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ALGEBRAICAL EXERCISES
— AN'l) —
EXAMINATION PAPERS
FOR PUBLIC SCHOOL LEAVING AND PRIMARY
EXAMINA TIONS
HY
C. A. BARNES, M.A.
Inspector of Schools, Lanihtov.
TORONTO :
THE COPP, CLARK COMPANY, LIMITED.
KiiUjred accortiiiig lo Act of the Purliaiiieiit of
iJarittda, in the year one thousand eight hundred and ninety-seven, by
TiiK C\)i'i', Clauk Oomi'any, Limitkd, Toronto, Ontario, in the Office of the
Minister of A^^riculture.
PREFACE.
This book of Exercises \v. Algebia h.is l)uen prepared to supply
a want felt by many teachers who are teaching elementary Algebra,
and for private students. The ordinary text has not a suflicient
number of Exan:ples to enable students to become thoroughly
familiar with the Principles of Elementary Algel)ra, and expert in
the best methods of solution. The aim of this book is to supply
that deficiency, and the Hhih giviju in the Answers on the methods
of sohition, it is believed, will prove helpful to many private
students, and enable them to lay a good foundation for more
advanced work.
The book is particularly intended foi' students wh.. are |)reparing
for either P. S. Leaving or Primary Kxaminationa.
THE A I THOU.
f I
EXERCISES IN ALGEBRA.
r*
EXERCISE I.
ADDITION.
1. 7a + 5/) + 3c ; 9a + (ih + oc ; 14a - 7h - Hr ; 3/) - Or ; 4a + 56 .
2. 8<(x-7hy + '3i/; ax' + 2hij-7if; 9ax + 4/-2; 3/>v-2w' + 7;
4aa!;-i/ + 6(ta!'.
3. ia-lb + lc; ]a-l/)-|c; ||a + ]6 + Jo ; 6-a + c.
4. 5a - 26 + 3o - 4d ; 36 - 4c + 5d - 2a ; oc - 6rf + 3a - 46 ; 7d - 4a
+ 56 - 4c.
5. p + <y-r;-(f/ + /--.s); r+p-,s',-(p-s-r-(i).
6. 2a^ - 3j-!/ + 4mn ; ^nm + '3xz + 7xy ; Smn - br^ + 2ap - 4ap -
4xy - 12 mn. '
7. 4a -106 + 13c -2^/; a + 66 - 14c + 5(/ ; 3a - 176 + Oc + 14cZ ; a +
146-c-17(/.
8. 4((,' + 6') + 37(a6 + ar) + 7(a^ + 6'') + 9((r6 + «,■) + d(nhc + a%-'c^) -
9. Sx + y+z;-7x-4y + Uz; 4z-x-,j; llx + y-z.
EXERCISE II.
SUBTRACTION.
1 . From 17a - 13ic + 27 take 15a - llx + 9.
2. From 4a"'' - 7xy - 7i/» + hz" + 13m - 11,
take 3aH7a-y-9y3+ ;;2 + 12m-lL
3. From
take
^a-7c^^xf-7^U-h\
12a + 7c-9a;y2 + GVa-62.
4. From;)H 3*/2+ll/-=' + 15/>^/-7y- 1,
20r/ + llrH17i>'i-7i/-lCK).
take
EXERCISES IN ALOKBRA,
5. What, fixproRsion must, betaken horn 2o'^ - ]^an> + Cmh^-2h''
to leave a"* - l7aVj - 'Adh'^ {
6. From fa^'- Jxt/ - •},)/,
take p-'^+' w'lf-'yi/.
7. From the sum of I4u + [)b Vh and 6a + 5/> - Ur take their
difference.
8. Subtract 3x^ + i.t'^y ~ 7. a/ + 20,/ from 4.r' -2jr:'y-{-4.nf+l4,/.
9. From |"'-ga+f,
take i - 3't- J.
10. Subtract the sum of ^^a-i^c and ic + J/>- «« from the sum of
tc-aand ^26-^c.
EXERCISE III.
ADDITION AND SUBTRACTION.
1^ ^:K'^*^'\iH~^+x' 2« + 2/> - 3c ;- 3a + 4/> + 14r together and take
14a + 56 - 15o from the num.
2. Add 12-^^4-.^^ 3.T-14-13x^ 15 + lLV^ + 7x - .r^' and fnmi
their sum take 11 - llx - 3x\
± ^^A^'W'^o}^^^^' 16-?/ + 18rxi;r; 15J/-6.,. 20f5^and
4^-14*/ + 9x-20 together and take 5x- ll./-5.~ + 2,^^^,- 15 from
the sum.
4. What quantity added to d^ - b^ will give j-''+ if^ /
5. If X = 2a + Sb', y=2a--2b', z = h- 4a, show that 3x + 81/ + 4z =
14a — 36. •
6. Add2a'^-62 + |; 3a^ + b'--; 6 '^ - a^+g together and from
the sum subtract 4a'- + b- - c'\
7 \W-\- ^- ^!l f "' *"' "' ''■'
'• ^^^ 4'*'3"2' 2"'"4~'>' 2"^4~^ and from the sum take
b' c^
2+4*
8. Frcmi
a^ + /,
take
a' - b''
% Add 13a+15/>-c; 2a-146 + .; | - 10a + | and from the
a b c
sum subtract o - .-j +n •
^ J o
10. Add a{a + b-c); b(b + c - a) ; c{a + c~ 6).
MULTIPLICATION.
EXERCISE IV.
Iloiiuivu bnickwts and collect like terms.
1. a-h-e- (a + 3/< - c) + 2rr - (/» - 2r + a - h - ,•).
2. 2(t - a; + c - (3a - j:) - {^x ~ 5a) + 3 - 2x - (a - .r) + Wx - 2.
3. a--{ 2/>-(3c + 2/^-(f) |-.
4. 2a -6-^ _(c_rf)-(_2a + /> + f0 }•.
5. '^a-{a-h-c)-2->, a + r-2{h~c) )..
6. 3a-[rt + 6--{ a + h + c-{a + }>-irc + d) \-].
7. -{ a-(6-c) I- - '\ h-(c-a) \- - J ,'-(a-h) y-(„^h + r).
8. 2a - [3h - -^ - 2o - (2a - 2e^>) ]■].
9. Up- ^ :iq + (2r-^r7];)-Hq \- -[4r+-^ 3.s-(2/> + 7 ;-].
JO. 4[a + 3 -( h - c - 4(d -e +/))•]- [3a + 4(3/> - 3c - I2d + 12t'
12/")].
EXERCISE V.
MULTIPLICATION.
A.
1. Multiply x'^ - 2xy + >/ + .'• - y by x - ij.
x'^ + y'^ + z'^ + xi/- x:: + i/;; by .*• - >j + ;;.
1 + 2a; + 3x2 + i\x^ by 1 - 2x.
IGu^* - ^x\j + 4xh/ - 2x,/ + >/ by 2x + ,j.
a^ - 2ab + h"^ + cM)y a' + 2ah + ¥ - r«.
1+ ic 4- .'-'^ + .v^ + .'■'* + X-' + .1'" l)y 1 - ;>;.
.*•» - 3xhi + 3aa- - a'' by ..•" + 3.v2a + 3xd' + "-.
.!•'"+</" by .r"' + j/".
a3-a2ft + <r//-'-/rM)y a+/>.
B.
2.
3.
4.
5.
6.
7.
8.
9.
10.
II
II
II
II
1. Multiply (f< + c)- + 2(a + <0 {h-\-d) + {h + <l)- by a + }> + i- + d.
2. M («*-2a^ + 4 l)y a* + 2u'''-4.
3. „ 9/^ + 4(/^ + ,r-2t/2 -3.0 (ixi/ by 3.^ + 2;/ + ;.
4. M !-«/-;+ if- - ijz + z'^ by 1 -f ;/ + 2.
KXKItCIHKH IN ALOEUKA.
5. Multiply <<.r-f aV-f-aV hy 1-nx.
0. II a* + 2aV>' + h* hy o* - 2(tV>^ + h\
7. .( tc» + 1/» + s'^ - 2(/-; hy x^ - y' - ;:» + 2yz.
8. M «"+a;" + a;< + j''Hl l)y u-'-l.
^' " fi + nix - nx' hy <i - ynx + nx''.
10. Find tlio iM.ntinued i)r()ducfc of x + Z - j;-" • x^a."* . -r«a. '^*
c.
1. Mult iply ,r« - .,.*(/* + i/« |,y ,,.< 4- ,,4_
3. Multiply yV (,,4-/,^,,. + ,,/, ],y .r^ + („+h)x + „h •md cx-itninn
What tho product hccon.e.s if in ,t either a or 6 is suhst^lLl C .
^^iv+ii'-uur::^-;;'''''';^ '"^"' -••^'pi-tion ti. product of
5 Multii.ly x + a~h hy x-a + h and show that tho product is
zero if the ddlbrence between a and b is e.iual to ;.. ^
6. Find the product of (,*• - n) (x - b) (x - c).
(x + iHlTkT-2)' '' ^"'^"'' «^^-"«i«ient ,>f x in the product of
^.^9. Multiply (..-3) C.,--5) (,.-7) using the principles of number
dn!:?'fl^"^'''i^ (■'■ + '') (.'•+/0 (r + r) x + <}) and deduce from the pro
cbict^the coethcient of . n, the product of (. + 2) (. + (5) (I^IO)
EXERCISE VI.
DIVISION.
A.
1. Divide 2{)f(3_i5,t2.^2r)a by 5a.
2. .. ^i*-2tiU' + x*hya' + 2ax + x\
3- •• ^-' -•'■* + .'-''-.'- + 2.r-l},yu.'H^;-l.
4. II y'^ + a.'' + bx + ah by ;*• + a.
5. I. ./•■■' - {it + 2)j;' + {%(. + b)x - 2/> by .1- - 2.
6. .. i<''~y'-e' + 2bchya-b + c.
DIVISION. 5
7. Divide *24.i» - mxij + 2 1 ,f },y ',\,r - 7y,
8. .. j'*+i(».i"''+;{rM"'+r»().i'-f24i)y x+4.
9. t - x* + '.h'hf^ + 2 If* I )y ,,••' + 2>/».
10. 1 1 j'^ + ;Ja.i;a -f. 3(,Vj. + a" + ^M)y a; + rt + 6.
B.
1. Divide (a + />)''-f 3(rt + hy + 2r' hy a + /> + c.
2. Slunv tlmh /. + :{.,.'-,.a_^2_2 j.^ aivisihlo hy r + 1 niul .; - 1
and writo down tliu (luotieiit. ' '
3. Write <lovvn the (|U()tient of ,. •'' + H/za - 27vT + 18.,.»/,^ hy .,• + 2v -
3^ Hndto«t the result hy putting .. = 5 ; «/=-4; . = 3in divis<.r,
dividend and (juotient. '
4. Divide (« + />)'^ -(,, + /,) (,; + ri)-(J(c + (/)-^ hy a + 6 + 2(o + </).
5. Divide tlie product of d' + ax^-x'' and aH^'^ hy a^ + aV + x*.
0. Divide J? + (a + /> + r)./^ + {ah + ac + hi')x + ahr hy ..- + a.
7. Tf the dividend he ^n'h' + 2{[^u' ~ 2}A)- ahVod'-llh') and the
• luotient 2a{u + h) + {n^ - h% what in the divisor ) '
8. By what must a' + n'^ + l he multiplied to make it aHa^H-l ?
0. 'I]he i)roduct of two algehraical exprt'ssionH in ^•fi + .'\+ iV-
tK»!/« + //- and one of them is ,,;'^ + .,,.j/+j/^ ; ^hat is the otlier )
10 What value of a will make Qx*-2x' + 2ax^ + 2x + a exactly
divisible hy x^ - j- + 1 ? •'^
C.
1. Find the value of a and /> so that y^ + ;V,/ + 4»/Mnay exactly
divide ..•H7r^^»/ + <M■^-/•^ + 5.1 V + r^.•■-y + /;..■,/■Hl2/A ^
of m?^^ •^•' + 15-^ + »'i is exactly divisihle hy a; + 7, what is the value
themis2.o + 5(/ + 3;:; hnd the other.
r •'i' ^l"!^''?^/f "•';+^' 'V + '-' + l '"1^^ .*^*-x' + l together, and
divide ./;'« + . ';H 1 hy the product.
5. Divide the difference hetween (2(t + 3/>)'^ and C:la + 2hY by
o(a + 6). /J
6 Three factors of ^c* - 4/>x' + 0/> V - 4/rV + //' when multiplied
together give {x - hf as product, Hnd the fourth factor.
7. Divide the product of (a + /._r); (a-/> + r); (h + c-,i) hy a^
EXKRCISES IX ALGKBRA.
