 depressions. Follow each line across and the numbers beneath 
 it are the coefficients of the equation with roots depressed. The 
 fifth figure of the root was obtained by the method of trial divisor, 
 i.e. I)y dividing .r)23404488 by 30.367756. 
 
32 
 
 1 
 
 o 
 _3 
 
 3 
 
 3 
 
 6 
 
 3 
 
 9.0 
 
 _-l 
 
 9-7 
 
 •7 
 
 10.4 
 
 ADV.^i 
 
 SIC ED ALGKBRA 
 
 — 12 
 9 
 
 <2, 
 
 — 8 (3-7587 
 
 — 9 
 
 
 — 3 —17.000 
 18 15-253 
 
 
 
 
 15.00 
 6.79 
 
 21.79 
 7.28 
 
 — 1.747000 
 I-481375 
 
 — .265625000 
 .242220^12 
 
 
 
 
 29.0700 
 
 •5575 
 
 29.6275 
 
 . 5600 
 
 — .023404488 
 
 
 II. 10 
 
 05 
 
 1 1 . 1 5 
 
 •05 
 11.20 
 
 •05 
 
 
 
 30.187500 
 .090064 
 
 30.277564 
 .090192 
 
 
 
 30.367756 
 
 
 11.250 
 
 008 
 
 11.258 
 
 .008 
 11.266 
 
 .008 
 
 
 
 5- 
 cedure 
 
 11.274 
 The example i 
 bv which we 1 
 
 n 
 nai 
 
 Art. 3 illustrates the method of pro- 
 / determine the real roots of any 
 
 numerical equation. We briefly summarize it in the following 
 statement of advice for any case: 
 
 Any numerical equation being given: 
 
 a. Transform the equation, if necessary, so that the coeffi- 
 cient of the highest power in/ (.rj is unity and none of the rest 
 irreducible fractions. Query: Why? 
 
 b. Search among the positive and negative factors of the 
 absolute term for integer roots, by dividingy"(.x ), or f ('.!'). by x, 
 or r, minus each factor by synthetic division. Use the nunieri- 
 
SOLUTION OF NUMERICAL EQUATIONvS. 33 
 
 cally smallest values first. Depress the degree of the equation 
 whenev'er a root is found. 
 
 r. When all integer roots are found, test for equal roots, by 
 noting whether the function and the derivative have a common 
 divisor. 
 
 d. Locate the incommensurable roots by II. Art. 33, and 
 approximate them as desired by Horner's method. 
 
 e. As soon as a quadratic is obtained, solve it in the ordi- 
 nary way. 
 
 /". Remember that if the number of the roots found does not 
 equal the degree of the equation, that the rest may be imaginary. 
 
 6. Examples. Find the roots of the following: 
 
 1. a' — 9.1:'+ 23.1- — 15=0. 
 
 2. x*-{-2.\'' — 2 1.1"' — 2 2. r+ 40^0. 
 3- a'— 51-'— i3.v- + 53.r+6o=o. 
 7. .r' — 6.i-''+8.v' — 17-1-+ 10=0. 
 
 5. -i-* + 4.r' — 2.1"'' — i2.v"+9=o. 
 
 6. 2.r' — 7.1-* — 9.i"'+33-v'+ 17-r — 30=0. 
 
 7. .v' — 3 1 J .v'-+ 2.1-1- 2=0. 
 
 8. .v' — 3=0. By Horner's Method. 
 9- -v'— 15=0. 
 
 TO, X' — 3.1-1-1=0. 
 
 / / . 4.1"^ — 1 3.r^ -f- 1 2.r — 30= o. 
 
 r2. 8.r — 25.v'-(- 24.1-— 75=0 
 
 7. ALCiEBRAic Solution of the Cubic, generally known 
 as Cardan's. 
 
 Suppo.se -r'-|-/>,.r'-f-/>,,i--|-/>3=o in the form 
 
 r'-f ^7.1-1- /^=o. II. Art. 42. 
 
 Let x=y+2, then 
 
 That is, /-\-2'+{xv"+a)(y+3') + b=o. 
 
 Now we will make another assumption with respect to r and 
 z. namely that 3i'.e-|-a=o. We can do this, since the only re- 
 striction placed upon the hvo unknowns, y and ,?, is that their 
 sum equals a root of the equation; and upon two unknowns two 
 conditions, not inconsistent, can be placed. 
 
34 ADVANCED ALGEBRA. 
 
 Therefore, /+z^+b=o. 
 
 A 1 • ^ •< ( « ) '^ , 
 
 And, since ^= , y+ -, .• +b=^o. 
 
 Or f+h^—{yiaf=o. 
 
 Whence, since this is a quadratic m terms of y, 
 
 z'=—b—/= - 1 2 b^V{y2br + {haf 
 Therefore, 
 
 A-= { - 1/2 (5 + >/( i^/^)2 + ,.3«):^ } 'o + I _ I . ^— ^[.a)-+ [^af } -' 
 
 Here jir is expressed as a function of the known quantities, a 
 and b. See Art. 2. 
 
 Now, if (ysaf is negative and greater than ( ^jl^f, the above 
 function becomes the sum of the cube roots of two imaginaries. 
 Since no arithmetical or algebraic method exists for finding the 
 cube root of imaginaries, " the roots are in this case presented to 
 us in a form which is very inconvenient for arithmetical pur- 
 poses." For this reason the solution is of but little practical 
 value. 
 
 8. Algebraic Solution of the Biquadratic, generally 
 known as Descartes'. 
 
 Suppose -r*+p^x'+p.,x'+p.^x-\-p=o in the form 
 x*-\-a x'- -hbx-\-r=o. 
 
 Suppose x'-\-ax'-\-bx+c=(x' + Ax + B)(x''+ Cx+D) 
 where A, B, Cand D are undetermined coefficients. 
 
 Then x' + ax- + bx+c 
 
 =x'+{A + C)x'+(B+AC+D)x' + {AD+BC)x+BD. 
 Equating coefficients: 
 
 A + C=o whence A= — C 
 
 B +AC+D=a " B+D—A'=a 
 
 AD-^BC=b " A{D-B) = b 
 
 BD=c " BD=c 
 
 Therefore D+B=a + A'; D—B= ^ ; BD=c. 
 
 A 
 
SOLUTION OF NUMERICAL EQUATIONS. 35 
 
 Finding B and D from the first two equations and substitut- 
 ing in the third, 
 
 Multiplying out, recognizing a product of sum and difference, 
 
 which is a cubic in terms of A' . Therefore A can be found and 
 then B, Cand D from the equations above. Finally, four values 
 of X can be found by solving the two quadratics, 
 .r- + .^r+^=o and .r+Ci-f /7=o. 
 
CHAPTER IV. 
 
 GRAPHIC REPRESEiNTATION OF EQUATIONS. 
 
 r . The Axes. Any point in a plane may be located by a sys- 
 tem of latitude and longitude ; that is, by giving its distances 
 from two fixed or standard lines of indefinite length. These 
 
 Y 
 
 + 3 
 
 X' 
 
 -r=+4 
 
 ■ stands for the adsn'ss(7S or /on<^tV//(/f: 
 stands for the on1i)iates or latitudes. 
 
GRAPHIC REPRESENTATION OF EQUATIONS. 37 
 
 Standard lines are called the Axes, and are distinguished as the 
 .V axis and the y-ax/s. In geography they are not lines but 
 arcs of circles and are known as the Equator and Standard 
 Meridian. In the figure A'A'' is the .x--axis and W is the j'-axis. 
 They are generally, though not necessarily, taken at right angles. 
 Their point of intersection, O, is called the Origin. 
 
 2. Co-ordinates. The distances of a point from the two 
 axes are called its Co-ordinates. The co-ordinate parallel to the 
 .v-axis is called the Abscissa ; the co-ordinate parallel to the j'-axis, 
 the Ordinate. These displace the words Longitude and Latitude. 
 Abscissas are always represented by an x, and ordinates by a y. 
 
 3. Signs. Abscissas, when measured to the r?^/;/, are reck- 
 oned positive, and when to the left, are negative. Ordinates, when 
 measured ///» are positive, and when doivn, are negative. 
 
 4. Examples. Draw the axes and locate the following 
 points. Use any convenient length as the unit of measure; for ex- 
 ample, a half-inch. 
 
 /. (.i-=-2,_i'= + 6) 2. (.v= + i,j=— 3) 3. (.r=o,j'=-2) 
 i. (.v= — 4, )'= I2) 5. (-v=-|-5, j'=o) 6. Locate the follow- 
 ing points and r(V?;/<'YY///<^v;/ /;/ order zcith lines: (-r= — 2, v= — 7) 
 (.v=-i,j'=-5 ) (.r=o,j'=-3) (.r=-|-i,j'= - I) (.t-= + 2,_>'=-f i) 
 ' » = + .v.i'= + 3) <-i"= + 4- J'= + 5) (-^"= + 5. J= + 7)- 
 
 5. When a number of points are to be located by x I y 
 their abscissas and ordinates, these are best given in a 
 tabular form as in the margin; each pair of values lo- 
 cating a point. Those here presented are the same as 
 the ones in the last example- 
 
 Now, it will be noticed that if the different values 
 of .V appearing in this table be assigned to x in the 
 equation 2X — 3= r, that the values of j resulting, will 
 be identical with those tabulated above. Hence we may 
 look upon the above table, and the points located therefrom, as 
 having come from this equation. In the same manner, other values 
 could be asigned to x and the corresponding values of i' found, 
 and the talkie therebv extended. Or, any number of fractional 
 
 2 
 
 — 7 
 
 1 
 
 —5 
 
 
 
 —3 
 
 + 1 
 
 — I 
 
 + 2 
 
 + 1 
 
 + S 
 
 + 3 
 
 + 4 
 
 + 5 
 
 + s 
 
 + 7 
 
38 ADVANCED ALCxERRA. 
 
 values of A" might be interpolated and intermediate values of r 
 obtained. In either case, all the points thus located by the pairs 
 of values would be found to lie along the same line. For this 
 reason the line is said to be a Graphic Representation of the 
 Equation 2x — 3=1'. 
 
 The student will now proceed to attempt the graphic represen- 
 tation of each of the following equations. Enough values of .v 
 and J' are to be tabulated, and enough points located, to satisfy 
 the student in each instance of the character of the graphic repre- 
 sentation of the equation. Each equation should be plotted on 
 paper about 8 X 10 inches, marked with the equation and its num 
 ber and kept for reference. 
 
 Also 
 
 6. Equations. Plot Nos. 3 and 
 
 4 on same a 
 
 Nos. 9 and 10. 
 
 
 
 /. 
 
 7-^— 3=.i'- /• 
 
 X-\-2 = _V. 
 
 2. 
 
 4.1 + 5=)'. 8 
 
 —3.1— 4= I' 
 
 3- 
 
 3-^'— 3= J'- 9- 
 
 — 5-1— 4=.i' 
 
 i- 
 
 3.1 +4= ]'. /n. 
 
 — 5.v+3=j' 
 
 5- 
 
 -\- — 3= 1'. //. 
 
 12.1-1-4 = 
 
 6. 
 
 '2-V+5=J'. /2. 
 
 ax+b=r. 
 
 7. Queries. These queries apply to the above equations 
 of first degree, or to any other equations of the first degree in the 
 form f{x)=-y. They are to be answered by studying the plots. 
 Of course the first conclusions reached will be merely probable in- 
 ferences leased upon the consideration of a limited number of 
 cases; but the student should afterwards endeavor to find the rea- 
 S071 why his statement is true, by which means he will probably 
 detect a rigid proof for his former inference. This will be a valu- 
 able course of discovery for the student. 
 
 The numbers in parentheses following the queries refer to the 
 equations and are intended to direct the student to such plots as 
 will suggest the answers. 
 
 /. What do equations of the first degree represent? 
 
 2. In case the absolute terms of several equations of the 
 finst degree are alike, how do the ])lots compare? {/and ?). 
 (S and 9), (./ and //). 
 
GRAPHIC REPRESENTATION OF EQUATIONS. 39 
 
 J. What does the absolute term tell about the plot? 
 ^. How do the lines compare if the coefficients of x are 
 alike? (j and ^), ig and /o). 
 
 5. What effect has the sign of the coefficient of -V upon the 
 direction of the line? (/ and S), (/, 2, j, /, 5, 6, and S, p, /o, 11). 
 
 6. What effect does increasing the numerical value of the 
 coefficient of x have upon the direction of the line? (5, J, i), 
 (8, g), (7, 6, 5, /, ^, /), (c?, g, 11). 
 
 8. Equations. Of course we might tabulate the values of 
 .r and y from any equation containing these variables, such as 
 .r'-f-.ri'+r"=i9, and attempt its graphic representation. It is our 
 present purpose, however, to confine our attention to the repre- 
 sentation of equations of the form /'(.r)= r, for then we will be able 
 to discuss geometrically many of the properties of /(.r) which 
 were considered algebraically in Chapter II. This is possible, 
 since, in such equations, y is always the value of /'(.i) correspond- 
 ing to a particular value of x, and, consequently, the ordinates in 
 the plot are graphic representatives of the different values of 
 /(-v) . As a consequence the assemblage of ordinates brings to 
 our e>-e the changes in value which f {x) undergoes when x 
 varies. This statement will be better appreciated as we proceed. 
 
 ^3- 
 
 .1"— 2.r — 3=)'. 
 
 18. 
 
 .V-— 6.v+8=r. 
 
 14. 
 
 .t"-4.r-5=i'. 
 
 ig. 
 
 .r' — 6-V+9=i'. 
 
 rS- 
 
 .r'— 6.v-7=r. 
 
 20. 
 
 .r'-— 6.v+i3=r. 
 
 16. 
 
 x' — bx =r. 
 
 21 . 
 
 x'^-^.x—:^=y. 
 
 17- 
 
 A-^-6.v+5=i'. 
 
 22. 
 
 .i-Hox— 3=1'. 
 
 g. Queries. 7. In what respect would you say the plots of 
 all these equations of second degree are alike? 
 
 8. What relation is there between the position of the lowest 
 point of the curve and the coefficient of .r? (/j, 21, 22). 
 
 g. Changing the absolute term has what effect on the plot? 
 {is— 20). 
 
 1 D. Equations- It will be fouud in plotting equations* of 
 higher degree that many times the value of j' are very large com- 
 
 *The wjrd " equation," unless the contrary is stated, is now being used for the com- 
 plete expression " equation of the form /(.») -y." 
 
40 
 
 ADVANCED ALGEBRA 
 
 pared to the corresponding values of x. This would require a 
 very long piece of paper for plotting. Such a requirement can be 
 avoided, however, by using a smaller unit of measure for the j/'s 
 than for the x's; for example, an inch as the unit for x and an 
 eighth of an inch for j; or ten-sixteenths of an inch for x and one- 
 sixteenth fori'; or an}^ such arrangement, the student exercising 
 his judgment in each instance. By this means the essential prop- 
 erties of the curves are retained; the effect is the same as if the 
 curve was plotted properly on a long piece of paper and after- 
 wards the paper shTiink a great deal in the direction of its length. 
 
 A PLOT OF AN EQUATION OK THIRD DEGREE- 
 
 23- 
 
 2.r'— 7.v-' + 4.v-f 4=)'. 
 
 27- 
 
 .v-'+2.i-+3 = i'. 
 
 2i- 
 
 .r-h3.r-f2.r=i'. 
 
 28 . 
 
 .1-'— 3.1 = 1'. 
 
 25- 
 
 .1-'— 5.v-+i = r. 
 
 29. 
 
 x + T,x^y. 
 
 26. 
 
 .r' — 6.v-'+ I i.r — 6=j'. 
 
 3(^- 
 
 .1'+ 2.r— 3.i'— 4.1+ 2=0 
 
GRAPHIC REPRESENTATION OF EQUATIONS. 41 
 
 11. Definition. Any portion of a curve having the same 
 direction as the .r-axis is called an Elbow of the curve. 
 
 12. Definition. Any point in a curve at which the curv- 
 ature changes from convexity to concavity, or vice versa, is called 
 a Point of Inflexion. The middle point of the letter S is such. 
 
 13. Queries. 9 In the plot of J\x)=y what represents 
 the roots of /(.v), or/(.i-)=o? 
 
 10. How is it shown that if different values be substituted for 
 -x-, giving results with contrary signs, that at least one real root 
 lies between tho.se values? 
 
 //. Can every curve of third degree be divided into two 
 equal parts? 
 
 14. Equations. 
 
 ji. x' — 4.r'-)-.i-'-j- 7.1- — 3= r. 
 
 32. x' — .r — 1 3.v-'+.i -f 1 2=y. 
 
 33. x' — ^2.v' — 7.1" — S.v-(- l6=J'. 
 
 34.. 2.r' — -x'' — 9.i" + 33.r-(- 1 7. r— 30=7. 
 33. X' — 3-1-* — 1 6.x -(-48= I'. 
 36. .r' — 5-1 = 1'. 
 3j. .r'+5.v=i'. 
 
 15. Queries. 12. What is the difference between the 
 cur\'es of even and odd degree ? 
 
 13. What shows that every f\x) of odd degree has at 
 least one real root ? 
 
 //. How many times can a line parallel to the r-axis cross 
 a curve representing any/(.v)=j'? 
 
 13. How manj' times can a line parallel to the .r-axis cross 
 a curve of any degree ? 
 
 if). What shows that imaginary roots enter in pairs ? (Note 
 eqs. 15-20 and query 9.) 
 
 77. How are hvo equal roots indicated in the plot ? (19, 23, 
 24, 30). 
 
 18. How many elbows does a curve of given degree con- 
 tain ? (Note 27, 29, 33, 35, 37. ) 
 
 16. Equations. In the following, the derivative of the 
 J\x) taken from a preceding equation (de.signated b}^ number) is 
 
42 ADVANCED ALGEBRA. 
 
 first to be found. This is then to be placed equal to r and plotted 
 on the same axes as the original curve, using also the same unit 
 of measure as was used before. When speaking of the new plot 
 with reference to the old one, it is called the Derivative Curve. 
 Straight lines are to be drawn parallel to the j'-axis through the 
 intersection of the derivative curve with the X-axis. Also, simi- 
 lar lines are to be drawn through the elbovy^s of the derivative 
 curve. 
 
 j8. Derivative Curve to 17. 
 
 jg. Derivative Curve to 26. 
 
 40. Derivative Curve to 30. 
 
 41. Derivative Curve to 32. 
 
 42. Derivative Curve to 36- 
 
 17. Queries, ig- What does the derivative curve tell 
 about the original curve ? 
 
 20. How can you find the number and locations of the el- 
 bows of any curve ? 
 
 18. Equations. 
 
 4J. x' — ]/^x=y. 46. {X — iY=y. 
 
 44. x'—.oix=y. 47. (.r— 1)^=1'. 
 
 /5. X' — o.v^j'. 48. (x — I )•'=_)'. 
 
 19. Query. 21. What peculiarity of the plot where we 
 have several equal roots ? 
 
 20. Equations. Take/(.r) from the equation designated, 
 transform it as required, place it equal to r and plot on the .same 
 axes with the original. 
 
 4g. Equation formed by decreasing roots of (26) by two. 
 ^o. Equation formed by increasing roots of (34J by one. 
 ^i. Equation formed by increasing roots of (32) by three. 
 
 21. Query. 22. What effect has increasing or decreasing 
 the roots upon the curve ? 
 
 22. Equations, not in the form, f(x)=y, which the stu- 
 dent may plot after tabulating the values of .v and )-. 
 
 a. x'-\-y'=2=^. c. 36.a''+iooi''=36oo. 
 
 b. .V-'— )-=i6. d. i'-' = 8.v. 
 
CHAPTER V. 
 
 DETERMINANTS. 
 