8. Divide tl*e product of 6x' - I7ax + \2a' and ix - 5- by 3.r - 4a.
0. Divide 49.r2 - 16-2 + 21xy + 12yz by 7x + 3y - 4z.
10. By wh.'i.; expression must a'~hr be niultip/lied that the
1 )roduct may be a» + a^b + a \- - abc - hV - he' ?
EXERCISE VII.
MISCELLANEOUS EXERCISE.
1. The product of two expressions is a* + 8((3 + *24r»H32a+ 16
and one of them is a^ + ^a. f 4 ; wliafc is the other I
2. Multiply (,,' - t,Y ,-ii by {,' + iif + .nj.
3. The sum of two expressions is x - ^^ - f^ and one is h'x + x) -i- '"'
what IS the other? " 5. ^/;i ;j
4. Find the continued i)roduct of x-'.]; x-1; x+i- x + 3
]• ^i f"id^ tlie/livisor by ^vhich 'Mr - 23x' + 12x^ + H must },e
divided so that the quotient will be 3x' + (ix + l, and the remainder
^x -f- I ,
6. Divide :| -4<(* by l + a.
7. What expression must be added to
-Tj - '. - 2 to make x'^ -2x-^l
M4-^V/^^' W much d..es a(m + n)-h(m-j>)-r{,.-n) exceed ni(a~
9. What value of ,i will make the ju-oduct of 'A -Ha and 3a + 4-
equal to the product of (5a 4- 11 and 'A -4a I
10. Divide /.','. +.!'l bv 1-+^
11 Show that the i)roduct ..f Hx' + 2ax-:ia' and 10./^+ lO(« + 5a'
may be written (3 ,- + a)' - {x + 2ay.
1 '2. Di vi.le -:* + 1 + F4 l)y ,:^ - 1 + ;..
•,nrV!li:M'^^i ^" ^''^^'':'i^i""^<^ i'^'"<lu^'t „f y, ., + 1, ,, + 2 and x + \
•md divide the result by ,.-' + [I, -\.\. '
14. Fiml /• and ,s i„ terms .,f ,/, A. ,,^ ,y. s.. that ,■' + p,^ + ,.,'>■ +
rx + . may be divided by .-' -f ax + l, yvuLL x may be
15. Divide x^~J, by .»•-.'.
l«i. From ,.■(.,-/, ,j.h..r) take „>-/,;.•,(/, f.,),^a.u! .livide the
uittercncu by ..• -j- i/.
17. Find the product of {a + h) (a' + ab + b') (a-b, {a-'-ab + b').
•2;
hounkh's method of division. 7
18. Determine .( and l> fliat iji the product of .>'M-.'- + l and ,'•''4-
ri.r' 4- /).,■ -f r- the coefticieiits of .»•■» aiid .»"'' may vanish.
10. Multiply - + ^r + - by i - 1 + 1.
^•'a he •'a h c
20. Find the value of a and h so tliat j-'^-(i.i- + l2 will divide j^~
h.r'^ + 2,i; + 24 witliout a remainder.
21. Divide ..-^ - (a + ^- );r -I- 1 by x - a
22. What value of m will make r..**-2.r'' + 2w.>;» + 2:v + »/, exactly
divisible by .»•'-'-.> +1 ?
23. What value must a, h, r, each have tliat ,»■" + </,>■- + /*./• 4-c may
have X — 1, :r — 2, x 3, all as factors '{
24. Tf n- = ;( + l, prove that x* + ((x^ + (i'\>-'^ + (i^x + <(* is exactly
(iivisible by x'^ + uax + a'-.
25. Find the expression which divided l)y (r + '2<i+4 will give
(i^ - 8 tor ([uotient.
2(). By what expression nuist .*■-!- 3 l)e multiplied to give .«:' + 2l87 ?
27. Find the value of <i which will -make x* - x^ - x'^ - «x divisible
without a remainder by x'^ + x.
28. Find the remainder when o,*'^ -8.*'^ + 8.'' + 7 is divided bv
;-).,• - :{. ^
29. If 4.i'^ + 3.'''- 18.'' + 27 nniltiplied by another expression is
equal to 4.''^ + il.''* + 81, find the other expression.
30. Tf a = 4, b=^-o, by what must ax^ + hx + 1 be multiplied to give
8.i-» + (5;< + 2)./- + (4/> ~ 3)x + 3 as product i
EXERCISE VIII.
HORNER'S METHOD OF DIVISION.
2. 5.-'^ ■ 4,."'' + :]x' + 22.»' + oo ~ X-' - 3x + 5.
;;. ,,.5 _ ;-,,,.3 ^ - ,.-2 ^ ,; ,, ^ I ^ ,,.. ^ o ,, _^ ^_
4. Cm--./- ll.r- + l(J,r-'-f.i'3 + 8./--ll). +204-2.r'' + ..'-' - 3,,' + 4.
5. C)x^ + bx* - 17.r' - Gj;2 + 10.«' - 2 -h 3.f'^ - 2./,'=' - 4.^: + 2.
0. .f^ - .'••»/ + ,/■•'*/' - .,'■-' )/3 + xi/ - //'> 4- .''^ - */l
7. O./-^ + 7x* + 7x'' + o..'2 + 2x + 1 4- 3.." + 2.r- + X + 1.
8. .!•« - 2x^ - 5x* + 20.r^ - 25.r2 + 14.r - 3 4- ,.■' + 2..' - 3.
9. x« - 29u;H55,.;* - 232.»=' + 351u- - 2004-..- + 8.
10. 21.*;5 _ 2x* - 70.i;'^ - 23.r'^' + 33x- + 27 4- 7.'-'= 4- 4.^; - 9.
o
EXERCISES IV ALGEBRA.
n.
1. Find the rernuinder when .•^-3.^+2.-7 is divided by .-2
7.-3^' ^" '""""^^•' ^'"'^ ^-i^i»g the following : 5.*-6.^-;
4. Find the value of 5.^ - 4..* + 3a-^ - 4..-^ + ^ + 4 when x= -4
^ ^ 5^ FnuUhe value of .e - 102.H 1()0..H 102.3 _ 99,. _ 20^
6. Find the value of 7..^- llc«H.._50.when .-^^^
^^7^ Find the value of x^- 98.* _ 98.3 _ 100.. + 98. +100 when
^^^8^ Find the remainder when 8.,.3 + i2,._4.-o is divided by
9. Find the value of 5.^-4^4 ,03 ^,.2 , . , . ,
^ ^-^ -t- -sx - 4. + ./• 4- 4 when . = 3.
10. Fmdtheremainderwhen.5_6^4, 5 3_4_.2 4. o ^. ,. ., ,
by X + 5, ^-^ -rojL ^x + d. - 2 ls divided
11. Find the value of a-J" - ^ .-T a. o -t n , i> i
vaiut. 01 X ,ix +a; -5. + (> when x=l.
,.l^- F"^^^ the remainder when (((4-1,4- A {,.h^i , v , .
divided by ,i + h. ^ +(> + t) {ab + he + ca) - ahc is
EXBRCISB IX.
INVOLUTION.
Write down the square of the following :
1. (a + h);(2a + 3h); (| +|) ; (^ + 3^^,
2. (m' + b); (15. + 14,/); (l^ + ^i') . (x y\
X (« - /,) ; (rt _ 2?,) . (2x - Gy) ; (?? - ^J!)
-" ' \ay 2x1'
■;' ^y •^•^' U 4/' I 4 r)-
5. (a + ft + o); (2. 4.3,/ + 4.), (l + x + x^).
6. (4a + 56 + 6c);(U|+|);(l + ^, + |,.).
7. (u + 6-c); (a-6 + c-); (a-b-c).
8. (-^-^iZ + l); (.r»-5. + 7); (.^-a.-6).
^- (|"+^ + |); (3x^-1 -3) ; /-"*-?!i)
10. ^a. + ^ + c-); (£ + ^ .£\
M ISCELLANEOUS EXKRf'ISE.
d
EXERCISE X.
Write down the culjes of
1. x + y; x-y; x+u + z; x + y-z.
?ft ' m ' n in
3. a-h + c; a-b-c; l + x+x\
4. Simplify {a + Sbf + 2{a + 3b) (a - b) + {a- hf
5. Simplify ((t + 6 + cf + {b + cY - 2{b + c) (a + 6 + c).
6. Show that {mx + ?i|/)^ + (na; - 17(1/)"^ = (m^ + vt,'') {x^ + jy'^).
7. n .1 {ax + byf-ir{cx-\-dyy + {a\i-bxf-^(cy-dxy^(a'^-[-
&2 + c2 + (i2)(x2 + i/').
8. Simplify {a + ^)2 - (6 + 0)=^ + {c + (i)^ - (d + a)\
9. „ (l_a-7 + (l + a2)3.
10. ., (3x-4i/ + 5,:;)=*-(5.^-4(/)3-3(3x-4|/ + 5;;y^ (5,':;-4i/) +
3(5r;-4*/)M3x-4|/+5,^).
EXERCISE XI.
MISCELLxVNEOUS EXERCISE.
1. Prove that (2a - &)H (26 - c)^ + (2c - a^ + 2(2a - ?>) (2fc-c) +
2(2« - b) (2c - a) + 2(26 - c) (2c - a) - (ct + 6 + cf = 0.
2. What will a^ + b^ + c^- 3abc become ifa + 6 + c=0?
3. If x+ -^^, prove a;^+ -5=p^-3p.
4. Complete the square in the following: x^ + y^ + z^ + 2xy-{-
+ etc.
5. Simplify (1 - x'^f + (1 + x^f.
6. Simplify (a -i-b-cf + 3{a + b- cfc + c" + 3(a + 6 - c)c^
7. If x = 2y + 3z, show that ^;3- 81/" -27;^^ -18x1/25.
^ 8. Simplify (2a - 36)H (46 - Sa)^ + (3a - 6)3 - 3(2a - 36) (46 -5a)
(3a — 6).
9. Simplify (1 + x + x'f -{l--x + x"")^ - 6x(H- xH x*).
10. Find the value of x^ - / ■\-z^ + 3x j/V when x^ - f + ^'^ = 0.
11. Simplify (a - 6 - c)3 + (6 + cf + 3 (a - 6 - c)'^(6 + c) + 3(a - 6 - c)
{b + cf..
12. Simplify (x'^ + xy + y'f + {x' - xy + iff + G(x'^ + y^) (x* + x Y + (/*).
10
EXKKCISES IX ALGEBRA.
FACTORTXG.
EXERCISE XII.
Factor
1 . <u: + hx + cdx ; ax + ai/ + (r:+ px +py+ pz.
2. ax^ - (I ,f - hx' + hif ; l-u-h + ah.
3. 9,rV + 33a^/>^-12//; d+^O (1 -x'O + ^i -i>a.'-' -,/.-+ p.
4. 15a?>V + I2a%c' - 2\ac^ ; 2rt V + 1 - 2a^ - x\
5. -iox + ay + Ahx + Inj; {u + l) {a-l)^-ab + l-l,--„.
6. 2nf+2hx + 2ax + 2hf', ^.:H 3;/ -3x' -.*•</.
9. '<- 6 + c-a/>-/*<^ + />'-'; a« + «'''-'( - 1.
10. ,ul + c?/; + tr - a>' + /^f- 'f + iif- i'd - hii.
■ a^x.
EXERCISE XIII.
COMPLETE 8QUARES.
Factor
1. a^ + Hab + UW; a' + Uab + 4%\
2. a^ 36(6 + 324; x'-10ax + 2ba\
3. x'lf - hjxy + ('4 ; 'ibx^ - 20bxii + 2ohyi
4. m*i! * + 2mW + 1 ; HJx^ + 16.e'^ + 4.
o. ii' - 18a + 81 ; 1 - 8x + IGx''.
0. ix* + 2xhi + 4r>f ; c^'" - 20'" + 1.
7. xhi* - 12x//^ + 30 ; (a + by + 2{a + b)(c + d) + (,• + dV.
8. ,/;•-■ + ,/ + z' + 2X11 + 2xz + 2 -/;: ; H]x' + 72x'if + 8iy'.
9. *)„■' + 4//^ + l(i. •-• - IQbc + 2iar - 12ab.
10. 9x'^ + 4//2 + ■■^ _ 4;/^ - (Ic + 12xy.
B.
1. I.'-* + liii/z' - ix'ijz ; «■- + b' + (•-' - 2ab + 2ac - 2bc.
on "■ f '.' ~ ^'^I "*" ^^' ' "^^^ + ^'^ ~ "^' + 2 (a - /.) (/; - (.) + 2 ((, - b) (c -a) +
li{b - (•) (c - «). ^ '
difkki<i:m(;k «>f .squakeS.