 I. lyCt US take two simultaneous equations of the first 
 degree and, from tliem, find the vahies of jf andjv. 
 
 a^x+b^y=c^ (i) 
 
 a.A'~\-li,y^c., (2) 
 
 Multiph' ( I ) by A, and (2) by /',, whence 
 
 a^h.,x+b^b.,y^c^h., (3) 
 
 a.,b^x+b^b,y=c,b^ (4) 
 
 Subtracting (4) from (3) 
 
 ( «| A, — a.,b^ )x=c^b., — c^b^ 
 
 , " " c,b—c,b, ^ . 
 
 hence x=~^ ^ (s) 
 
 a^b.^—a.J)^ 
 
 Now multiply (i) by a., and (2) by a^, whence 
 
 a^a.,x+a.,b^y=a.f\ (6) 
 
 a^a.y-\-a^b.,y=^a^c., (7) 
 
 vSubtracting (6) from (7) 
 
 («, A, — a.Jb^ ) )'= a/; — -a,/, 
 
 , ax., — a.,c. 
 
 hence r= / (8 ) 
 
 It is to be noticed that the denominators in the values of .v and 
 y (equations (5) and (8)) are just alike and contain only the co- 
 efficients of .r and I in the given equations (ij and (2). This ex- 
 l)ression a^b.,—a,b^ iscalled the Determinant of the four quantities, 
 <?p a.,, /?,, A,, and we agree to write it in the form of a square, 
 where the numbers «,, ^,, «'.„ A,, are arranged in the same order as 
 they appear in the equations ( i ) and (2) with a vertical line before 
 and after the square, thus 
 
 Now in order that this shall mean the same as ajj., — a.,b^ it is 
 evident that we must take the product of the diagonal terms run- 
 
44 ADVANCED ALGEBRA. 
 
 ning downward to the right and from this product subtract the 
 product of the diagonal terms running upward to the right. 
 
 Returning now to the value of .r in equation (5) it is seen 
 that the numerator ^-jA^—r/^j is like the denominator except that 
 Tj and c, are written in place of a, and a.,, and this expression is 
 called the determinant of the four quantities r,, b^, c, b.„ and, 
 like the other, may be written 
 
 U. K\ 
 
 provided it is agreed to interpret this square array of numbers the 
 same as the previous square array was interpreted. In the value 
 of J' in equation (8) it is seen that the numerator a^c, — a.,c^ is like 
 the denominator except that c^ and c.^ take the place of b^ and b^, 
 and, just as before, the expression a/, ~^'f\ ^^Y be written in the 
 form of a square, provided the square array of numbers be inter- 
 preted as before. 
 
 The values of .t" and )' written in the notation just explained 
 are 
 
 ''", 
 
 b^ 
 
 c, 
 
 b., 
 
 «, 
 
 b, 
 
 a., 
 
 /^„ 
 
 and r 
 
 I a, ^ 
 
 I '^\ ^\ 
 a., b.. 
 
 Each of these determinants in the numerators and denomina" 
 tors of the values of x and r is expanded (i. c. written out in 
 the ordinary form) by the same rule, viz. : Multiply together the 
 numbers in the diagonal running downward to the right and from 
 this product subtract the product of the numbers in the diagonal 
 running upward to the right. 
 
 The student should carefully notice these values of x and y. 
 In each case the denominator is the determinant obtained by writ- 
 ing the coefficients of x and y in the same order as they appear in 
 the given equations, while in each case the numerator is obtained 
 from the denominator by writing the right hand members in place 
 of the coefficients of the quantity ivhose value is sought. The results 
 in this form are easv to remember and convenient to use. 
 
DETKRMINANTS. 45 
 
 Ex \MPLES. 
 
 \ 3.V— y= 7 
 
 i 2.r+3j'=i2 
 
 Here .v= 
 
 7 
 
 -I 1 
 
 12 
 
 3 1 
 
 3 
 
 -I 
 
 2 
 
 3 
 
 (—12) 33 
 
 2 
 
 3l 
 
 \3 
 
 7 
 
 1 2 
 
 12 
 
 3 
 
 -I 
 
 2 
 
 3 
 
 9— (— 2) II 
 
 36 14 _ 22 
 
 9— (—2.) ~ I I = 
 
 In this manner find the values of .v and i' in 
 
 <^ 3-t'+7J — 22 (3A'+27=i6 
 
 ■ ( 2.i-+5j'=i5 ^- ( 7--^'+5i'=38 
 
 I 3.V— 7)'=io ^ i 3-^'+5,i'=i3 
 
 ( 5-^'— 3J= 2 ^' ( 7-^'+3J'=i3 
 
 2. Definitions. In the determinant 
 
 the four quantities <;?,, (6|, a,, d.,, are called Elements.'^- 
 
 The elements composing any horizoyital line are called a Row, 
 and those composing any vertical line are called a Column. 
 
 When the determinant of two rows and two columns is ex- 
 panded, i. e. written out in the form «,A, — a.fi^ each term is 
 seen to contain two factors and hence the determinant is said to be 
 of the Second Order. When expanded, each term is seen to con- 
 tain one and only one element from each row and one and only one 
 from each column . 
 
 3. Let us now find the values of .v, y and z from three equa- 
 tions of the first degree containing three unknown quantities. 
 
 ^7,.r-f/vr+r, -=«',. (i). 
 
 a,x-\-b.,y^c.,z=^d.,. (2). 
 
 a..x-\-b..y-\-c..z^d... (3). 
 
 * These quantities are called Constituents in Salmon's Modern Higher Algebra, in 
 Chrystal's Algebra, in Aldis' Algebra, in Bnrnside and Pauton's Theory of Equations, and 
 in 'rodhiinter's Tli-ory of Equations, 
 
46 ADVANCED ALGEBRA. 
 
 Multiply (I) by b^c.^ — bj:.,, (2) by — {b^c.^ — b.f^) and (3) by 
 b/,- — A/"j and we obtain 
 
 a , ( b.,c.^ — b/.^x + b^ ( b^c.,^ — b.f.^y^r c^ ( b^c^ — b^c^)z= d^ ( b^c^ — b.^c^ ) (4) . 
 —a.Xb^c.—b.f^)x—blb^c—b.f;)y—clb^c—b^c^z=—dlb^c.—^^ (5). 
 «,(/^jr,— V,)-^i"+Ai('^i^,— Vi)J'+^/'^i^2— Vi)-=«^^C'^i^2— Vi) (6). 
 Adding (4), (5) and (6) we obtain 
 [«, ( b.f.—b.x., )—a,{b^c.—b.f^) + a.Jyb^c.^—b.f^)\x 
 
 =d^{b.f—bf.^—d^{b^c,—b..c^ ) + d.Ab^c.—b.,c^) 
 
 d^{b./..—b./.,)—d.,(b/.^—b.f^)-\-d.ib^c.^—b.f^) 
 
 (7)- 
 (8). 
 and 
 
 a^{ A/... — b./., ) — a,{b^c.^ — b./^ )-\-<^J b^c., — A/, ) 
 Similarly multiply (i) by — (a.f.^ — ^/a^' (2) by a^c- 
 (^3) ^3' — (^^1^2 — '^■f\)^ si^d add the resulting equations and we obtain 
 — d^{a.f.^ — a..c.^)-\-d.^{a^c.^ — a/J — d^{a^c^ — a/,) 
 
 — b^ {a.f^ — a.,c.,) + b^{a^c^ — a/,) — b.Jya^c.^ — a/,) 
 Again, multiply (i) by aj).^ — a.Jb.,, (2) by - — {a^b.— 
 
 (9). 
 
 ./, ), and 
 
 (3) by «/., — «./i, and add the resulting equations together and 
 we obtain 
 
 d^{ajK^ — a.^b.^) — d.^iaj).^ — a^b^) — d^{a^b.^ — «./,) 
 
 z= — ' '■ — (10). 
 
 ^li^iK — ^i^^ — (^2(<^\K — "■ib^} + cj,a^b^ — a.,b^) 
 If we remove the parentheses from the denominators in equa- 
 tions 8, 9, 10, each of these denominators is easily seen to be 
 
 Now this expression (11) which contains only the coefficients of a, 
 I', z in equations (i) (2) (3), we wish to write in the form of a 
 square array where the letters appear in the same order as in the 
 three given equations, if it is possible to interpret such a square ar- 
 ray so as to give the same result as the one here written. We 
 must then, if possible, interpret 
 a, 
 
 SO as to give the same result as (11). The quantities composing 
 the square array will be called Elements, any horizontal line will be 
 
 a^ 
 
 K 
 
 c^ 
 
 «2 
 
 K 
 
 ^2 
 
 «3 
 
 K 
 
 ^3 
 
 (12) 
 
DETERMINANTS. 47 
 
 called a Row and any vertical line a Column, and the square array 
 itself, or its equal (11), will be called a Determinant. As each 
 term of (11) is the product of three quantities, the determinant is 
 said to be of the Third Order. Now it is easy to see that the ex- 
 pression (11) can be obtained from (12) by taking the algebraic 
 sum of all the products obtained by taking one, and only one, 
 element from each row, and one, and only one, from each column; 
 using the sign -f or — before each product, according as the order 
 of subscripts in that product is obtained from the natural order 
 I, 2, 3 b\'an even or an odd number of interchanges of consecutive 
 subscripts. Hence we may write 
 
 «., h., c, =a^b.,c.. — a^b^c.,-\-a.,b.f^ — a.,b^c..-\-a^b^c., — aj).,c^. (13) 
 I «. b.^ ^,. 
 Each of the denominators in (8), (9) and (10) is equal to the 
 determinant ( 12); and, since in (8) the numerator is seen to differ 
 from the denominator only in having «j, a.„ a^ replaced by d^, d^, 
 d.., therefore the numerator in (8) is equal to the determinant 
 
 d, b, c, 
 d, b^ c^ 
 d, b, c. 
 
 As the numerator in (9) is seen to differ from the denominator 
 only in having b^, b.,, b.„ replaced by a',, d„ d.^, therefore this 
 numerator equals the deter minint 
 
 I «, d^ <:, i 
 
 I 'h d, <-. ' 
 
 I a., d.^ c^ \ 
 
 And similarly the numerator in the value of z in equation (10) 
 equals the determinant 
 
 a, b, d^ 
 
 a.-, b., d.^ 
 
 a„ b.. d„ 
 
48 
 
 ADVANCED ALGEBRA. 
 
 
 
 Hence we have 
 
 
 d\ b^ cA «! ^1 c^ 
 
 
 ^1 -^i d,\ 
 
 d., b., c, j 1 a., d^ c^ 
 
 
 a, b, d. 
 
 
 
 d, b.,c.^ 1 a, ^3 c^ 
 
 
 «3 K d. 
 
 
 -^ = 
 
 ^: K c^ ^~ 
 
 aj b^ Tj 
 
 a^ b, c. 
 
 
 
 ^■l K ^2 
 
 fl, /^, r. 
 
 
 a.,_ b, c. 
 
 
 
 «. ^3 ^. 
 
 a., <^3 ^., 
 
 
 a.^ b.^ c.^ 
 
 
 It is to be notic 
 
 ^ed that in these values of x, jk and z the denom 
 
 inator in each case is the determinant formed by taking the coeffi- 
 cients of X, y and z in just the order in which they appear in the 
 three given equations, and that in each case the numerator is ob- 
 tained from the denominator by substituting therein the right-hand 
 members of the equations in place of the coefficients of the quan- 
 tity whose value is sought. 
 
 Examples. Find by determinants the values of .v, y and z 
 in the following equations: 
 
 ( +1-+ 6r— 3.-=i7. 1 x+ j'-h2.-=34. 
 
 -r+ 
 
 35- 
 
 3- 
 
 ( 5a--f-i3)'-f 4.2-=82. 
 ( .r+ 2>/-|-35'= ( 
 - .r+ 3.r+4--= I 
 (2.1-+ 5,'+8,c-=p 
 
 2. - .r+2r-f 5=33. 
 (2.V+ y+ -=32. 
 
 \ -^'+3J'+ 5~=4- 
 ^. - 2.i-+5r+ 3-"=- 2. 
 ( 3A--f 9_rH-io~= 
 
 5, There is no difficulty in extending the process above em- 
 ployed to a greater number of equations, but, after these illustra- 
 tions, we prefer to treat determinants by themselves, independent 
 of their application to a set of linear equations. "^ We will, how- 
 ever, return to the solution of a set of linear equations when we 
 have obtained a number of properties of determinants. 
 
 We have seen how a determinant of the second order, com- 
 posed of four elements, is written: 
 
 I «, b^ I 
 
 I ''. ^ I 
 and defined as being rt, A, — a.b^; also how a determinant of the 
 third order, composed of nine clenienls, is written: 
 
 Linear equations" are eqiiati.ms of first degr 
 
DETERMINANTS. 49 
 
 a., b.^ c 
 a.^ b .^ c 
 and defined as being the expression, 
 
 Similarly a determinant of the 7zth order is composed of n^ quanti 
 ties, called elements and naturally written: 
 ! a, b, c, d^ . . /, 
 
 a., b, c, d., . . /, 
 
 a., h.. c. d.^ . . l.^ 
 
 a, b, c, d, . . /, 
 
 a^^ b^^ r, d^^ . . /_ 
 
 The elements in a horizontal line are called a row and those 
 in a vertical line, a column. 
 
 This determinant is defined as being equal to the algebraic 
 sum of all products that can be formed by taking one and only 
 one element from each row and one and only one from each 
 column, the sign + or — being written before each product or 
 term according as the order of subscripts in that term is derived 
 from the natural order by an even or an odd number of inter- 
 changes of successive subscripts, it being understood that the let- 
 ters preserve the natural order. 
 
 The collection of terms written out according to this defini- 
 tion is called the Expansion of the determinant. 
 
 6. It is to be noticed that the elements are here represented 
 b}^ letters with various subscripts. Obviously other symbols 
 might have been chosen to represent the elements, but the ad van 
 tage of this notation c(msists in the fact that the position of each 
 element is indicated. The letter shows the column and the subscript 
 the row to which any given element belongs. Take for example 
 the element, <f,; since </ is the fourth letter, the element belongs 
 in the fourth column and, the subscript being 3. it belongs in the 
 third row; thus its position in the determinant is completely in- 
 dicated. Of course the position of an element would not be thus 
 
50 ADVANCED ALGKBRA. 
 
 indicated if there should be any disarrangment in the rows or 
 columns of the above determinant, unless we knew exactly what 
 disarrangement had taken place, so that we could allow for it. 
 
 7. Since from the above definition each term must contain 
 one and only one element from the first column, therefore each 
 term must contain a with some subscript; and since each term 
 must contain one and only one element from the second column, 
 therefore each term must contain b with some subscript. In the 
 same wa}' it follows that each term must contain c with some sub- 
 script and so for the other letters Hence all the letters appear in 
 each term of the expansion and no letter appears more than once 
 in the same term. 
 
 Again, from the definition, each term in the expansion must 
 contain one and only one element from the first row; therefore 
 each term must contain some letter with a sub.script i ; and since 
 each term contains one and only one element from the second 
 row, therefore each term must contain some letter with a sub- 
 script 2; and in the same way each term must contain some letter 
 v/ith the subscript 3, and so for the other subscripts. Hence all 
 the subscripts appear in each term of the expansion and no sub- 
 script appears more than once in the same term. 
 
 From this it is seen that every term in the expansion of the 
 determinant contains all the letters a, b, r, . . /with all the sub 
 scripts I, 2, 3 . . «, and if we please we may keep the letters in 
 their natural order and the subscripts will be attached to the.-e 
 letters in every possible order. To illustrate, a dtteiminant of 
 the fourth order 
 
 .?, ^, r, d^ 
 
 \a.^ Ik c < 
 a.. A, ('.. d.^ 
 
 a, b^ (\ d^ 
 expanded becomes 
 
 a^b.f/l^—a , bj\d..^—a , b./.,d ^ + a , bj\d.. -f <? , b/:,d,—a , b^cd, 
 — a.,b^(:.d^ + a.,b^c\d, + aju^d^ — ^"Afi'A — <'■,'''/',''':•,+ ''■/'4'VA 
 -f a.^b^c.d, — (i::b,i\d., — a..b/^d^ + (^A/\^\ + ^^J'A^' ~ ''^A/Ai 
 — a ,b,c,d.. -\- a , b,c,d, -\- a , b.,i\ d — a , />,/■,//, — a , />./-, d., -\- a , bj .,d^ 
 
5^^- 
 
 iry^ 
 
 I 
 
 9 
 
 3 
 
 4 
 
 I 
 
 2 
 
 4 
 
 3 
 
 I 
 
 4 
 
 2 
 
 3 
 
 4 
 
 I 
 
 2 
 
 3 
 
 4 
 
 2 
 
 I 
 
 3 
 
 4 
 
 2 
 
 3 
 
 I 
 
 db:terminants. 51 
 
 In this expansion the signs are determined according to the 
 definition by the order of subscripts. Take for example the term 
 afi.,c,d^, where the subscripts appear in the order 4231. This 
 order can be determined from the natural order as follows: 
 
 Natural order 
 First, interchange 4 and 3 
 Second, interchange 4 and 2 
 Third, interchange 4 and i 
 Fourth, interchange 2 and i 
 Fifth, inte: change 3 and i 
 
 As 5 is an odd number, the sign before aj?.f.^d^ must be — . 
 
 8. The rule for determining the sign of an}' term in the ex- 
 pansion of a determinant ma}' be simplified by noting that the 
 interchange of any two numbers, however far removed, is the 
 sime as an odd number of interchanges cf successive numbers. 
 For suppose any number of numbers, 
 
 1234....//....;;/.... 
 
 Let there be /' numbers between // and in. Then if we wish to 
 interchange m and // we have to pass m to the left successively 
 over the r intervening numbers, and then over the li: we have 
 next to pass li to the right over the r numbers that originall}' sep- 
 arated // and w. In passing in to the left we have made r-f i 
 interchanges of successive numbers, and in passing h to the right 
 we have made r interchanges, so in all there are 2r-\- 1 inter- 
 charges of successive numbers, and this is an odd number what- 
 ever be the value of r. We may then strike out the yNOx^siiccessive 
 in the rule for determining the sign of any term in the expansion 
 of a determinant. 
 
 The order 4231, derived by five succesive interchanges, 
 may be derived by a single interchange of 4 and i. 
 
 9. By using subscripted letters for the elements of a deter- 
 minant it can be expanded with considerable ea.se, but how can 
 the terms be written out when other symbols are used to denote 
 the elements? vSuppose we wish the expansion of 
 
52 ADVANCED ALGEBRA. 
 
 a b c 
 
 d e f (i). 
 
 g h k 
 Now we do know the expansion of 
 
 a, K '-'■z (2). 
 
 a., d^ q ! 
 
 It is a^djC.^ — a^d^c, — a.,b^c.^-\-a.,b^c^ + a^l>^c^_ — a^b./^, and to obtain 
 the expansion of (i) we must of course substitute for each of the 
 elements in (2) that one which in (i) occupies the same position. 
 The expansion of (i) then becomes 
 
 aek — a/if- — dbk-\-dhc-\-gbf^get. 
 A better method will be given further on. 
 
 10. Theorem. The expajisioji of a determinant of the n\\\. 
 order contains 123. .n terms. 
 
 From the definition the terms are obtained by taking the let- 
 ters a, b, c, . . . in their natural order and the subscripts in every 
 possible order. Whence the number of terms is the same as the 
 number of ways of arranging n things taking all at a time, which 
 is 123 . . n. 
 
 11. Theoreij. If in any determinant the rows are changed 
 into colum7is and vice-versa , the determinant is not changed. 
 
 Thus we have 
 
 I fl, 
 
 /;, 
 
 (^ 
 
 d. 
 
 1 '^■l 
 
 b, 
 
 c, 
 
 d, 
 
 • ^., 
 
 b. 
 
 c, 
 
 d. 
 
 ! ^4 
 
 b, 
 
 c. 
 
 d 
 
 b. 
 
 , , . , , I \ d, d] d.^ d^ i 
 
 For convenience represent the first determinant by J and 
 the second by J'. Now it is evident that J' involves the subscripts 
 in the same way that J involves the letters and vice versa. J' 
 equals the sum of all the products formed by taking one and only 
 one element from each row and from each column, hence the 
 terms in the expatision of J' are the same as those o^ J. More- 
 over the signs are alike because the signs prefixed to the te-ms of 
 -I' are determined by the number of interchanges of letters from 
 natural order abed while the signs prefixed to terms of J are deter- 
 mined by the number of interchanges of sub.scripts from the nat- 
 
DETERMINANTS. 53 
 
 ural order 1234, and it requires the same number of interchanges 
 of letters to pass to a given order as it does of subscripts to pass to 
 a corresponding order. 
 
 Thus a.fi^c^d.^ or, what is the same thing, d^a,d./^, is a term of 
 both determinants and it evidently requires the same number of 
 interchanges of letters to pass from order adcd to the order bade as 
 it does of subscripts to pass from order 1234 to order 2143. 
 