11
9 25 "^10 '^~b"
**• "'■ +'25 + Hi +^~ + '•'''' +
5. 4//^ + 9r^ - 41 > - (io + 12/>c + ] .
i» .» v" z^ %n
2X2
7. (p + ,^ + ,.)2_2.s.(p + ,^ + ^) + .,2.
8. 2= - 2 + ^ ■ ; (2.*- - :^;/)^ + (2x + H./)^ - 2(4r^ - V).
9. 4a*-12(r'' + 2r)a'^-24a + l().
K ). a' + />" + (•* + 2a^/>-' - 2tt'''o''^ - lU'cK
C.
1. 4(a + 6)'^ + 12(a + />)(o + (0 + »(c+<i)2.
2. 4a- + />2 4.<y2_4,,/,_^4,t,._2?><;.
Ji. 2r)(te2"'-448a;'"+'* + 196a;^".
4. If ,<;'^ + 1/2 = ~2 and .-• // + /;; + ijz = 0, show that (x + y + zf =- 22^.
r». Write down the scjuare of «■' - ia^ _ g^^ ^ i
G. Arrange the six factors of (6rt2 + a-2) (3a2-7a-(5) (2a2-7a
+ 3) in the form of three .s(juares.
7. Find the two e(iual factors of «l + 9 - 4a; + M' + 3a _ ««
8. Show that 2(rc - ./) ({/-,:) + 2(|/ - ,~) (;; - cc) + 2(:; - .*:) (,<; - y) is the
Sinn of tliree S(£uares.
9. Find two equal factors of {x^ -- j-aj)'- -2{/^ - x,\j) {;tij - ]p-)-\-
10. Find two equal factors ,,f 'I: +'!!+ e! _ 2- - 2^ + 2-.
EXERCISE XIV.
DIFFERFNCE OF SQUARES.
A.
1. 4x2 _ 9,y2 . 144^2 _ 289,/2 . 16^* _ |_
2. (2a-?>)2-o2; (4rc+,/)2-~2; -;, +2n)2-^l
8. 1992 - 1 ; x> - (1/ - ;~)2 ; (<t - 2?>)2 -(/>-■ i^'f.
4. (a:2 + 1/2 ^ .2)2 _ 4a.2.2 . ^2 _ 2a;, + 12 _ x-2 - 2X1/ - •/.
12
EXRKfMSKS IN ALfiKBRA.
5. 4{H,I + bi'f - {a' - />■•' - r- + dj ; 4(f-//' - {<r + U' - r")\
7. (.' ■' + i/ + rS" - xy - ur; - yzf - (xy + yz + xz)\
8. 15.r-()0//^; 243x*-48/; l-4r('^//'«
1». ((f-2/>)2-(2a-3/> + 4c)% aH2a» + a2-/>* + 2fe'-?>'.
1 0. (x' + ,/ + ~2 - 2x-|/ + 2xz - 2;/;^) - (y + zf.
B.
1. Write down tlic quotient of
(4x - 3y - 2zf - (3x -2y + 'dzf by x-y- bz.
2. Write down the value of {x^ + y'^ + --") (a;^ + y^ - ~2).
3. Factor a* -^¥ - c* - ti* + 2a -7/'' - 2e'd'^.
4. Sliowthat(5.r-3i/-4)2-(3.x + 7|/ + 4)-is exactly divisible by
2x 4- 1/.
5. From a'-?)''^ = ((( + 6) {a-h), find the difference of the squares
of 118] and 121|. ^
6. Prove 1, - 4 + ,l- U (^ -!- +M(1 -1 - 1).
a- ab ' b- c- \a b ^c /\a b c /
7. Sliow that i+<ll:^l>'^^(^+b+<^)(^+b-c)^
'lab '2ab
8. If a+1> + c + d = 2s, show that 4 (ab + cdY -(cr + h'^-c'^-ifiY
= m{s-a){s-b){s-c){,-d).
9. Show that (a- + Ir + -iahy-{,r + h-'y = 8uh{a + by.
10. Show tliat (.<■ + j/)2 - - ■-' + ( ,/ + zy - a-2 + (~ + .,•)•-' - y^ = (x + y + zf.
EXERCISE XV.
EXTENDED APPLICATION OF (x±yy AND x" - y\
A.
1. Factor .i-' + 4//* ; x^- 15x- + 9 ; x* + x- + 1.
2. 25l)u* + Idt;- + 1 ; x' + ox- + 49 ; a^ b' - lla^t'^.
3. 4a*-37(r7*^ + 9&*; 9,c* + o.i-y + ;/* ; -t"* + ;/^ - 18x y.
4. mH>i*-18m-/<2; a-» + .i-» + l; c* + c''a' + a*.
b. (( * + (U/>* ; ()25«^ -f- 25(1'^ + 1 : (r^ - 19«2?,« + 9H
6. 9<i< - 4( ( -7>-' + ^:; ; 4x^ - i'^J + 9.
'y.
i ])y
ares
Trinomials.
lfia;2 . 256
7. .-+i';f^ + ^;x^ + 25.^ + 025.
8. 16ff.*-17a26H6<; x*™ + e^" ; a;*-7x'+l.
9. 16m*-28m'/i2 + 9u*.
10. (a; + i/)*-7«%x + |/)H2*; (a + 6)4-3cV + 6)2+c*.
B.
1. 9a* + 3a2&H46*; »:* + 7j'' + 16 ; 16a-* + 36xy + 81]/*.
^' 9
a* a262 64
266' "^ 144 '•' 81'
3. tt* + 6* + c* - 2a"6'^ - 26^02 - 2c'a\
4. x* + 4(.v + 2)*; (a + 6)*+(a-6)* + (a2-67.
1_ 9 3,1
' X*
5- A+i
a-x-
a;V^J/**
6. 4(a + fe)* + 9(a - 6)* - 21 (rt^ _ ^2)2^
7. 16a* + 4(6 - c)* - ^(,\h - c)\
8. {x' + ,f - a;i/)* - 7(a;H 1/)-' + (x + y)*.
9. {a + h)* + 4(a - hy ; 4x'' - 13xV + 97/.
ft 16 . 1 . 4
10. 2+-^
■«»4 ' a*'^ b*"^ a'-'b-^'
13
-(/2)2
EXERCISE XVI.
TRINOMIALS.
A.
1. x2 + 8a; + 12; x' + ^x + ^O; a3H47x- + 370.
2. .r2 + 89x+1960; a;2-27x + 182; x-^ - 19a; - 150.
3. u;2 + 16x--80; a;^ - 88x + 1612 ; i.;^ - 37aj - 120.
4. 15.t^^fl7x + 4; 6u.-2-5xi/-6j/2; 16c2-16ac-21a».
^ .,.2_9* 1 . -" 21a;
«^' •<- on ~ ■'^ J
X^
-1; a:'^-^+;4.
35x , „
20 -.' •- 10 - ' - 18 ' 18
G. a-'' + 33x + 252; ce^ - 02.*- - 693 ; ^-^ -37^-528.
7. 21u;^-55xj/ + 14,/- .'* - -|; , , „ ,^
8. 6(2.*; + '^ijf + 5(6x2 + 5x-]/ - 6 ./-') - 6(3x- - 2i/)l
9. 4(x + 2)*-37x%r + 2y'' + 9x*.
10. (a-6)''''»-44(a-6)"»4-363.
14
KXKHCrSKS I\ AUiKHRA.
B.
1 . 72 J -' - 145,1- -f 72 ; Hx'-- .'Wr + :ir>.
2. 24./^ 29.t|/-4|/^; ](>./'- 17.' -f.'l
a. Lm'^ + 1 14..' + m ; 12j-^ + 19.I- - 21.
4. Cnr - (»/; - 15//- ; 32z^ - 24ac - 20.1-2.
5. AV6/' - (Smxy - 29\hf • 204./^ - :^29a-|/ - 2(M)i/2.
(). 45ic^ + fil4x- - 2249J) ; 80^- + 859.i- + 5247.
7. 78.«-' - 4231:f + 48015 ; 5<)..'^ + 1:>,7..- - 27H85.
8. 42r'^ - 135;r - 1 1877 ; OHr^ + 580.r - 9019.
9. :i4y^- 200b -30745.
10. 28(f'^- 411a -8.3467.
1.
2.
3.
4.
5.
r-
8.
10.
1.
2.
3.
4.
5.
EXERCISE XVII.
POLYNOMIALS.
A.
20.i-'^ + 2a;i/-G»/2-8.i; + 4(/.
0f«''-7a/>-2062-6a + 156.
7j'-42i/-2..=* + 9.i;!/+18i/^.
3a,-''' + 19.,'(/ + 20i/'^ + 2.,- + 1 Oj/.
u-'^ - xif - f)//'^ - 4x~ + 12iy;;.
18a;'^ - 24^-1/ + 8?/'^ + %z - 6yz.
6y' - ih/ - 2i)z' + 22;/;; + 7xz - bxy.
65a^ + (W' - 12(;'^ + 34/>c - 8a<- - 71a&.
da' - 23(»/> + 10(«- - 25/>c + 21/>'- - 4c-.
'it)i'^ + 2nin - 11^ - or^ - &nr - 2mr.
B.
7j-'^ - xy - Qy^ - 6.x- - 20i/ - 16.
20/^ - Wxy - 5(/- - 68a; - 42j/ - 88,
20y2 - 20*/'^ + ^xy + 28..- + 35 J/.
.i-2-.n/-12,/''-5.r-15j/.
6..'^ + Gj/'^ - Uixy - 82''' - 2yz + Sxz.
APPLICATION OP^ ir'^±yK 15
«). 4f<'^ - 15/>2 - 4a/> - 21c"^ - 306c - Sac.
7. 15^''^-18a:-28]/H42i/-23a;]/.
8. iV-^ -l<>a/; + 126- -2« + ()/;.
9. 18.i'^-42xj/ + 20|/2 + 9x-15i/.
10. 20^2 -24x + 12i/''-l 81/ + 31x1/.
0.
1. 4u^-5a/)-2162 + 4rt-12/).
2. (i..'2 - 7a- 1/ + 2x;: - 20|r + Mir. - 48,r.
3. 2x^ + 5xy + 2i/ + Uxz + i7yr. + 2UK
4. 24x- + 37.r!/ + 12x-5i/''^ + 2a(/.
5. Sx^- xy + 4:XZ-^\/-3yz + z^.
6. 2x'^ + 5mx + Sm"^ + 2sx + 5nt,s - I2s\
7. Show that 7^-3)1 is a factor of 28m^ + 21jpm-75m» - 9/>n
+ 27 /«-'•* and write down the other factor.
8. If x+p and x + g are factors of x^ + (/) + m),«; + m/> and x' +
(q + m)x + (pn respeetiv^y, show that both are factors of x^ + {p +
q + m)x'^ + (pq +jyni + qm)x + pqm,.
9. Show that one of the factors of Gx^ - 7xi/ + 14x - 20j/''^ - 35 1/ ia
also a factor of Sx'-^ - IQxy - 10y'\
10. Show that the factors of 16x''^ - 46xi/+ 15i/ + 76x2 - 54t/2 + 482-
niay be written as the difference of two squares.
EXERCISE XVIII.
APPLICATION OP x'±j-^
1. a^ + h^; {a+xf + y^; (m + nf + (p + qf.
2. {m^-vm + n^y^ + (m^ + mn+n^f; a^ + h^.
3. ai2 + /,>2; a^^ + b'''; Sa^ + 27bK
4. xi« + i/"; 125.«2i + 512;/2^; a''~(h + cf.
5. Sx'' - 64i/' ; a^^ - IF ; x=* - 3ax''' + 3a2x - d^ + h\
6. Show without uuiltij)lying out that (x''* + .«•[/+ t/^yt -(;)'- xi/ +
i/^y is divisible by x and y.
7. Show that the sum of the cul)es of 2x''^ - 5x - 9 and x- + Gx - 5
is divisible by either 3x + 7 or x - 2.
^^* EXEUCISES IN ALOEBUA.
8. Wriho the quotient of :*;^ + 3j''a + .Im^ + «•'« + />'' l)y x + a + b.
(^^A^''""^ *'""*"' («^' + '^'/ + '-)•' + ('••^•■-^ + «<':)•'' is divisible by (a+c)
10 Show that 2(6 + rO is a factor of (a + h + ,- + df ~ U -h + c- d)^
Hud that (2xH5^ - 9)'' - (x-HOx - 7)' is divisible by (.« - ^2)(x + 1). '
EXERCISE XIX.
GENERAL EXERCISES IN FACTORING.
A.
1. n(x + yy~bc(x + y); 15p'^ + (i7p-24.
2. (a + b-cy-(a-b + cfi (('' - b'' - c"" - 2n ~ 2b - 2c - 2bc.
3. 2x^ + llxy + 12f + 7jcr. + 13yz + 3z''; x'' + 3x'yT -4,/\
4. (b - cyx" + 2{ab - ac)x + a^i - ah ; 2(a + bf + r)(a + />) + 2.
5. a;4 + xhj - xf -y"', a'- h^ - 3«2 + y^ _ i.
6. (ac + bdf + (ad-bcf; (x'' + oxf + 10(x^ + 5x) + 24.
7. aJ» + x*y + xY + xhf + xy* + ,/« ; a;H cc^ + a: - 1.
8. a=^-2a2 + 2a-l; cc^ + 4a;2 + 5a; + 2 ; m;^ + 5a;-^ + y^; + 3.
10. a;=^ + 2a;2 - 5x- - C ; a;» - a;^ - 22a; + 40 ; x-" - 2a;2 - 5.^ + 6.
B.
1. 2x^ + lla;2 + 17a; + 6 ; Sx' + lla;^ + 12a; + 4.
2. 3a;3+5a;2+7a; + 5; 2a;3 + 7a;2 + 2a; - 3.
3. 4x' + 8a;2 - a; - 2 ; 9a;3 - 45a;2 - 4a; + 20.
4. 6a;3 - lla;2 _ 313, ^ 3Q . ^s ^ g^o^ _^ ^gai^ + g/A
5. Prove that (6 - c)a^ + {c - a)b^ + {a - by is exactly divisible by
a + 6 + c. •'
6. Express a;* - ^ja;" + ga;^ - xHpx - (^ in the form of three factors.
7. Factor a{b + bc-c) -f- ?>(c + ca - a) + c{a + ab - 6).
8. Express (x + l) (.,- + 2) (u- + 5) (^ + 10) - 3(Sx - 136 in two linear
and one <[iiadratic factor.
9. Resolve into three factors x* - 11^- + 10.
lO. Resolve into three factors x^ - 2x^ -x+2.
if
H. C. V,
17
1
2.
II
3.
II
4.
II
5.
II
6.
II
7.
It
8.
It
9.
It
10.
It
EXERCISE XX.
H. C. F.
A.
1. Find H. C. F. of (Ka - x) ; 4(<t2 - x') ; (a + hf and a"" - b^.
a.2 _ (a _. if,).;. _ ,(h and x^-(a + h)x + ah.
a;2 _ vsx + 14 and x" - Wx + 28.
x^ - 15j- + 30 and x^-Qx- 3(>.
a;4 + a;'^-fianda;^-3.r2 + 2.
ar* - 2j? + 3a; - 6 and x* -'j?- x^ - 2x.
6^2 + I7.r + 12 and lOx^ + 3.,; - 18.
24X'' - 22x'' + 5 and 48^;'' + 10x« - 15.
6a3 - Ga^ + 2a - 2 and 12a'' - 15a + 3.
3«4 + 8r(:' + 4a^ ; 3a'' + Ha^ + eas ; 3a^ - IBa^ -
12a^
B.
FindH.C.F. of
1. a;2 + «-30; a;'^ + lla; + 30 ; x2-x-42.
2. 21x^ + 8x1/ - 4j/2 ; 21.<y^ - 20.r*/ + 4;/^ ; ^^x' - 28u;|/4 4iA
3. ar* + xV - Sx-)/^ + j/* ; x" + 'ixhj + xi/- - f.
4. x" - 2^2 - 15x + 3G ; 3x-2 - 4x - 15.
5. 3x5 _ 3.^4 _ 53x3 _ 43j;2 + 34^ + 30 and ^.v} + 3x* - 53x3 ^ 43^2 + 34^
-30.
6. x3 + 3x^1/ + 3xj/^ + 1/* ; x' + x% 4- xf + j/3 ; x' + xhj - x,f - vl
7. Find H. C. F. of x^-Gx + O and x3 + 4x'^-9 and also what
value of X will make both (juantities vanish.
8. What value other than zero must ])e given a so that a?-x-as
and x'^ + x - a may have a counnon factor ]
9. Find H. C. F. of 2x^ + x^ - x - 2 and t' -v?- 2x2 ^ 2x and show
that its s(iuaro is a factor of the latter expression.
10. Show that (x-2)2 is a common factor of x' - Ox'^ + 13x'' 12x4
+ 4x"' and x"' - 5x« + 8x-' - 4x^, and tind H. C. F.
2
18
KXEIU ISKH IX ALOKURA.
EXERCISE XXI.
L. 0. M.
Find L. C. M. <>f
1. «j;^ 2run(l lU-y - 3x-^.
2. Sx^ + lU + tiMidx^ + dx + H.
•A. .'• + 2; /^-l; x' + x-2.
4 . ,'^ + ("ij; + 4 ; x'^ + 2j- - 8 ; x^ + 7^+12.
5. a» -1; f»'''-<); «» + 2^-15.
♦>. x-'-^V2x' + 47x + {\{); ay^ + V.ix' + nih: + HO.
7. J-"* - ijx' + llx - « Hiid u."' - yx^ + 2«u; - 24.
H. rr' - 1 ; ,t* - 1 ; ,f» - 1.
IK (x + ;/) ' ; {x - yy ; x' - ,f nud ,/r' + ,/.
10. x' + iix - 27 ; j;»+ 17.« + 72 ; x' -M; x'- 1 1.,- + 24.
EXERCISE XXII.
GENP]KAL EXERCISE— H. C. F. AND L. C. M.
1. Find l)y factoring tlio H. C. F. and L. C. M. of (c' - 'A<i'^ + a + 1
and (i^ - Ha + 2.
2. Of .r=' - x^ -4x-4 and x^ + (i^^ + H^; + «>.
.3. Detorniino the algebraic oxprossion which involves the lowest
possible dimensions of ./• tliat can be exactly divided by u-* + 5./' + fi
and .i,-2 4- 7.,. + 8.
4, l( X- :i measures x'^ - 7x-4-rt, find a.
5. If a -I- 4 measures a^ _ j; _ ^^^ gjj j ^^
<). Find the vahies of a and /> in order that a: -3 may be a com-
mon factor of x^ - 7x + it, and x^ + x- - h.
7. What value of a will make x- - 7 a measure of .'•'-' - ax + 2\ ?
8. Tf 4,i:' + l>x - 20 is i nndtiple of 2.'' + .5, find h.
9. X- + 3 is a measure '<f " - Gx-^ + Ux - r, find r.
10. If X- + 9 is a meas- it ' x'' - tr, find n.
11. TheH. 0. F. of two expressions is ./ and the L. C. M. abx,
find the product of the two expressions.
H. C. F. AND L. ('. M.
19
M
hi
12. \{ x-\-4 and .r + 7 (iro >)()Mi mofiHuros (if r' + ll.i-f »h, fiinl m.
13. Tho II. C. F. (livi.liMl into L. 0. M. given 2<if> fur (|H(»ti«iit.
Tf II. r. F. is '2./', llnd two sets of exproHHions that will .sutisfy tho
cuudition.s.
14. Tf j+4 iind .' -f .'i inuHHUio x' + <f/'' + *2<»' +*i4, tiiul a, and iiIho
what othor cxprosHiuii will 1h) a iiieaHuro of tho given ono.
15. Sliow that if a (juantity divido A and li exactly it will divide
pA±<iH.
10. If j:^+nu + H is a nieaHiiro of x» -f /JJ^* + '/*• + '", prove tluit
n>i -u^ — rm.
17. Show that, we ean ol)tain L. C. M. of two algebraical expres-
0118 A and /> by dividing their product by the H. C. F.
18. Tho H. C. F. of two expressions is .'+3 and the L. C M.
■ji^ + Jj^^ + l2x'^-9x-''^, ono exi)re8sion is a"' + 4.t;'^ - i), find the <.ther.
10. If a number bo a measure of two others, ])rove that it will
also bo a measure of the ditlerence of any multiples of these
numbers.
20. If x'-/>x + 7 is divisible by ./• - /• and r - .s, show that ;> = r -f a
and 7 — rs.
21. Find L. C. M. of /'-ox- 14; ./'-4x--21; and ,r'-t^f-
26x'-21, and for what value of .'■ will all three expressions vanish.
22. Find H . C. F. of x-'' - x^ - 2.<' + 2 and .>-* - '^x^' + 2x' -\-x-i. What
value of X will make both exjiressiona vanish i
Sions
JSt
n-
X,
20
EXERCISES IN ALOKBRA.
FRACTIONH
EXERCISE XXIII.
1.
2.
3.
4.
5.
6.
Sinii)lify tlie following
n--nb 3
6
u"b - ahc+b^c - ab- ' (x - 2) (x - 3) (x - 1) (x - 2) (x - 3)*
x"-(a + b)x+ab
a--9h- a- + nh-\<lb"
x"-\-{c-a)x-ac ' a" - 'lab - lab- ' U'-bab
a"Jrb^ +c^-Mbc ^ x--ax x--V'ix-\-ax-\-2a
a-+b'' + c" -hc-ca -ab ' x'--A x--a'
(?/ + ^-^C)2^-_(?-fXj;;^2,V)-. b fb _2/.2_ .v\\
(x+y+zy-i-ix+y-bzy- ' c\c bA '^j'
(3.r-2?y)g-(2a- + 2.'/)- _ x-+x-(i x-+4x
'2x-si/ ' x- + ix-S^ "a;--9 *
a* -«■■' -(T + 1
a^-'Aa-b+Sab--2h^
8.
9.
10.
a< -2«-' -a--2a+l ' ^ ' a<+a-t"+6*
1 . 6.V 1
X + Sn ;).•--*)(/- 3I/-X*
n-3 «-3 l + 3a
rt-2 «-l rt--3(( + 2*
1 1 1
X- - ix-rS X- - 3x+2 x" - 5.T+6 "
2J-+10 :\x + l _ i).c-rl3 247 - 8a;
8 +""7" 4 + 14 "
EXERCISE XXIV.
Simplify
3x+:
fi
;?.r - 2
,, <»••-.(•■••, (r ^j- a-x \
"' ax \((--.r- a'-irax + x-/'
, 12.r-4 -r-l _^14-0j-i ';>J--
"^' 1 - 8J-+ Itii^ ' 10.r--l *
^1
FRACTIONS.
21
4.
5.
6.
7.
8.
9.
10.
a;2 - 7xii+12y^ _^ x'-hxy+Ai/^
lc'^+5xy+6u- ' x^+xy-2y-
2a
x-a
(x - 2aj" x^ - 6ax+6a'^ ~ x-'3a'
a;^- 6a; 4-8 a;''-5a;+6 ^ (x - 2)-
x^'-ix+-i ^ x'^--2x-8 ■ 'a;'-! *
2
H-a;+a;2^1-a;+a;=' l+x'+x*'
a-b
X
a'' + 6'' a-'-b" a"-b'
- X-r
/_2£_ , J/_ J^" \ . / 1 ■ ^\
\x+y x-y X'-y") ' \x+y'^ X'-y'^)'
EXERCISE XXV.
Simplify
1
2.
3.
4.
5.
6.
7.
8.
9.
10.
a b c ' ab be ac '
_a x_ , 3jM-2j/ _ 3a; - 2y
a-x a+x ' 3a; -2(/ 3a;+2i/'
a;"+a-+l a;''=-a;+l ' a+a;'^a-x a--x'*
1 . 1 . 2/j
a+t^a-^^a^'-^^*
~z. — r
b .'d-b
IT "^ a'-b- '•' a^ "^ a+6*
1 1
+ ,
{a-b){a-cyib-a) (b-cy(c-a) (c-b)'
(x-a)(x-S) x+4
(x+4)= x-a
2X--8X+6 3x=-27x+60 . x=-10.r+21
3x^-15x+12 2X--10X
111
X
x^-7x
1 + 1 - a - -
'a n a
6a" - a?» - Vlb-'^ _^ Vla"^ - 16ab - 'Ab-
6a^T23a"fti + 20<;^ " Ga - + lab - 2i)b '' '
oo
EXERCISES IN ALGEBRA.
EXERCISE XXVI.
\l+x'
l-xj •
\l-x
a ^ h
2rf+3
3«+4 "
4rt+5
5«+«
X X'
a I
a +2
2rt + 3 '
3«+4*
4rt + 5
a c
h 7i
— V — •
/ -•'• .
k •"-
4rt* 2rt- fl.
a+a; a-x'
S.
<>.
10.
,+,
(a-b){a-c) ' (h-c) {b-a) (c-a) (c-b)'
(^'+S)(^;)•
i:
a+b , a-l>
<>^a + <»/ ■ Ifl-^ a+bf'
X
x-a
? <.r+(T j--a\ . , x+a x-o •,
a ~ \x-a~ x+a ( ' \x-a x+nf'
x+y
{jc ~ If ^ -r .'/ '.'/ " ■> }/ ~ ■^
x*+n"x"+a*
x + a
x--a'^ x-+ax+a^
x+a
-y
x" -ax+a^
x-a
j..-!_„i x'-nx+a^
_ X '
x^+a-'' x-a
%
^''+ax+a^'
Ij
E(^l ATIONS.