 12. From this it follows that any theorem that can be 
 stated about the rows of a determinant can be stated about the 
 columns, and z'ice versa.- for we may write side by side two deter- 
 minants in which the rows of one are the columns of the other 
 and any theorem about the rows of one is a theorem about the 
 columns of the other. 
 
 13. Theorem If hvo nnvs or two columns of a determinant 
 arc interchanged the sign of tJic determinant is changed. 
 
 Consider the two determinants 
 
 I «, /7, ,, d^ 
 
 I a., d., c, d., 
 
 I a'. /;■ c. f 
 
 I ^^ ^ <i '^'. 
 
 Represent the first by J and the second by J', J' being de- 
 rived from J by interchanging the second and fourth rows. Now 
 take any term of J' as a^ l\ c, d.. This is found in J and as in the 
 expansion of -1 the term a., b^ <■; d.^ has the sign +. So in the ex- 
 pansion of J' the term a^ h^ c, d.. has the sign -|- but a^ b^ c^ d^ is 
 also a terra of J and as a term of J it has the sign — . In the 
 same way every term in the expansion of J' appears in the expan- 
 sion of J with an opposite sign; therefore J= — J'. 
 
 General Case. Represent the determinant of the wth order 
 by J and from J form another determinant by interchanging the 
 /v'th and rth rows and represent the new determinant by J'. Now 
 select a7iy term in the expansion of J' differing from the selected 
 term of J only in having the -^th and rth subscripts interchanged, 
 the sign being the same in each case. This term in the expan- 
 sion of J' also appears in the expansion of -I but with an opposite 
 
 and 
 
 b, c 
 
 \ ^U 
 
 b^ c 
 
 \ d 
 
 b.. c 
 
 :. d 
 
 b., i 
 
 •., d 
 
54 ADVANCED ALGEBRA. 
 
 sign, being determined from the previously selected term of the 
 expansion of J by interchanging two subscripts, which of course 
 changes the sign. In the same way every term in the expansion 
 of J' is found in the expansion ot J with an opposite sign. There- 
 fore -l=— J'. 
 
 14. Corollary. If a determinant has tivo rows or two col- 
 umns identical the determinant equals zero. For if we interchange 
 the two identical rows or columns in the determinant represented 
 by J, we get a determinant represented by — J ; but interchang- 
 ing two identical rows or columns cannot change the determinant 
 at all. Therefore J=— J ; 2J=o and J=o. 
 
 15. MikoR Determinants. " If in any determmant we 
 erase any number of rows and the same number of columns, the 
 determinant formed with the remaining rows and columns i^ 
 called a Minor of the given determinant. The minors formed by 
 erasing one row and one column are called first minors; those 
 formed by erasing two rows and two columns are called second 
 minors, and so on." — Salmoyi s Modern Higher Algebra. 
 
 If the given determinant be of the «th order, the first minors are 
 of the {n — i)th order, the second minors are of the [n — 2 )th order, 
 and so on: so we may speak of rth minors or minors of the order 
 n — r indifferently. Minors of the first order are the elements 
 themselves. 
 
 The elementsat the intersection of the rows and columns erased 
 also form a minor of the given determinant called the Complemen- 
 tary of the minor which is left. In any determinant the comple- 
 mentary of an rth minor is a minor of the rtli order. If the de- 
 terminant is of the nth order the complementary of a minor of 
 the order r is a minor of the order ;/ — r, or the complementary of 
 an rth minor is an {n—r) th minor. 
 
 In any determinant there are as many ist minors as there are 
 elements and the first minors obtained by erasing the row and 
 column intersecting in any given element is called the first minor 
 corresponding to that element. In the determinant 
 
DETERMINANTS. • 55 
 
 «:, A3 ^3 
 
 which we represent by J, if we erase the first row and first column 
 there remains the determinant I b.^ c^ I which is the first minor of 
 
 J corresponding to «,, and similarly I Z", <:, I is the first minor of 
 
 I ^ c, I 
 J corresponding to a.,; also I a, q I is the first minor of J corres- 
 
 I ^, ^:i I 
 
 ponding to b^; and soon. 
 
 16. Expression of a Determinant in Terms of the 
 Elements of any Row or Column. 
 
 j^, b^ r, d^ 
 
 I ^2. ^2 ^2 ^2 
 
 I a.^ b^ <:,, d^ 
 
 \<^^ K ^, A 
 
 Represent the above determinant by J. Since, by the defin- 
 ition of a determinate, every term in the expansion must contain 
 some element from the first column, a certain number of these 
 terms will contain a,, while other terms will contain a.^ and so on. 
 Collect together all the terms of the expansion of J which contain 
 (7,, and after taking out this common factor «,, there will remain 
 an aggregate of terms which we will represent by A^, so that a^A^ 
 will represent the algebraic sum of all those terms which contain 
 a,. Referring to Art. 7, we see that 
 
 A=b/.^d^ — b.f^d.^ — b^c.^d^-\- b.f^d.,-\-b^c.^d.^ — ^3^2- 
 
 In the same manner we might collect together those terms in 
 the expansion of J which contain the element a.^ and the algebraic 
 sum of all these terms would be represented by a^A^ and from 
 Art. 7 A.,= — b^cj^+b^c^d^+b.^c^d^ — k.f^d^—bf^d.^-\-b^c^d^. 
 Similarly the algebraic sum of all the terms in the expansion of 
 J which contain a^ would be a.^A.^ and the algebraic sum of those 
 terms containing a^ would be a^A^. 
 
 Now as every term in the expansion of J must contain some 
 element from the first column and if we collect into one group 
 
56 ADVANCED ALGEBRA. 
 
 those which contain a^ and into another group those which con- 
 tain a.^ and into another those which contain a, and into another 
 those which contain a^ ; then in these four groups we will be sure 
 to have all the terms of J. 
 
 Therefore, A=a^A^-Ya.,A.,-^a..A..^^a^A^. 
 
 This is an expression for J in terras of elements of the first 
 column; in a similar way we could have obtained an expression 
 for J in terms of the elements of the second column or any other 
 column or any row. 
 
 If we select the terms in the expansion of J which contain 
 any one of its sixteen elements and, after taking out this common 
 factor, represent the remaining aggregate of terms by a capital let- 
 ter of the same name and with the same subscript as the element 
 we are considering: then J may be expressed in any one of the 
 following eight ways: 
 
 A=a^A^+a.^A.,-\-a.^A,^-\-a^A^. (i). 
 
 J= b^B, + bfi.^ + b,B,, + b,B,, (2 ). 
 
 J= ,-, C, + cC + qC, + r^C,. (3). 
 
 J = d^l)^ -j-dM.^-hdM,-j-d^D^ (4). 
 
 J=rt,.-i, +/7,^",+ f,r, + </>,. (s). 
 
 J=a,A., -f b.,B.+ c, C+d^n.,. (6). 
 
 J=a^,Al + b,B: + 'c,,C,-^d,D,. (7). 
 
 A = a^A^ + bJ^^ + r,C, + <A- C-*^)- 
 
 The explanation is here given for a determinant of the fourth 
 order. It is so evident that the process applies to a determinant 
 of any order that we omit a separate explanation for the general 
 case. 
 
 17. Expression of a DE;TERMrN.\.xT ix Terms of the 
 First Minors Corresponding to any Row or Column. 
 
 As we usually represent a determinant by J, so let u-; repre 
 sent the first minor corresponding to «, by -J«i and similarly repre- 
 .sentthe first minor corresponding to any element by J with that 
 element used as a subscript. 
 
 We will prove that in the eight equations above the factor 
 
DETERMINANTvS. 57 
 
 that multiplies any element is either + or — the first minor cor- 
 responding to that element. 
 
 First, to prove .-J,= -''',. 
 
 The terras of .4 J are obtained from the terms of -I that con- 
 tain the element a, by striking out this element; that is, they 
 consist of the letters ^, r, ^/ in the natural order with the sub- 
 scripts 2, 3, 4 attached to the letters in every possible order. 
 Moreover in those terms of J from which the terms of A^ are de- 
 rived, the element a, stands at the head, and hence the sign is de- 
 termined by the luimber of interchanges of the /ast three sub- 
 stripts. But the terms of -l«j also consist of the letters b, c, d with 
 the subscripts 2, 3, 4 attached to these in every possible order and 
 the sign of each term is here also determined by the number of 
 interchanges of the subscripts 2, 3, 4; hence, ^-4, = J",. 
 
 Second, to prove A.,-^ — ^a,^. 
 
 The terms of A., are determined from the terms of J that con- 
 tain the element a, by striking out this element; that is, they con" 
 sist of the letters /^ ^, i/ vi^ith the subscripts i, 3, 4 attached to 
 these letters in every possible order. The sign of any of these 
 terms is the same as the sign of the corresponding term in the ex- 
 pansion of J and in this term of J it requires one interchange of 
 subscripts to begin with to get the subscript 2 or the element a^ at 
 the head, and so the sign of any term of A.,_ is determined by a 
 number* one greater than the number of subsequent interchanges 
 in the subscripts i, 3, 4. 
 
 The terms of J"?,, also consist of the letters b, c, d with the 
 subscripts i, 3, 4 attached to the letters in every possible order; 
 but the sign of any term of Ja.^ is determined by the number of 
 interchanges of subscripts i, 3, 4 ; hence the sign of any term in 
 J"., is opposite to the sign of a corresponding term in A^_ ; and 
 consequently .-V.,= — -!«,, 
 
 ■■Plus if tliis miinliei' is even and iiiimis if tliis number is odd. . 
 
58 
 
 ADVANCED ALCxEBRA. 
 
 In exactly the same way it may be shown that 
 
 ^3= + 4; 
 
 A = — J^, - 
 
 B,= + J/u , etc. 
 
 Hence we see that the nmltipUer of any element in the ex- 
 pansion of a determinant is either + or — the first minor corres- 
 ponding to that element, the sign + or — being used according as 
 the element is removed an even or odd number of steps from the 
 element in the upper left hand corner ; where, in counting t'-.e 
 steps, we pass along the first row to the right until we are in the 
 column in which the element is found, and then downward until 
 we come to the element, but never pass along a diagonal line. 
 