2;i
EQUATIONS
EXERCISE XXVII.
A.
Solve the following etiuations :
1. o.r - (3x- 7) = S5-2x.
'1. 9,,-3(5x-(J)= -72.
:} . ( 2.,' - 7) (x + 5) - 2x{x - 8) - X + 2G5.
4 . 5('( -H .'•) - Tu' = 3((< + ().'•).
r>. 8(.'- i)4-17(.''- 3) = l(U'-32.
(1 . 2./' + 34 - 'M)x + 95 = 3./; - 119.
7 . i>x - 1 7 + 3x - 5 = <»./; ~ 7 H.y _f- 1 iC.
^ 7x+5 f);C-l ^j^9 '2a-- 3 _ J Q
iu.
4x+l r).i--3 7a--4 (i.r-5
1.
li
.".or - 1 7j- - 2 66 - 5a;
13 •
B.
10 10
2. 4(x-3)-7(.'' 4)-<;-^.
3 . J (Ou; - 10) - i^o ( 1 2.t; - 13) - 4./; + • ('Zx - 7).
-^^ a--I -""Aa•+•J/+•^'•'■
5 . (X- 3) - 3(..- - 5) + 5(r - 7 ) = 0.
. (.<• - 5) {X - 3) - (.f - 5) - (,f + 7) (x - 2) - 0.
r. a-+l
24
EXERCISKS IN ALGEIiRA.
H.
■r 1 8 !).r \1 Ox t 1 29 -Hat
8 10 10 ~ 20 •
-^ 4a: -17 ll-(!6ar ,.(54 + a;)
10. — -- ,,y =0 .^-.
a;-5
-3-.
C.
1.
2.
3.
4.
6a;+13
15
3a:+5
5x - 25
5 ■
(i-r+l
15 ' '
2a--
"7a;-
4
-16
2a; -1
5
3.rfl _
9.r+8
a--fl
4
12
a;+8'
9a; +20
3(3"
_4x-
~5x-
-12
-4
^ 4-
5.
6.
7.
8.
9.
10.
l()x-l-17 _ 12.T-f 2 _ 6a; -4
18 TSx -• 1(3 "" 9 "
9a-+5 , 8a--7
14 "^ Gx-f-2
3(5xfl5
6x-f-7
15~
2x+3
x+l
2x-2
7x-()'
56
2a; -K
~5 •
_4 - -i
4x+5 3x+3
'4x+i'^ 3X+1'
7.rj-^ _ 7x - 26
x-1 ~ x-3 '
x-l_7x-21
X - 2 '" 7x - 20'
lOi
1.1
1 4x4-17 3.1- -10 _^
^- 'x+3"'^ x-4 -'•
„ 6x+8 _2x-f38_.|
"^^ 2.rrl Xil2~-^-
3.
4.
x-4 _i«-i'_J^-i _--^
a^ -^5 X - C) ~ X -^ 8 a- - 9"
x-1 x-2_x-4_x-^
x-2~x-3~x^5 x-8'
K 3x-l , 2x1 r , 96
s- x-.r+-x+4-=^+x^^-
D.
EQUATIONS.
25
IS
i
,, fix -3 x-l 1 f.
" • ., u - .-1=0.
3a; - 8 a; - 4
Q a;-1 a;-3_o
9.
4a; -3 _ 4a;-7
SaT-^l ~~ 2a; - 5 ■
10 a^+lO , ^^ a;+ll , a;+2
E.
1.
o
3x'f4..- +3 _ 2^2 + 8x+3
3a;+4 ~" 2i+8 *
3+a; . '-5_i I ^''zl
l+a;'^a;+7~ "^7+8a;+a;'-*
a , h d ^ X ^ I 1
X c c ' a -6 a+b
ah be ca
?*a; aa; '
__2aa!_
a-b~ a+b~a^-b^'
3 . ((( + J') (?> + X-) = (c + u;) ( J + x).
4.
5.
6.
7.
8.
9.
10.
a b _rt+ft a- - b-
X a X u'+ab'
(m+n) (n -x)
m-n
+ aj = 0.
20x+lla bx+1()a _4a; 61
2ua
a;4-w
a;-7n
m<
t'^+mx+m'^ x--mx+W- a<. '+7»^a;='+)rt*)*
F.
'-f'+^i-'+'-i-'=10-
o
3x--4-^^ 9-(2a; + 7) + 3x'}.=13.
« _ 5^ _ ^0 -X ^-z _ 12 - a; _ ^
a 5 6 " 4 3
(2x' - 3)''' - {4x' - 28x' + 49) - r>x + 15.
2f5
KXKUCISKS IX AI.OKHHA.
3^*
18j: -.S
(;. ](.'•- 1)+ "^.-•'.-i(2.'-+7) + •'■■**
7. 7.S;i - .')(!>./• + 'A) 4- 1 L>.. + 42 - 48./' = ( ).
o * - 7 ,'te - 5 , , „
10 t" 4.r+44 _ _ 3
xhlO ^-4-22x4-120 ~ a- +T2''
EXERCISE XXVIII.
PROBLEMS.
•uuAvh.rns'i,;'Tl' ^""f ^f^^^^'^!, ^o In^ersoll at 8 nnles per hour,
.uul lutuins by Ihaniesford, 5 miles fartlier, at 9 mili-s i,er \umr i,
lo nnnutes nn.re. How far from Loudon to lugersclll /
2. A liorse .-ukI buggy cost !|!280, au.l 5 tiuies tl.e price of tb..
ho,^^wa. e<,ual to 9 tiuies the price of the buggy. l^^ld'h^'Hc:
.siiai e, ana L 4 times Ji s share.
4 A passenger leaves M<mtreal for T.,r<.ulo. a distance of 'XVS
ndes at the saiue time a.s one leaves Toronto for Moutival The
tram from Moutival runs 42 unles per lu.ur, and tee from
loronto 32 nnles per hour. How loug l,efore they n.eet 'l
; 5. A received a legacy of 84000. and 1? .^.'JC.OO ; after each hid
... vii:\^'''^*Ii'' ^ ^''T '''^. l^'"" '■'' '^ '"^ ^^''^^ •■^'"^ contains Or.OO
M . >ds. less tluui another field which is the same length, but 40
yds. w uler. Find the size ..f Hrst field.
then has only tw.ce as much. How nmcl, had eacli at first .^
8. T l»<.>.ght 23 cows, sou.e at .^.'38 each aiul the rest at $50
eaclK ^ Ihe whole amount p>a,d was .Sl»70. How many of each dtd
. 9. A mercliants selling price is 20 / advance on cost l)ut he
uien ?t'i..iu. I'lnd the cost price.
I
i
KQUATIONS.
27
dill
10. A man gives to a beggar 81.00 less than | of his money, and
then there remains $1.00 less than | of it. How much did he
give the beggar ?
11. A
niu
when his income is §176.00.
-1. A speculator owns $!5000 stpck, some at 3%, 4 times as
ch at 3V/, and the rest at 4%. Find the amount of each kind.
12. A man bought 80 yds. cloth, some at 50c. and some at 75c. i)er
yd. He tinds by selling all at 75c. per yd. he would gain $2.50
more than by adding 12k". per yd. to the price of each. How
much did he buy at 50c. ? "
13. A father gave his boy a certain sum of money every Monday.
During the week he spent h of all he had at the beginning, and at the
end of the 3rd week he had $1.40. What was his weekly allowance ?
^ 14. My income, i960, is derived from money invested, some at
3% and some at 9%, but if the rates were interchanged my income
would be doubled. How much is invested at 3% ?
15. Two men begin business with equal capital. The first year
•me gains $250 and the other's capital is reduced J, and the first
had then twice as much as the second. How nmch had each at first?
1<'>. A man sold to A J, of his cattle, J to B, I to C, and the rest,
27, to D. How many had he at first ?
17. Divide 90 into 4 parts so tliat if the first be increased by 2,
the second diminished by 2, the tliird multiplied by 2, and the
fourth divided by 2, they shall be all ei^ual.
18. A farmer sold a number of bags of wheat for $72, and a
second lot of 5 bags less at the same rate for $03. Find, the
number of bags in each load.
19. Divide $7400 among A, B, C so that A shall have $120 more
than B, and C $106 less than A.
20. There is a number of two digits whose sum is 14, the unit's
digit is the greater and ^\ of the number is half as much again as
the unit's digit. \Vh;it is the number ?
21. A merchant increases his capital every year by ^ of itself,
but spends $10(J0 for expenses. At the end of the 3rd year, after
deducting $1000 for expenses, his capital is doubled. What was it
at first ^
22. Divide 192 into two parts so that the larger divided by 7
may be 4 less than the smaller multij)lied by 3.
OQ
23. A man divides a sum of money among A, B, C, giving
$120 less than ^ of it, B $40 less than |, and C $32 more thai
part. What did each get '?
A
»an I
28
KXERCISKS IN AI.OEUKA.
24. A man receives $6000<). He invests i>;irt in a liouse, ^ of the
remainder at 4%, and the rest at 5%. His income is $V.)m. Find
the cost of the liouse {
25. A house and garden cost |3400, and 5 times the i)rice of the
house e(iuals 12 times the price of the garden. Find the price of
each.
26. A farmer has horses worth $62.50 each, sheep worth $11.25.
The total number of animals is 35, and the value $957.60. Find
the number of horses?
27. A person mixes tea at 60c. per lb. with some at $1.00 per lb.
He wishes to sell the mixture at 73k. per lb. and gain 10 < on
every lb. sold. H(nv many lbs. of the inferior must he mix with
each lb. of the superior 1
28. Divide 150 into two parts so that one jtart divided by 23 and
the other by 27 will give 6 when added together.
29. The stones which pave a square court w<»uld cover a plot 6
yds. longer and widtli 4 yds. shorter than the side of the square.
Find the area of the square.
30. A after spending $50 less than J of his income found that he
had $225 more than ^ of it left. Find liis income.
31. The left hand digit of a nimiber consisting of two digits ex-
ceeds the right hand one by 4, and when the number is divided by
the sum of the digits the quotient is 7. Find the number.
32. The length of a floor exceeds its breadth l)y 6 feet, but when
each is increased 1 foot the area of the floor is increased by 31 sq.
ft. Find the original dimensions,
33. Divide 40 into two parts so that 3 times one part and 5 times
the other shall be 168.
34. The sum of $380 was raised by A, B and C together. B
gave $50 more than A, and C as much as A and B. What did each
give ?
35. The price of a work of several volumes is $13.60, but if each
volume cost 26c. more than it does the price would be $16.20.
How many volumes are there 1
36. The sum of $2500 was divided among 4 societies so that the
first and second together got $1400; the lirst and third $1300;
the first and fourth $1100, Find the share of each.
37. A general, after battle, found only ^ of his army and 3000
more fit for duty, ^ and 600 more wounded and the rest, ^ of the
whole, slain or prisoners. Find the nuixiber in his army.
i
EQUATIONS.
29
EXERCISE XXIX.
on
1. 3x-7!/= 7
11 ^+5!/ = 87.
2. J- +11 = 10
x-y= 4.
.'{. x+ J/ -10
ArfJ!/
3;/:
4. 4j-- j/= 7
3a.' + 41/ = 20.
5. 3.t.-ll]/= 4
5^ -12)/ =13.
<;. ^+ 1=41.
4o
1/ - oo^' ■■
0.
13. ?±^' = a
a--!/
14. x + ]i = 'm
ax +IJ— H.
15. - + - = «.
71 . ttl
10
17.
- +
X+ )I = C
ax
hy = 0.
+
— 5
9 + ^ = i
K a-
by
leii
les
B
1
4-?^=ll
^ 4- ?' = 7
8. 8x-7;/=12
J- - 2.1/ , 'Ix - y
+
+ - =31
-"^=31.
10. ^ +-F = 8
? - ^ = 0.
= 1.
11.
+ ^=18
_ ?i - '>i
12. ^+-^-^6
a;-?/
4 '•
18. 7.r-ll!/-3 =
5i/- 6x-f7 = 0.
19.
x+xj- :
a; - 1/ + ;
5,. + y + ;; = 20.
20. a;+ I/+ 12 =
x + 2(/ + 3," = l
x' + 3j/ + 4:<; = 2.
21. a' + 6(/ + 5s =
2,,- 9// + 3^ =
x + 3j/+ 2 = 3.
22. x- + 2i/ + 32 = 14
2x- + 3»/ +
11
3x+f/ + 2s=ll.
23. 7x- + 8:; = 53
92-5^ = 21
12.c-5i/ = ll.
30
KXKKCISI S IV Al.cfKBUA.
24.
M-<>
i^H
X - i/ + ;=4.
25.