 Calling that diagonal running from the upper left hand corner 
 to the lower right-hand corner the I^eading or Principal Diag- 
 onal, then the rule just given to determine the sign may be simpli- 
 fied. The sign + or — is used according as the element taken 
 is an even or an odd number of steps from any element in the 
 principal diagonal. 
 
 All this is given for a determinant of the fourth order, but a 
 careful examination of the discvission will show that it is equally 
 applicable to a determinant of the ;^th order. 
 
 1 8. The above gives us a new way of expanding a deter- 
 minant. Take for instance the determinant of fourth order and 
 express it in terms of first minors corresponding to the elements 
 of any row or column; say the first column: 
 
 
 «, ^: ^, ^1 
 
 b, ^■, d., 
 
 J= 
 
 a.,d.,c.,d^ =ci, 
 
 b,c,d. 
 
 
 iicM 
 
 b, c, < 
 
 
 a^ b^ <r, d^ 
 
 
 b., r.. < 
 
 b, c, d, 
 
 b,c,d. 
 
 ^a, b.,c^d., 
 
 b,c,d, 
 
 ■ bj,d, 
 
 b, c, d, 
 b.,c,d, 
 bled.. 
 
 Now we can expand each of these determinants of third order 
 in terms of their first minors in the same way: 
 
DETERMINANTS. 
 
 59 
 
 J=^ 
 
 -a, I b, 
 + a., { b. 
 
 •«. I ^ 
 
 c^ < I 
 c, d., I 
 
 c.dA 
 c, < I 
 c.dA 
 c. d.. 
 
 
 d., 
 d] 
 d, 
 
 < I I ^:, 
 
 < I +'^. I '-. 
 
 < I k. 
 ^1 1 +^' I ^, 
 
 d, I 
 
 d, I 
 
 ^, I 
 ^, I 
 ^, I 
 
 Each of these twelve determinants of the second order we can 
 expand and we then have the expansion of the determinant -J as 
 follows: 
 
 I a J b^c^d^ —a^b.,c^d.^ — a , b.f.^d^ + a^b^c^d.^ -\- a , b^c^d.^ — a , b^c^d^ 
 
 I _ J — ^iPx^id^ + aj)^c^d^ + a.p.f^d^ — aj}.f^d^ — a.p^c^d.^-\-a.p^c.,d^ 
 
 ~ I + afi^c^d^ — a.Jb^c^d^ — a.Jt\c^d^ + ajy,^c^d^ -\- a.b^c^d^ — a.pf.^d^ 
 
 I — aj)^c.4.^ + ajj^c^d^ ■\- aj).f^ d^ — ajb^c^d^ — ajb^c^d., + ap.^c^d^ 
 
 The result agrees with the expansion given in Art. 7. 
 
 By the process here given we can expand a determinant when 
 the elements are represented by any symbols whatever as easily as 
 when the elements are represented by letters with subscripts: thus 
 
 a 
 d 
 g 
 
 =a(ek — hf)—d{bk—hc) +g-{b/~ec) 
 = aek — ahf^dbk-\-dhc-\-gbf—gec. 
 19. Examples. /. Express the determinant 
 
 a b c 
 
 d e f 
 
 ,g h k 
 
 in terms of the first minors corresponding to the second column 
 and expand the resulting determinants of second order and show 
 that the result agrees with that given in the last article. 
 
 2. Express the same determinant, in terms of minors corres- 
 ponding to the elements in the second row and show that the 
 final result is the same as before. 
 
 c 
 
 f 
 
 = a 
 
 e f 
 h k 
 
 — d 
 
 b c 
 h k 
 
 + g 
 
 b c 
 c f 
 
 k 
 
 
 
 
 
 
 
6o ADVANCED ALGEBRA. 
 
 J. Find the value of the following determinants 
 
 2 
 
 3 
 
 
 I 
 
 3 
 
 I 
 
 
 I 
 
 2 
 
 3 
 
 
 I 
 
 3 
 
 2 
 
 3 
 
 3 
 
 
 2 
 
 3 
 
 4 
 
 
 I 
 
 4 
 
 4 
 
 
 I 
 
 4 
 
 4 
 
 4 
 
 4 
 
 
 3 
 
 3 
 
 4 
 
 
 3 
 
 3 
 
 3 
 
 
 3 
 
 3 
 
 3 
 
 20. Theorem. If all the elements of any roiv or column can 
 be expressed as the sum of two or more quantities, then the deter- 
 minayit can be expressed as the sum of two or more determinants. 
 
 Take for example the determinant, 
 
 I C«i + '^^) b, c, 
 («,+/5j b.. c, 
 
 I r^+;j;) b, q 
 
 Expressing this in terms of minors corresponding to the elements 
 of the first column, we get 
 
 I b.^ c, I I b,_ c.^\ ^ I b,, c, I 
 
 ^ ; "■\ \ ^1 ^1 I ~'^-!. I '^i '"i I +"•:! I <^i '"i I * 
 b.. c. b.. c, />., c, 
 
 ( 
 
 b. C| 
 b. c„ I 
 
 \ 
 
 b, c^ I +;-.. I b, r, I ) 
 b-, c, I I ^., C I ^' 
 
 But the quantity in the first bracket is evidently equal to the 
 determinant 
 
 b. 
 , b., 
 
 . b, 
 
 and the expression in the second bracket equals the determinant 
 
 Hence 
 
 I (^ + '^',) b, c, 
 I («..+/5,) b,, c. 
 
 K b. 
 
 c 
 
 
 
 
 /A, b., c, 
 
 
 
 
 K K ^i 
 
 
 
 
 
 '^ b^ ^> 
 
 
 >\ ^ c. 
 
 = 
 
 "■; b^ C, 
 
 + 
 
 K b, C, 
 
 
 a 
 
 A, r. 
 
 
 K b, c. 
 
DETERMINANTS. 
 If the given determinant had been 
 
 ('^.+,^.,4-;- J A, c. 
 then, by what has just been given, this determinant 
 
 («,+'5,) b^ r, 
 
 
 /-, /', ^-, 
 
 (S+/^J b. c, 
 
 + 
 
 /'. A, .; 
 
 («.+/5^) ^ ^. 
 
 
 ;-., b. c. 
 
 and suppl3'ing the value of the first of these from above we have 
 
 ("■, + /5, + r:) b^ 
 
 i";+f^A-r^ b, 
 ("■.+>',+rd b. 
 
 c^ 
 
 
 a, b^ i\ 
 
 ^2 
 
 = 
 
 "■■I b., c, 
 
 c, 
 
 
 'h b-f, c.^ 
 
 i:^ 
 
 b, 
 
 b, 
 ?, b. 
 
 r, A, ^1 
 
 + r, ^, ^-2 
 
 r, b., c. 
 
 and so if all the elements of the first column were the sum of any 
 number of quantities, then the determinant equals the sum of the 
 same number of determinants, the forms of which are evident 
 from the example here given. 
 
 Evidently this peculiarity might have presented itself in any 
 other column as well as the first or in any row. 
 
 A precisely similar discussion would show that if all the ele- 
 ments of any row or column were expressed as the difierence be- 
 tween two quantities, then the determinant could be expressed as 
 the difference between two determinants. 
 
 21. Example. Express the determinant 
 
 2 3 I 
 
 3 3 3 
 
 4 4 I 
 
 as the sum of two determinants in three different ways; find the 
 value of each of the resulting determinants and compare the sum 
 with the value of the given determinant. Also express it as the 
 sum of three determinants, find the value of each and add. 
 
62 ADVANCED ALGEBRA. 
 
 22. Theorem. If all the elements of any row or column be 
 multiplied by a common factor the determinant is rmdtiplied by that 
 factor. 
 
 Take the following determinant, which we represent by J : 
 
 \ a b c 
 \d e f 
 \g h k 
 
 and multiply all the elements in the first column by m and 
 we obtain 
 
 ma b c I 
 7nd e f 
 ?no- h k 
 
 Calling this J', express J and J' in terms of the minors corres- 
 ponding to the first column and we get 
 
 = <7 
 
 c f 
 h k 
 
 -d b c 
 h k 
 
 +^ 
 
 b c 
 e f 
 
 -ma 
 
 e f 
 h k 
 
 — md b c 
 h k 
 
 + mg 
 
 b c 
 c f 
 
 from which it is evident J'=wJ 
 
 Corollary i. If all the elements in any row or column 
 contain a common factor that factor may be taken out of each 
 of the elements and placed as a factor of the remaining 
 determinant. 
 
 Example. Verify the corollary in this determinant: 
 
 I 
 
 2 
 
 3 
 
 3 
 
 3 
 
 3 
 
 r 
 
 4 
 
 4 
 
 Corollary 2. Multiplying any row or column by any 
 number and dividing another row or column by the same number 
 does not change the value of a determinant. 
 
DETERMINANTS. 
 
 63 
 
 23. Theorem. If the elements of any row or eolumn, each 
 multiplied by the same number, be added to or subtraeted from, the 
 corresponding elements of another rozv or column, the deter- 
 minant is not changed.^^^ 
 Take the determinant 
 
 a b c \ 
 d e f I 
 g h k ; 
 
 and add to the elements of the first column the corresponding ele- 
 ments of the second column each multiplied by m and we get 
 
 ( a-\-mb) b c 
 
 I i^d-\-me) e f 
 
 Ig+mh) h k 
 
 Now, because each element in the first column is the sum of 
 two quantities, therefore 
 
 {a-\-mb) b c 
 
 
 a b c 
 
 
 {d-\-me) ef 
 
 = 
 
 d e f 
 
 -f 
 
 (g-\-mh) h k 
 
 
 g h k 
 
 
 b 
 
 b c 
 
 e 
 h 
 
 <^ f 
 h k 
 
 
 a b c 
 
 
 b b c 
 
 = 
 
 d e f 
 
 4- 
 
 e e f 
 
 
 g h k 
 
 
 h h k 
 
 Taking out the factor m from the elements of the first column 
 of the second determinant on the right side of the equation we get 
 
 {a-\-mb) b c 
 (d-j-me) e f 
 {g-\-mh) h 'k 
 
 But the last determinant in the equation has tv,^o identical 
 columns and therefore vanishes, whence 
 
 I (a-\-mb) b c 
 I (d+me) e f 
 ! [g+mh] h k 
 
 *rhe wording of the theorem sliould be carefully noted, for if the elements of any 
 row or column be added to or subtracted Irom the corresponding elements of another row 
 or column multiplied by the same number the determinant is changed. 
 If from the determinant 
 
 ' a h <■ ' 
 
 . d e f 
 
 g h 'k 
 
 we make another by adding the elements of the second column to 111 times the elements of 
 
 the first column we get 
 
 (;« +b) h c , 
 I mil+e ) ,- f 
 { mg^h \ li k , 
 and this is just m times the first one. 
 
 
 a b c 
 
 = 
 
 d e f 
 
 
 g h 'k 
 
a b c 
 
 
 b b c 
 
 
 a b c 
 
 = d e f 
 
 — m 
 
 e e f 
 
 = 
 
 d e f 
 
 g h k 
 
 
 h h k 
 
 
 g h k 
 
 ADVANCED ALGEBRA. 
 Similarly 
 
 {a — mb) b c 
 {d — me) e f 
 ig—mh) h k 
 
 Scholium. — When dealing with a numerical determinant 
 where the elements are large numbers we may combine rows with 
 rows and columns with columns according to this theorem so as 
 to reduce the elements to smaller numbers and thus obtain a 
 determinant easier to compute. 
 