\^r-^
»-^>^
^^-•
26.
X y Z
I -I + U4
X U 2
•^+l + i--20
X ^ y ' z
27.
.r + j/==3
J/+ ■: -h
:+/)-= 7
a + /' --= 5.
28.
.'■+»/ + ;; -29.25
.f+|/ ,:-^ 18.25
j; -i/ + 2--l.'i.75.
2t>.
'? + - + - -9.
J- y z
'1' - '-i!- : ^4
..m
^- + .'/ + -• -- 0.
x + y-'Z^b.
X - If - z = c..
EXERCISE XXX.
1. A man bought two pieces of cloth for ^^oO.OO, one piece at
$1.00 and tlie other at $il.80 per yd.; lie sold them at an advance of
40c. per yd. and gained $12.00. Find the lengtli of each piece.
2. Twenty-eight tons of coal are t(j be carried in carts and
wagons. It is found it will retjuire 15 carts and 12 wag(>us, or 24
carts and 8 wagons. How much can each cart and each wagon
carry ?
3. A certain number of two digits is equal to five times the sum
of the digits, and if 9 be added to the number the digits are
reversed. Find the nund^er. .
4. A person buys 8 lbs. tea and 3 lbs. sugar for $5.28, and 5 lbs.
tea and 4 lbs. sugar for $.'1(54. Find price of each per lb.
5. A farmer sold to one man 30 bus. wheat and 40 bus. barley
for $54.00, and to another 50 bus, wheat and 30 bus. l)arley for
$()8.00. Find price of each i)er bushel.
G. An account of $105 was paid in $5 l)ills iuul silver dollars ;
and 4 times the number of ])ills exceeded twice the number of
silver dolhirs by 14. How many were there of each {
7. Two men, A and B, received $23.40 f<»r work done. A
worked 15 days and B 14 days. A received for 4 days $2.20 more
than B did for 3 days. Find the daily wages of each.
j
EQUATIONS.
31
]
8. A farmer sells to A 9 horses and 7 cows for 81200 and to B,
at same price, 6 horses, 7 cowh fur the siime amount. Find the
price of each.
9. Find two numbers so that ^ the first and i^ the second shall
be 9, and ^ the first and j^ the second shall be 6.
10. Two purses contain togetiier 8.'iO(), and if ^MO are taken from
one and i)ut into the other there will be the same in each. How
many dollars in each purse ?
11. Find three nuuilKTs such that the sum of Ist and 2nd =7,
1st and :h-d =8, 2nd and 3rd =9.
12. There are three numbers such that the 1st and ^ the second
^14, the second with j( of the third =18, and the third with ^ the
first =20. Find the numbers.
13. A and B work together for 50 days at $1.20 each per (ky.
A spent 12c. per day less than B, and at end of the time had safed
twice as much as B, and the expense of two days over. What did
each spend per day (
14. Find two numbers such that the sum of 7 times the greater
and 6 times the less may be 332 and 51 times their ditt'erence 408.
15. If John gives Tcmi $10.00, Tom will have three times as
much as John. If Tom gives .John Jj^lO.OO, John will have twice as
much as Tom. What lias each ?
16. The sum of two numbers divided by 2 gives 24 for quotient,
and the difference divided by 2 gives 17. What are the numbers ?
17. The cost of 7 lbs. tea and 5 lbs. coffee is S7.04, and 4 lbs. tea
and 9 lbs. coti'ee $0.48. What is the cost of 1 lb. of each /
18. A farmer bought 100 acres of land for $4220, part at $37 and
part at $45 per acre. How many acres had he of each kind '(
19. The smu of two digits com]»osing a number is 0, and if the
number is divided by the sum of tlie digits the (piotient is 4. What
is the number (■
20. A certain fraction })ecomes 2 when 7 is added to the numera-
tor, and 1 when 1 is subtracted fiom the denominator. What is
the fi'action /
21. A grocer bought tea at $2.(K) ]>er lb. and coffee at 60c. for
$125. He .sold the tea at .Sl.<»() and coffee at 90c. j.er lb., and
gained $1,'0. How many lbs. of eacli did he buy '.
22. What fraction Is that to the numerator of which if 7 be added
its value is |, but if 7 be taken from the denominator its value is | <
32
KXKHCISKM IN AI.iiKHIlA.
2.'{ Thrco times the KWitef of two nunibuiH excoedH twico th«
lusH i)y 115, and tvvici) tlio Kic'iUir tdgothor witli M fcimoH tho Iuhh ih
250. Find tlio lunnlitTs.
24. If B fjfivoH A .1*50 thoy will luivo i!(|iml suinH, hut if A ^ives B
$(44 B's nionuy will l»o iMpial to t.vici) A'.s niont^. How nmch Iiiih
oacli I
25. If X givos Ti .^5 hv will luivo SJifC. Iuhs tlmn B, l»nt if B givoH A
85, tluMi ;{ tinu's A"m nionoy will ho !S<20 nioru tlmn 4 tiuius lis.
How nuuli liUB ouch i
20. A fraction hecnnioH tuiual to h v/\wn 'A w adtled to tho numora-
tor, and 'j when ;{ in added to tho donoininator. Dotermino tho
fraction.
27. A fraction becomes e«|ual to I when tho denominator is in-
creased by 4, and ecjual to 'i'{ when tho numerator is decreased by
15. What is tho fraction '(
28. Finu two fractions whoso numerators are 'J and 5 respectively,
anil whoso sum is U ; and if the denominators are hiterchanged the
sum will be 2.
29. The sum of tho two digits of a number is 8, and if 36 be
added to tho number the digits will bi- reversed. What is the
number ?
30. Tf a certain numboi l)o divided by the sum oi its two digits,
the ([uotiont is (> and remainder 3 ; if the digits bo reversed and the
resulting number divided by the sum of the digits, tho (quotient will
be 4 with a remainder of 9. What is the niunber 'i
31. The first digit t)f a number when doubled is .3 more than the
second, and the number itself is i> less than 5 times the sum of the
irits. What is the number »
32. A boatman rows 30 miles and back in 12 hours. He finds he
can row 5 miles with the stream in tho same time as he can row 3
against it. Find the time going up and down respectively.
33. A man has filOOOO invested, part at 5% and the rest at 4%.
The income from tho former is $50 more than from tho latter. How
nuich has he in each hivostment ?
34. A ninnber consisting of two digits is e(iual to 7 times its
unit's figure, and if tho digits be reversed its value is increased by
18. What is the number i
35. A man travelled 240 milos in 4 days, diminishing his rate each
day by llu; same distuiice ; duniig the lir.sL iw.. days he travelled 13{>
miles. Ht)W far ditl he go each day /
IC<^IIATH»N.S,
33
.%. A and B uiigiiKod in tmdo, A with $1100 .md Tl !?I'J0O. A
loHt half as much a^aiii as H, and B had then loft half as much
again us A. How much did each lose {
'^7. Divido HO and *M) into two such parts so that tho sum of ono
from each pair may l)o 100 and ditlert'nco .'JO.
38. A farmer iiought HhL'o{) at $4 each and found ho was $H short
of money to pay f<»r them, hut had he only given l^.'J each he would
have had $4 <tver. Ilow many sheep were tiierc, and how much
money had he i
39. Find three lunnbers sudi that tlio Ist with h tho other two
= 34, the 2nd with i tiie other two ===34, and tho 3rd with | tho
other two ^^ 34.
40. A man has two kinds of coin ; 12 of tho first and 8 of the
second are worth ■£2,'',,, and 5 of the first and 10 of the second are
worth £1§. Find the value of each kind of coin.
41. There are three munhers whoso sum is lU. Tho third is I
less than three times the first, and tlie second eipial to the dilleronco
between tho Ist and 3rd. Find the nund)i'rs.
42. Tlie sum of PM\0 is lent to A and B so that 4 times A's
interest at 5/„ and 7 times B's interest at 4% is $!240. Find what
each had.
43. Tho sum of the heights of two towers is 5 times their differ-
ence, and half tlio height of the higher one is 4 feet less than 1: the
height of the h)vver one. Find tho heiglit of eacli.
44. Sold fi lbs. tea and ."") lbs. coflee for $4.88, and 10 lbs. tea and
12 lbs. coffee for $U.(>0. Find price of tea and coffee per lb.
■m^'
34
EXERCISES IN ALGEBRA.
MISCELLANEOUS EXERCISES.
A.
1 . Divide as^ + 24x + 65 by x- + 4a- + 5.
2. What is the value of aH 6' + fS - 3a6o if a = -16 = 3 and c =
3. Multiply x2-4 by uj2_4^. ^nd divide the product by .vH 2a;.
4. From a;HllaJ take Ga;2 + t> and divide the remainder by x - 2.
«n ^i "^^^A^i^® product of 3x-2!/ and 2j/-5a: to the quotient of
28x!/3 + lObxhj by 7xi/.
C. Subtract the square of a-\-h from the square of a-h.
7. Divide the sum of {x + y) (x+z) and (x-y) (z-x) by 2 + iy.
8. Factor a^ - 2a + a6- 26.
9. Write down the product of x^ - 9 and xH 17.
10. Find the product of m^-vm + n'^ and m + n.
11. What number is that from the double of which if 17 be sub-
tracted the remainder is 69 ?
12. Simplify 2.^2 + 4ifi/- 37/2- (x-- 2 (//.
13. Factor (a^ + ah + 62)- - {a ' - ly^y.
14. Factor 3x2 + 2x]/ ; 15a'6 - 106H 156c.
15. Find H. C.F. of x^^ - 1/" and x^ - /.
16. Find tlie co-efficient of x in (x - 5) (x - 6) {x + 7).
17. Multiply a + 6 - ^ by a. - 6 - -.
18. Factor 4a*6c - 3a62c H- 2a6c* making one factor a monominal.
19. Jamts has 2^ times as many dollars as .Tohn, and tlie differ-
ence between their sums is ,|40 ; how many dollars has eacli ?
20. Write down the cube of m - n.
i
!
21. Find the value of |5L x i?^ x '""
lex"
~x*'
22. Factor 8x2-f-7x- 46.
23. Divide the difference of the squares of 3x2-4.i-|-o and 3x^
+ 4x - 5 by the sum of the quantities.
24. Solve *±* - 9 = ^^ - i^.
c =
I
X.
2.
of
ub-
Fer-
3x2
MISCELLANEOUS EXERCISES.
35
2b. F-Actor X +5xy + 4y^ + 5x + 5y.
^^ J6. Find the value of (77x + 19;/)3 + (23x + 81y)» when :r = 73 and
27. Multiply 6a - 86 by 6a - 76 and divide the product by 3a - 46
28. Factor 28^2 - 109.T + 88.
29. Solve -'^
- 5 +
.5
= .x + 3.
30 Divide 100 into two parts so that if one i.art be divided bv 6
and the other part by 4 the sum of the quotients will be 20
31. If x ■= 4a + 6, 1/ = 5a - 36 and z = 26, find the value of 2.c + 4y + 5::.
32. Write down the co-efRcient of x' in (x' + 3x - 5) (.^'-^ - 5.». _ i).
Sr^+4d' ''^^' ^^'^' ''"^ ^"'^ '^"^' ^"'^ ^^'^ ""^^"^ «f a2 + 26^ +
34. Factor aV- ay.
35. Multiply ^!±^^ bv -^-^i::^
36. Divide x^ - bx^ -x + U by x^ - 3x - 7.
37. If fn,m 3 times a certain number we subtract 8, half the
is re'num bei ? "^"'^ '" ''" """^'"" '^"^'^ ^^i'-"i«l-d by 2 ; wl.;^
38. Find the sum of -;r-4- and ^
x^+x-+x+l
a--'
-a-'-'+.T-r
39. Simplify r.+^^'f'+Sab^+b:' a-^-2ah+b^-
40. Divide x* + 10x^ + 35x:' + 50x + 24 by (,i- + l) (,,. + 4).
41. Divide ^!£±£^t^^ bv -^
42. Divide a;* + x^ - 19x - 4x^ - 15 by x' - 2x - 3.
43. Simplify _62lzli±hl2_ , •i7ff^+i8ff-24 25^.= - 25^ +6
12«^--25a + 12 ^12a=;+ 7a-12 "^20rt^-'23^f,-
hatt"^'!?o'''fr*'S^ '^^''"i^^ ^^'V" ^''''' ^"" ' '^f your n.oney 1 w,ll
have ^1/0 then J„hn said t.. dames if yen will mve me k of vour
money I will have $170. What did each" have ? ' '
45. Put (a:'''-5.^ + 4) (x' + ox + i) into 4 linear factors.
46. Simplify 35!±lar?^f3y^
47. Write down the product of (x + iii- z) {x' + if + r' . .,. j/ . ^.~ . y.)
48. Find two equal factors of ix' - I2xit + 9*/^ + i^z - 6tr. + z\
36
EXEHCISES IS AUJKIJKA.