 24. Theorem. If all the elements but one in any row orcol- 
 um.71 be zero the determinant is reduced to one of the next lower order. 
 
 Take, for example, the determinant 
 
 «, 
 
 ^ 
 
 d, 
 
 a.. 
 
 b., 
 
 d. 
 
 a.^ 
 
 b\ 
 
 < 
 
 a. 
 
 h^ 
 
 e, d. 
 
 fl, 
 
 b. 
 
 d, 
 
 a.. 
 
 b.. 
 
 d., 
 
 a^ 
 
 b^ 
 
 < 
 
 a 
 
 l\ 
 
 d^ \ 
 
 a 
 
 J.. 
 
 d., 
 
 a 
 
 '.K 
 
 d. 
 
 Express this in terms of minors corresponding to elements in 
 third column and it equals 
 
 a^ /;, d., a^ b^ d^ a, b^ d^ \ a, /', d^ 
 
 a.^ b^ d.^ — o a^ b.^ d., -fo a, b.. d^ — r, | a., b., d, 
 a^ b^ d^ a^ b^ d^ a. b. d. I a., b., d.. 
 
 which equals 
 
 a, b, d 
 
 25. To Compute the Value of a Numerical Deter- 
 minant. If we have to compute the value of a numerical deter- 
 minant it is well (i) to see if all the elements of any row or column 
 contain a common factor so that as many common factors as pos- 
 sible may be removed in order to reduce the elements to smaller 
 numbers; then (2) to seek, by some combination of rows 
 with rows or columns with columns, to still further re- 
 duce the elements, especially aiming to transform the de- 
 terminant so that in some row or some cohimn all 
 
DETERMINANTvS. 
 
 65 
 
 the elements but one shall be zero, when the determinant is 
 reduced to one of a lower order. We then treat the new deter- 
 minant in a similar way and thus by continual reductions we may 
 find its value usually much more easily than by expanding. 
 Let us compute the value of the determinant 
 
 from last and we get 
 
 Subtract two times first column from second column; also subtract 
 fourth column from third; and subtract fourth column from fifth: 
 
 
 6 12 6 3 9 
 
 
 3 631 2 
 
 
 49412 
 
 
 5 i^ 3 2 5 
 
 
 12 24 6 4 12 
 
 Take factor 3 from first row and 2 fi 
 
 
 2 4213 
 
 
 3 6312 
 
 6 
 
 49412 
 
 
 5 10 3 2 5 
 
 
 6 12 3 2 6 
 
 2 o 
 
 3 o 
 
 4 I 
 
 1 I 2 
 
 2 I I 
 1 I I 
 
 = —6 
 
 5012 
 ,601 2 4 I 
 Subti'act fourth column from first: 
 
 I o I 12] 
 
 -6 ' ' ' '\ = 
 
 2 I 2 3 I 
 
 I 2 I 2 4 I 
 Subtract third row from fourth : 
 I o I I 2 
 
 1 I 2 I I 
 — 12 
 
 I I I 2 3 
 
 I O O O I 
 
 Subtract first column from second; 
 
 ! o I 
 — 12 I I I 
 
 2 1 
 
 3 2 
 
 5 I 
 
 6 I 
 
 I 2 
 
 1 I 
 
 2 3 
 
 2 4 I 
 
 I 
 
 1 2 
 I I 
 I I 
 
 1 1 
 
 I 2| 
 
 1 I I 
 
 2 3l 
 2 4l 
 
 1 1 1 
 
 2 I I 
 I 2 I 
 
 I I O 2 I 
 
66 
 
 ADVANCED ALGEBRA. 
 
 Subtract first row from second ; 
 
 O I 
 
 I 
 
 I o 
 
 o 
 
 I o 
 
 2 
 
 I I 
 
 O 2 
 
 H 
 
 A^ 
 
 A..=- 
 
 
 li. c. d. 
 
 J,„= 
 
 i^: G < 
 
 
 ^ ^\ < 
 
 
 b^ r, ^, 
 
 
 
 b., c. d.. 
 
 
 b\ c\ d\ 
 
 etc. 
 
 whicli is the value of the determinant of the fifth order that we 
 started with. 
 
 26. In article 16 eight different expressions were given for 
 
 the determinant 
 
 \ a^ /?, r, d^ 
 
 1 a., b., c, d., 
 
 I a^ b.. r., d^ 
 
 each in terms of the elements of some row or some column, and it 
 was noticed that 
 
 Keeping carefully in mind the meaning thus given to the cap- 
 ital letters with various subscripts it is evident that 
 
 b^A^ + b.^A.^+b.^^A.,_-^b^A^ 
 
 I b, (\ d, I b^ r, < I I /;, r, ^, I 
 
 -b., I b., q d.. I -\-b.. b.. c. d., -bA /'„ r, d., \ 
 I b\ c^ d\ I ■■ b\ f] d\ ! I b.._ a d[ I 
 
 and this is evidenth^ the expression of the determinant 
 I /7, b^ r, d^ 
 I b.. b., c. d.. 
 I b, b.. <■.. d. 
 I '''. ^', '\ < 
 
 in terms of the minors corresponding to the elements of the first 
 column. Now this determinant, having two identical columns, 
 equals zero; hence 
 
 b^ A^ + b.^A.^+ b.,A.^+ b^A=o. 
 
 In the same wav we could obtain a relation connecting the ele- 
 
 
 b.. c, d., 
 
 =^ 
 
 b., c. d.. 
 
 
 b. c\ < 
 
DETERMINANTS. 
 
 67 
 
 ments of any row or column with minors corresponding to the ele- 
 ments of some other row or column. There would be in all twen- 
 ty-four such relations given by the determinant 
 
 rt, 
 
 l\ 
 
 c^ 
 
 < 
 
 a., 
 
 i 
 
 c.. 
 
 d. 
 
 a.. 
 
 Ik 
 
 c. 
 
 cl, 
 
 '^^ 
 
 ih 
 
 r. 
 
 d, 
 
 In the same way if we were given the determinant of the ?zth order: 
 
 a, b^ c\ . . . . /, ; 
 «2 b., r, . . . . /, 
 a., b., c, . . . . l. 
 
 we could obtain 2;/ different expressions for it, each one as multi- 
 ples of the elements of a row or a column and we could obtain 
 2n{,n — i) other relations connecting the elements of one row or 
 column with the minors corresponding to the elements of another 
 row or column. As samples we write two equations of each kind 
 and leave the student to write others. 
 
 a.,A.^+b.,B.,+c.,C., 4- 
 a..A.,+ b.B.,+ cC + 
 
 + a^A~A. 
 
 a^B=o. 
 l.L=o. 
 
 Examples /. Write the six different expressions for 
 
 and verify each expression. 
 
 2 Write the twelve other equations expressing the relation 
 between the elements of one row or column and the minors corre- 
 sponding to the elements of another row or colunui of the determi- 
 nant in E.K. /. 
 
68 ADVANCED ALGKBRA. 
 
 J. Express the value of 
 
 I a h r I 
 
 I <^ ^ f I 
 
 I K h k I 
 in six different wa^-s. 
 
 4. Write the twelve other equations expressing the relation be- 
 tween the elements of one row or column and minors correspond- 
 ing to the elements of atiother row or column of the determinant 
 
 in Ex. 3. 
 
 28. Application TO THE Solution OF Stmut.taneous Equa- 
 tions OF First Degree. 
 
 Let us take ?2 equations of the first degree containing « unknown 
 
 quantities. 
 
 a^.x-\-l\y-\-c^z-\- . . . .-f/|T'=w,. 
 
 a.^x-\-b.^y-\-c.,s+. . . . -{- /-c>= ff/ ... 
 
 Here we suppose that the determinant formed by the coeffi- 
 cients of the unknown quantities, viz: 
 
 a^ b, (\ . . . . /, i 
 a.^ b, c, . . . . /,\ 
 a., b., c, . . . . /., ! is not zero. 
 
 «„ "„ ^„ • ■ • • ^„ 
 Multiph- the first equation by A^, the second by A.,, and so on, and 
 we have 
 
 a.,A.^x^b..A,y+c.,A.,z^ .... -\-/.,A.,v=7n.,A,. 
 a^,A^,x+li^A^y-\-cA,z-{- .... -j-LA..y=nL,A,,. 
 
 a„ + Ax+b^A^y+c„A^2-^ . . +/^Av=w^A,,. 
 
 Then adding we obtain for the coefficients of a- the determinant 
 
 of the coefficients of all the unknowns, which we will represent 
 
 by J; the sum of the coefficients of each of the other unknowns 
 
 become zero (see Art. 25 >; and the right-hand meml)er is what J 
 
DETERMINANTS. 69 
 
 becomes when the a's are replaced by the n/'s, i. e., by the right 
 hand members of the given equations. Let us represent this 
 determinant by J,. Then 
 
 •J, = -',; therefore .v= 
 
 J, 
 
 In the same way, if we multiply the first equation by B^, the 
 second by B.,, and so on, and add the resulting equations, we get 
 
 __ J.^ 
 -' T' 
 where J.^ means what J becomes when the d's are replaced by the 
 right members of the given equations. 
 
 Again, multiply the first by C,, the second by C, and so on, 
 and add the resulting equations, and we get 
 
 J 
 
 where J., is what J becomes when the r's are replaced by the 
 right members of the given equations. 
 