4J>. Find the value of — r when (t=^}f and 6 = i.
a- (/ - "
50. Simplify ^^^.H--^?:^,.
51. Find the value of a^ - P + c^ + oabc when a— '03, b='l and
c=-07.
CO o. 1 17-3aj 29- 11a; , 28a;+14
52. Solve x-="ir— +-"2r •
53. Multiply a^ + a'^ + a + lhy cv' - a* + a-l.
54. Solve 6(-« - 1) + 8(x + 2) = 27(i*; - 3).
55. a'^6 + 4 is a factor of (1*1"^ + a% - 12, find the other.
50. Tf a = 1, /> = 2, c = - 3, find value of (t^ + 8/>'' + o'' - Cmbc.
57. Simplify 42 {' ^
fix-Sy 3x-4v\ p.,. /3a; -21/ 2a;-3//\
58. Solve
7.r+l 17 -2a; f)a-+l
20
12
IC
59. The product of two factors is (9j +5j/)2-(5:«' + 9i/)^ and one
factor is a; - ;/. Find the other.
a;*-f2a;'-+0
GO. Reduce to its lowest terms v ,~rri7r;:~"T^ ..•
X* -4a;'' + 10a;--12a'+9
61. If x + d will divide x^ + l('>jc- + ax + IS without a remainder,
find a.
G2. Factor 24ic''' - 70.«;i/ - 75|/'^.
03. Find the sum of the sciuares of ■»(,,'• + (///; qx — viy; imj + qx
and qy- ))u\ and exjjress the result in factors.
04. What number must be added to a'^ + 9.:t: + 4 in order that it
may be divisible by x + Gl
65. Solve ?_2^-i^-^^^0.
2 3 4
,>n c- ^•e 2ita;+3rt" ^ax-' - Sa-x- a^-x-
^ •' 4a;--3aa; a-x--a* 2a'-+3aa;
07. M ultiply (('• + a"' - a - 1 by 1 - a + (r -a^ + a*.
08. Solve -^^~--^^"-'' = 3.«;-20.
09. Solve , , - ., ,- =2.
•j.x-1 3j- 1 1
70. Find tlie L. C. M. nf /^ - 9..- + 20 and .r^ + Oir-dS.
71. Divide .SaOO ])etweon A and V* so that for every dollar A
gfts li shall get. !:52.oO.
72. Factor 2,.'-' -2l,.M-55.
MISCELLAXKOUS EXERCISES.
37
73. Solve '^-l + '-J' 2-^-^-2 -^+2
7^4 12 ^ 28 •
74. Factor 21:1 - 2vm - k^ - I' + m' + n\
75. Factor a{a -h) + vi(a - h) + l{a - h).
B.
1. Find co-efficient of ,.• in the product of (,i; + 8) (.«:-|-3) (./• + 2).
2. Simplify ^;^+ '-^ .
3. Find pi'oduct of (.>■ + </)(.*; + ?>) (.«-f).
4. From result in number three write the product of (.*;-|-8)
(a- + 9) (j;-7).
5. Divide a^o + a' + l bya^ + ^ + l and multiply the (.uotient bv
(5. Simplify ^I^y-^ll^'-^l,
* 4 5 6
7. Solve f +11 = «^±i" + 6^-16 +i
2 ' 2 4 ^ 8 ^12"
8. A farmer sold 2 calves and 3 sheep for |50 ; and .'{ calves
and 1 sheej) fen- S4(). Find cost of each.
9. Multiply ^'*--^':>-^ by ^■'^ .
10. What value of x will make ,r'^ + 9 equal to 58 ?
11. Subtract {b - n) (c - d) from (a - h) (c - d) and find value of
result when (( = 2h and d = 2('.
12. Solve {x 4- 5)''^ - (4 - x^ = 21,i-.
13. Simplify -I +,'• ---^..
'■ •' 1+,'/ 1 -.1/ i-y
14. What number is it whose half, third and fourth parts taken
together are e(pial to 78 ?
15. Find value of x* - x^ - 4:X^ - Hx - 5 when x = 3.
16. What does x* - 4rt.r'' + ivf'x'^ - in^c + <(* ecpial when .*• =^a'i
17. Simplify {<i + h + 3c)' + (/' + 3c) ' - 2{b + 3c) {(( + h + 3c).
18. Multiply <(■ -b-\-c by a (piantity which will give (("'-(/<- <)^ as
product.
19. From the (piotient »>f (x'' - if')~{x - i/) take the quotient of
38
EXERCISES IN ALGEBRA.
20. A person has two kinds of wine, one at 40c. per qt. and the
other at 24c. per qt. How much of each must he take to form a
gallon worth |l.l2?
21. Find the value of .r^ + f/^ + 'J - lUiiz when .<• = 4a + 5, y = 4a - 5,
z=-8a.
22.8i,nplify(l+l)(^)(-).
23. Find H. C. F. ( >f x^ + 4x^ + 4./' + .'J and x^ + IW + 4x + 12.
24. What value of .*• will make (< -f 3) (.<H-4) <,'reater than (.*- + 7)
(x - 6) by 102 'i
25. If x=:2, v = 3, andc = 5, Hud tlie value <.f " ' ' + ^ -"- + ^"^ " ''^^-
26. Write down the cube of (4i/ - [iz).
27. Sunplify x , - -r - ., .
28. Tf nr - be = ,r, b"^ - ac = (/, r' - a/> = i, prove that ax + by-\-cz =
{x + y + z) (a + b + c).
on c<- IT 7a;-10 3x-7 27a;-30
29. 8nnplify ^ — -
30
30. Solve I +1-1^1
X 'dx 3x 3
31. Simjdify (2r»,^^)-(7^T7^3 + („rrT^
Xw - 2)
32. Reduce to lowe.st terms
j-''-3x- + 7.r-21
2.T< + 19a;'' + 35 '
33. Factor m^ - n^ - p'^ 4-7^ + 2 (m q + p}i).
34. Divide ;rH 4 + 2 by .*•+-•
X- -^ X
35. Factor (.r + yf + (x f //) {<t +b) + ,ib.
36. If a + 1 and a + 2?> are factors of V>a~b + 'A<r + 12ab'^+l2ab +
3a + 126'' + 6/>, find the other factor.
37. Divide x« - 20«».*'=' + 343a« by y^ + ax + 7a- and then divide the
quotient by x'^ + 'iax + 7<i'^.
38. If .*'2 + 47.<-m is divisible by .) -1-17, find ni.
39. Find the value of a; that will make x^-{-3ax'^-\- 4nh- - 9n* equal
to the cube of x + a.
40. Take r--A~.. fi-""» 1— V-.-
41 Reduce to its lowest terms
^^TT 4^HT xQa; a -^ 46X+T26 ■
/( - 1
n-\
42. Divide 1 + '-^ by 1 - '^^
n + l
n + 1'
MISCKLF.ANKOUS EXKKC.'ISES.
39
43, Which of the factors x2-2x + 3 or 2j;' + a;-4 is likely to be-
long to 2.*-* - 3x^ - 2x' + lOo: - 8, and why ? Find the other factor.
44. Divide -^-^-^lill* by *-+^-'-^«
' j;=+7a;+12
x^+7x + 12'
45. When .K- 5 and y = 3, show that {2,r + yf - {2:>' - y)^ is eciual
to 8x'(/,
46. Prove that (a ~ 2)' - 2(a - 2y + 3(a - 2) - 4 = a» - Sa^ + 23a - 26.
47. Find value of (,<: - //)■' + (.r - 9j/) (.*• - y) when x = 5 = ';/+ 1.
48. Multiply 1 + a6 + a^b' + a%3 by 1 - ah + a*h* - a%\
49. Divide :,'* + xh/ + y' hy x' + ry + y% and hence find factors of
50. There is a number of two digits of which the unit digit is
three times the other, and if 54 be added to the number the digits
are reversed. Find it.
51. Multiply i-^, ^ and 1 + ^^ together.
52. a* + a^iy' + b'is divisible by a' + ab + ¥ without a remainder.
Apply this to divide x^ + lO.i* + 25(5 by x* + 4x2 ^ |(;
53. Write down the co-efficient of x"^ in (x^ - 4ic + 9) {x^ + 3x - 5).
54. Divide "p^^^^ by ''^^
a- -ac+ad-cd
55. Simplify „^^^ X
a + d'
4a2a;2+2aa;'''
2ax-x^ x^-ax+a" ' ia^-x-
56. Find value of r— - +, — when x = ^.
l+x 1-x -
57. Divide a^-4ab + 4¥ by
a--2ab
1i+2l}
58. W>ite down the cube of r +- •
a
59. Find product of ^^-^/and-'4- + ^ + -?^.
•^ a b a^ ah ' ftz
60. Simplify --— L_., + ^^^^-L--.
61. Find L. CM. of a^-l; a'^ + 2a-3; a'-7aH6a.
62. Simplify ( ^^-+J-)-f_
' •' ^a + & a-b' a
i- li-
ar - cp
63. Find value of (f,c + 6i/ when;^— ^^^ — ;- and u j- .
aq-bp ^ aq ~ bp
64. Reduce to lowest terms
(4 + 12X+ 9a;«
2 + 13!c+15a;2'
65. SimpUfy(^4^,-';;e|:)(j,. + ^.)
40
EXEUCISKS IN ALGEBRA.
<i7.
()«.
()!).
70.
71.
72.
7.'}.
74.
7r,.
( i( ;. Fuc-t. )r .. ■■' - 8//' + ;? + <M'//^.
i^a^ + 27h^-r^ + lSabc.
Hn^-27h^-c'-\Habc.
:'•■' + 8|/H <).!•// -1.
;,.3 _ 8yS _ *;,,.,^ _ I
X* + 8,i:3 - 1 0,r2 - 104.'' f 105.
X* - 14,r3 + 7U- - 154,f + 1 20.
.,.4 4. 4^.3 _ 49_,.2 _ IJ.^;,,. 4. j^^Q
7(). Prove the following identity by fficttjrinc- • (9,r^-4(/'^) ('81a;* +
16|/^ + 3Gx-^/yO = (27.>"'' + 8//^') (27.*-' - S,/). ' ^
77. Divide <(^ + b* + 2aW - c* - (Z* - 2c-Vf-^ ] >y ,,-' + //^ + ,.2 + ^^2.
78. Show that (((2 _ bcf-h (62 - ac)3 + (<- - ohf - 3(a' - be) (6- - ac)
(c^ - ah) is an exact square.
79. Factor a;"* + lO:.-^ + 17^:^ - 40x- - 33.
80. Show without simplifying that x + 1 is a factor of mn(/^ + l)
+ {h' + nr) (..■* + ,«■)- (n' + 2wn) (x^ + x^).
81. Simplify (<* + h) (a + 2h) (a - 2h) (a - h).
82. Simplify—!^, x'-^^^^.
n 1)1
83. Factor 4a^ - 12a^c + 13a 2.^2 _ ^-^^^^z ^ ^i
84. Solve -+-=-5- + __. + i.
85. Factor x^ + Ghim + 5?n2 _ 12^,1;^ _ 97^2^
8(;. Multiply itj-i-l' + i ])y ;,.-i + £' + i.
87. Factor ((6 + 26/ + (2a + 6)3.
88. Show that U-W+^±^-J^+^\
89. Find an expression that will exactly divide ./•' + 2aa''^ + o2 ,.
+ 2rt3 and «=* - 2ax^ + a'x - 2a\
90. Tf 4.,"' + 28.>'3i/ + R;ry + 42r/ + 9/ is a perfect s.juare, find R.
^
i
a
bi
MISCELLANKOUS KXEHGISES.
41
EXERCISE I.
1. Solve
f +!+?=- + «.
2. n
1 + i _ i_5 .
3. .,
zx+i r.
2X-1 "" 4 '
4. „
128 216
Sx-4 5.r - 6
5. Find the value of (2r+|/) (x + |/) + (2!/4-.;) (i/ + ~) + (2:; + x)
«^: ^i^^l' ^^ Horner's method of division, the quotient of 2W
- 2j:* - 70x3 _ 23^v! ^ 33 ,. _^ 27 divided by 7x' + 4u- - 9.
7. Show how to find the product of (tn - ti+j) - r/) (?/i, -n~p + q)
without multiplying out, and write it down.
8. Multiply a2 + ah + h'' ; a^-ab + 6'' ; and (t^ - a''¥ + b\
9. Factor a' - ah - 66^ . a' + b'- 23a'b' and 15 /^ + Sxj/ - ICnf.
10. Show without multiplying out that 7^ - 1 is divisible by 6.
1. Simplify
EXERCISE II.