 It is now evident that the value of an v unknown quantity in the 
 given set of equations is the ratio of two determinants, in which 
 the denominator is the determinant of the coefficients in the given 
 equations and the numerator is what the denominator becomes 
 when the right hand members are put in place of the coefficients 
 of the <7A (?;////!■ -a'/iose value i^ soiio/il. 
 
 29. Another Method. 
 Form the determinant 
 
 I (rt,.i-f- /',.)'+ . . +/,'■—'",) ''', i\ ■ ■ A 
 
 i (a.,x-\-h.,y-\- . . +l.,v-7n.,) A, C . • L, 
 
 I (a^x+bj'-\- . . Iv—m^^) b^^ r^ . . /„ 
 
 Each element in the first column is formed by transposing the 
 
 right-hand members of the given equations, and hence each of 
 
 these elements equals zero, therefore the determinant itself equals 
 
 zero. Now each element in the first column is expressed as the 
 
70 ADVANCED ALGEBRA. 
 
 algebraic sura of ;/-|- I quantities, hence the deterrainant can be 
 expressed as the sum of «+ i determinants. Whence 
 I a^.v b^ r, . . /, I I b^y b^ c\ . . l^\ I w, b^ l\ . . /, 
 
 I ^''2-^" ^-l ^l- ■ ^A _|_ I '''•-'J' ''•'2 "^2 • • "^2 I 1 I '^^2 ^2 ^-l- • ^' 
 
 1 ax b,^ r„ ..". /, I I b^j b] c[ . . /„ i I m„ 'b,^ r,. ." .' /„ I 
 
 All these determinants, except the first and last, vanish; for after 
 taking out the common factor from the first column we have left 
 a determinant with two identical columns. Moreover after taking 
 out the common factor x from the first column of the first deter- 
 minant we have left the determinant J and the last determinant is 
 evidently what we have called J,; hence J.v — -11=0, orJr=J,, 
 or 
 
 the same as before. Similarly the other unknown quantities may 
 be found. 
 
 30. If the determinant called J should equal zero, we cannot 
 obtain a definite, finite set of values of the unknown quantities. 
 If all the numerators are also zero, the unknown quantities are 
 indeterminate and the equations given are not independent. If 
 however, some of the numerators are not zero then the equations 
 cannot be satisfied by any finite set of values, or the equations are 
 said to be incompatible. 
 Thus the equations 
 
 2.1- -f j'4- 55-= 1 9. 
 
 3.r+2_r4- 42'= 19. 
 
 7 r+4j'+ 125=49. 
 are independent and compatible and therefore form a solvable set 
 of equations. In this example -1= — 9 J, = — 9, J,= — 18, 
 J.,= — 27, whence .1= I, r=2, z=2)- 
 The equations 
 
 2r+ _)'+ 52-= 1 9 
 
 3-V+2i'-f 4z=ig 
 
 7.1- 4- 4_v+ 145=57 
 
DETERMINANTS. 
 
 are compatible but are not independent, the third being derived 
 from the other two by adding the second to twice the first. 
 In this example J=o, -l,=o, -l.,=o, -';i=o, so that the values of 
 
 each unknown quantity assumes the indeterminate form . 
 
 o 
 
 The equations 
 
 2.r+ 1'+ 5.3-= 1 9 
 
 5x-\-2j'-{- 45-= 1 9 
 
 7X+4J/+ 145=50 
 
 are incompatible with one another. If we add the second to twice 
 
 the first we get 
 
 7.r+4j'+ 145=57 
 but this contradicts the third equation. In this example J=o, 
 -I, =42. J.,= — 49, J.^=— 7, so that no finite values of x, y, 2 sat- 
 isfy the equations. 
 
 30 IvCt us now take n equations of the first degree contain- 
 ing « — I unknown quantities: 
 
 a.x +b,v-\- c,3-\- .... 4-/,=o 
 
 ^n^'+^ny+ ^n^+ .... +/ =0 
 
 The absolute term is written on the left-hand side of the equations 
 because it is better for the method here pursued. It is to be noticed 
 that the number of equations is one more than enough to enable 
 us to find the values of the unknown quantities. Represent the 
 deteirainant of the known quantities by J, whence 
 \a,b,c^ . . . . /, 
 i_ ^.. b., c^ . . . . l, 
 
 I '^n ^n ^n ■ ■ ■ ■ ^„ 
 
 Now if we add to the last column x times the first, jy times the 
 second, z tiraes the third, and so on, the determinant is not changed 
 in value; therefore 
 
 b^ c^ . . {a^x-j-b^y-\-c^3-{- . .-(-/,) 
 
 b^^ r, . . {a^x+b^^y-\-c^z-\- 
 
 ■0 
 
72 ADVANCED ALGEBRA. 
 
 But from the given equations it is evident that every element in 
 the last column equals zero. Therefore 
 
 J=o. 
 
 We have tacitly assumed that the given equations were compat- 
 ible with one another and have shown that the determinant equals 
 zero; whence the following theroem: If n equations of the first 
 degree containing n—\ unknown quantities are compatible with 
 one another the determinant of the known numbers equals zero. 
 But this determinant may be zero ^^hen the equations are incom- 
 patible, as in the set: 
 
 2.1-1- J'-f- 45- 16 = 
 
 3,1--!- J'-f 22 11=0 
 
 8X-I-3V+ 8.S-— 38 = 
 7X-|-3T'-|- \oz— 40=0 
 
 Here J= 
 
 Subtracting twice the second row from the third we have three 
 identical rows, hence J=o. But if two times the first equation be 
 added to the second we get 7-i"+3)'-(- io5 — 43^0, which contra- 
 dicts the fourth equation, whence the .system is incompatible. 
 
 Let us see if we can tell when several equations are compat- 
 ible. For the sake of definiteness in language let us take four 
 equations, 
 
 a.^x-\- b.^y-\- c,z-\- d.,=o 
 a.p: -f k.^ y -\- c.z -{-d.=o 
 a ^x-\- .\y-\- r^z+ d =0 
 
 and suppo.se three of the.se, as for instance the last three, are in- 
 dependent and compatible. Then from these three we find the 
 value of .r, )', z to be 
 
 2 I 4 
 
 — 16 
 
 3 I 2 
 
 — 1 1 
 
 8 3 8 
 
 -38 
 
 7 3 10 
 
 —40 
 
DETERMINANTS. 
 
 73 
 
 < K c. 
 
 \a.,-d,c.,_ 
 
 
 a,b,-d. 
 
 -d., b. c. 
 
 a,-d,c. 
 
 
 a,K-d. 
 
 |-< b^ c, 
 
 l«4-<^. 
 
 
 «4 b, -d, 
 
 
 
 
 . 'V = 
 
 «. 
 
 ^ 
 
 ^2 
 
 
 a. 
 
 b., 
 
 ^3 
 
 1 
 
 a^ 
 
 b^ 
 
 c^ 
 
 
 ' a., b.^ c^ \ <^-, b., r, 
 
 a^ b^ c^ ' «. b, <r. 
 
 Changing numerators so that the column of a'' s shall be the last 
 column in each case and taking out the factor — i if it occurs 
 we y-et: 
 
 b, c, d., 
 
 
 fl'j c^ d.. 
 
 
 a.^ b, d., 
 
 b, r, d., 
 
 
 «3 ^:i d.. 
 
 
 a, b, d. 
 
 b, c. d, 
 
 
 a, c. d^ 
 
 
 ^4 b, d. 
 
 ^., b„ c, 
 
 a, b., c, 
 
 , ,i 
 
 a, b^ c. 
 
 a., /'., r.| 
 
 a, b.. r. 
 
 
 a.^ b, c. 
 
 ^4 b^ c^ 
 
 
 «, b, c^ 
 
 
 a^ b^ c^ 
 
 These determinants are easily seen to be minors of 
 
 I «. b, r. d, 1 
 
 I *^i b-t i\ ^2 I 
 
 I «:, b., c.^ d.^ I 
 
 I «4 b, r, < I 
 
 Representing this by J , we may write according to a previous 
 notation, 
 
 
 Now if these values be substituted in the first of the given equa- 
 tions we get 
 
 
 + 
 
 <'',J/ 
 
 r,J, 
 
 -J<A 
 
 + «', 
 
 but this is evidently J; therefore J=o. Hence if the values of x, 
 y, z that satisfy the last three equations also satisfy the first then 
 
74 ADVANCED ALGEBRA. 
 
 -"=o, but if the values of x, _r, z found do not satisfy the first equa- 
 tion then J is zero. 
 
 Our conclusion may now be stated in general language as 
 follows: If we have n linear equations containing yt — i un 
 known quantities and if some n- i of these equations are indepen- 
 . dent and compatible then the determinant of the known numbers 
 equated to zero shows that the n equations are compatible. 
 
 30. If we are given four homogeneous^ equations of the 
 first degree containing four unknown quantities, as 
 
 X _y z 
 we may divide by zc and then consider -, -, - 
 
 It.' 7f w 
 
 the unknown quantities and we have the case previously con- 
 sidered. Hence if we have n homogentious equations of the first 
 degree containing ;« unknown quantities and some ;/ — ^i of these 
 equations are independent and compatible, the determinant of the 
 coefficients equated to zero shows that the // equations are com - 
 patible; or thus — if the determinant of the coefficients equals zero 
 the equations are compatible provided some n — -i of them are in- 
 dependent and compatible. 
 
 Homogeneous equations may always be satisfied by making 
 each of the unknown quantities zero, but this solution is ex- 
 cluded and the determinants of the coefficients equal to zero shows 
 that the equations are satisfied b}- values other than zero provided 
 some n — i of the equations are independent and compatible. 
 
 In the apphcations of this theorem (which are numerous) it is 
 as.suraed without statement that some w-i of the equations are inde- 
 pendent and compatible and the usual statement is that if the deter- 
 minant equals zero the equations are compatible. 
 
 * The word "homogeneous" is here used in its strict sense, meaning an equation whose 
 terms are of the same degree and Zf/VA «oaA.?o/«/f /crw. Thus u i-+Av--o is homogeneous, 
 but ax~by ^c^=j) is uol homogeneous. 
 
vA \ 
 
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