S(x^+x-2) 3(a;2-a;-2) Sx
X--X-2 a;2+x-2 x"-i
2 Factor 1 - a^ - b'^ + c^ + 2ah + 2c.
3. Divide j^ + }/ +z^- 3xyz hy x + y+z.
4. Show tliat {a - bf + 3ab(a -b)^ (a + bf - 3ab(a + b)- 2b^ if
a = 3, 6 = 2.
5. Simplify <4:i^M+^^^l2^(^+2/_)'.
6. Express (a'-' - //') (c^ - cZ^) as the diflFerence of two sc^uares.
7. Factor 6x'' - bry - Gy"^; x^- + y^' and 2a' + 5a + lab + i\b' + 8/> + 2.
8. Simplify " -J -^«+^^ --"-««_.
9. Simplify ^!^.\^ •
10. Express rt.2((^-/)) + /)2(a-c)+fXfe- a) as the product of three
binomial factors.
42
KXEKCISKS IN ALOKliRA.
EXERCISE III.
1 . Solve 3^ - 4 ^ 9 - (2^ + 7) r + 3r - 18.
2. Solve -8^+4-^rT^-l-
'A. Divide $2280 among A, B, (', giving A g more than B, and B
!$24() less than C.
4. Divide the ])roduct of {.^'^ ~ -I'y + >/) hzuI («'!'^ + jj/ + i/) by the
(juotient of x^ - 1/ by x - y.
5. FindH.C.F. of H.c^ + . rhj + ih; f' + :m, / &nd i^.t'^ -2'Axhj + o2xy'
-If.
6. Add 1 to the continued product of x, x' + l, cc + 2and x' + 3
and divide tlie result by .>:;'■' + 3,r + 1.
7. Factor 4..C* + A^i^y' + 4?/* ; //'' + c' - a' - 26c, and 2a2 + Sax - 2x-^.
8. Solve ^*^^^+-^ + 5 = x.
15 6
rt c< 1 3j;-13 4a:+6 . x-l
9. Solve — g— ^— = I - -^(^ •
10. Show by Hoi-ner's method <>f division that when x^ + 1 is
divided by x^-\-x-\-\ the reniainder is x + 1.
EXERCISE IV.
1. Simplify 3(x + :;) -(«(/- c)- 2-^ x- (2;/ + 2)-(i/-3;) }>.
2. Find by factoring the L.C.M. of x2 + 5x + 6 ; x''' + 2x-3 ; and
x* + x3-2x2.
3. Distinguish between an identity and an equation. What
value of C makes (x - 2f - (x - 1) (x - 3)- an identity ? Will any
value which does not involve x make it an ecjuation I
4. Show that the difference of the squares of any two consecu-
tive numbers is equal to the sum of the numbers
5. Multiply ^' + 1 + ^ by x-l+\-
7. Write down quotient of (a-^ + ab + b^ + (a^ -ab + b'^f by a» + &=».
8 CI 1 2j; r\ ox -lb
. Solve 9-2 =-yg •
9- Solve ^-:-H-l=0-
10. Divide (i + v/ - 3(a: + yfz +3{x + yy -z' hy x + y-ss.
s
I
MISCKIJ.ANKOUS KXKIU'ISKS,
l.J
EXERCISE V.
1. Show thiit (m.r.+ nij + i>:f + {f,.,' ini + m-f is divisible bv
(m + />)(..; + , :). ^
,.J^- ^*'""' {"' - f')-*-- {h - c)>j take (a + b)x + {h + c)ii and divide the
dineroiice l)y .* ' + |/.
.'3. What is meant by a co-efficient? Find without multiplying
out, the co-etiicient of :*■ in the expansion of {x + 4) (r-f-5) {x + V)). "
4. Find the co-efticient of x when x^ + 4x' - 110.»; - 03 is divided
by 05-9.
5. Multiply a2 + //' -c2 + 2«Z> by cHaH?)''' + 2rt6.
6. Divide x^ + 3ax^ + 3a'x + a^ + b^ l)y x + a + h.
7. If « = 1, 6=: 2, (• = .'], fZ = 4, find the value of
a+h 4h-c c+d _ Id -ja+h )
a-f)'^ b+c "*" c-d c+^r^~''
8. Find by factoring the L.C.M. of x'^ + x-2', x^-4x + 3 and
x"^ - X - (J.
0. Sill, plify (x + y + ::)■' + (x + y - zf + (^ - J/ + ::)' + {~x + y + zf.
10. S(»lve.«-^-ji-| = ^;^-^' + ^^
EXERCISE VI.
1. What is the value of (i when x* + ax'^ -('tx'^ + 3x~2la vanishes
if u; = 4?
2. The product of two algebraical expressions is x''-64x and
one is .<;- - 4, lind the otlier. ^r \ \
3. Factor iri^-Jii^s + Sx'* -,'•=' -8.
4. Factor SOx^ + 27.' ' - 20,.' - 15.
5. Write down the cube of x + 8y.
. _ 1 . _ 1 . _ 1
6. Solve ^_i "___ '^ , 1 ^=0
<! <f b
•v©©-'
gWO^
7. Simplify 'LL^(.^ + //-.^) + 'i;::(/.^ + .^-a^) + ';;;)lH^-/.^).
8. A train can-ies fu'st-class passengers at 4c., second at 3c., and
third at 2c. a mile. There are 12 times as many third-class as
second-class ; five times as many second as first. The whole fare
was $11 . 12 per mile. How many are first-class ?
9. Show that (n + b- cV^ -(<i-h + c^ = 4<t(b - c).
10. Prove by division that , = a' + 1 +-.^ '
a —
a
r<.i
II
KXKItCISKS IN ALCKFUIA.
EXERCISE VII.
1. Find value of^'J^;^ ''^'^■'^ ^^J"'" '' ^' 'J-^ ;: = 6.
2. Einj>l<)y factors to tiiul tho result of dividiiij^ <»' + //'c'^-a'-'c'
- <rh' \)y <u' + 0'^ - he ~ nh.
.*{. VVrito down tlio (luotiont of (4.i' + .'///- 2-.)' - (3x - 2(/4-.'i':)2
divided \\y x + iiij- 5;;.
4 Sinndifv ^^""*'^^'+^""''^"^-^^"+''^^"-^)K
A -^ ((■'+a-h+ah-+ly'>
5. Tf (.'•- S)''' -(,.;- 7) (.'•-3) = r«, tind tlie value of a in its
simplest form.
6. Find tho value of (^ -^) (^„ + E)^(±' + >£') K- Ii\ when
X — 4, y — <>.
7. Prove that x'^ - 3ic - (/ ; and .i'^-4a-r» are both divisible by
tile same <iuantity if a = 10, and Hnd it.
8. S<.lve -^_^_+.--^_^ =3.
9. The sum of two numbers is 35, the dilierence exceeds \ of
the smalh-r by 2, find the nund)ers.
10. Solve 7(.'' - 2) - o(2x - 9) = \{,v + 13).
EXERCISE VIII.
1 . Solve {2x - '6y - {2.V - 7y - 5(.f + 3).
2. Factor ifz - yz'^ + z\r - zjr ; x' - (a + -)x + 1 and ■/' - 3,t;(a + h)
-^{a + hy. '
3. Find the value of (1/ - zf + (^ - xf + (a; - xj)' when .,•:==- 1, y = 0,
4. Find by factorhig the L.C.M. of 5a;^- 15.»-flO, (u''^-6x-]2
and Vlx'' - 12.
5. Reduce to lowest terms
x*-\-i z^- 19x- - Wx+VM)
3.4 _ 26a;- +144
2a
x~a 2
I y (a; -2a)- «--5aa;+6a- ' x-3a
7 Simplifv ^'^+^)('^+^<^) a<a;+a)(2a:+a) ^
6a
8. Deterrrune a and b such that in the product of x^ + x + l :\d
x^ + ax'^ + bx + c the co-efhcients of cc* and x^ may vanish.
9. Show that (Sic- 3 (/- 4)2 -(3x4-71/ + 4)^8 exactly divisible by
2x + y.
10. Express ^-^(l-l\.->i^ill\ as the difference of two
squares. a ^ y y ^ Ji
MISCELLANEOUS KXKHCI8ES.
46
EXHJRCISE IX.
1
X
J. Dividt) .»•*- \. by ,,
2. Find the imxluct ..f yt ^h) (a^i-ah + h^) (a - />) («« - ah + !,•').
.*{. WlK.vv that (x - ,,f + (,, - -.)3 + (., _3.)3 = ;j(_,. _ ,^) ^,^ _ .,) ^., _ ^.^
4. Solve lli' + ^' = 2.
5. Divide (.,■■' - (/,';)••' + 8//';;3 i,y .,,1! ^ ^^.._
«. At yvlmt times between 4 and 6 o'clock are the hands of a
watch at right angles to each other '{
7. Show that liia - h) (a - c) + 2{b -c){c- a) = 3((f - hf.
8. S( )1 ve a{x - a) + h{x - h) + 2ab = 0.
{>. Use Horner's method to tind the quotient of r'H.'-«-2
divided by ,*'^ + .»•'-' +1, ^ -r *
10. Find the continued product of x-\-a, x4\ x-\-v, and from
tlie result write down the product of a - m ; a - n ; a -p.
EXERCISE X.
1. What must be added to {n+h + ry that the sum may be
[^ti — f> — cy ( *'
2. What (quantity must be multiplied by x + l to give x^ + 3x'' +
fix -f- 1 E
3. What must be multiplied by, i:- - to give x^-\-(x ~ 1)" ^
4 The price of barley per bushel is 15 cents less than wheat,
't J'-S^ r'r. 'i 50 bushels barley exceeds that of 30 bushels wheat
Dy ^o.50. mnd price ot each per bushel.
5 Examine whether x^-dx + H is a factor of :»;» - 9,/^ + 20 r - 24
and hnd the other factor.
6. Show that a' + cc'b' - aU' - b' has a' -has a. fact<.r, and Hnd tlu'
other.
7. Solve 1+ ?>=^+d
8. Solve - +'r + - =1
9. Divide m'' - (^ u + ,-jm + l by m-n.
10. Solve ^^-''^''±^^^^'K
20 21a; - 2 5
46
KXKa<MSKS IN AI,(il{|tUA.
EXERCISE XI.
1. Divido tho product of <r-\-ax + .>'' ;uid a» + a;' by a* + aV + x*.
2. Show that (l+.'+^;^+....x«-')(l-.'' + .'C»-x»+....a:"-0 "
o.iuiil to l+x' + x*+ ';'"-'.
',\. Find what vuluo of x will nmko 5(.i;-3)-4(j +1) t'<iual to 76.
4. Find by factoring tho (luotiout of a» + a% + ii^c - abc - b'c - hc:^
by (i^ - '"'.
5. What must bo tho vahio of ac, y, z tlmt u^ + a'x + ay + z may
havo a - 1, <t - 2, a - 3 as factors '^
<). Divido ,^ + {m + n + p)x' + {>nn + inp + ni>)x + mni) by :«+p.
7. Find without actual niiiltii)licatiou tho [)roduct of -g- - -cy
•*■.'/
1
+ 9byf + 3.
»* - 6a; 4-8 g'-Sg+fi ^(a;-2)a
8. Simplify ^^rrj^:;^ ^ x''~u-s~ x'^ - 1
rt7> + rt
9. When x = :,^y and ^ = ^1;^^ roduco ^;;j;; to its sin.plost
form in terms t)f a and b.
10. Prove *'-^ = (x + 2|/y^-3!/(.«-i/).
EXERCISE XII.
1. By what binomial must x»-3u;-2 be multiplied that it may
be a perfect square i
2. Show that X* + 1/ + (x + y)* = 2(x^ + xy + iff.
„ ^, .1- i.1 .«'+''* "^^ (a+b+c){a+b-c)
3. Show that 1 + —gal, — "^ 2a6
4. A cistern is J full of water, but 220 gallons are run off, and
it is then ^ full. How many gallons will it hold ?
5 Find the value in its simplest form of
W'^ 3a- 32a;-/ ^a-'x-' ' a-'x"-!
6. Find H.C.F. of (Sx^ - lOx^ - 16x - 3 and 2x^ - llx'^ + llx + C.
7. Factor i(;'H4x' + 4- 'aH4a«/-i/-.
8. If y= -2, tind tho value of y in the equation 7x-f 18|/-=4.
9. If I of my money is ecjual to ^^ of it and $47, what is the
sum ?
10. Factor x' + Gu;^ + 27.c'^ + l^%c + 729.
M ISCELh A N ICOUS K\ K KCISKS.
47
?t
BXBROISB XIH.
1. Factor l(ur« - 24.i-' - Hb:^ + V2x + 4.
2. If (,, /, ,' l>i> throo o.msocutivo nmiihors, r boiiig M,o groatost,
Bh«>vv tliHt the (liHort^ncu botweoii tho H(iuare8 of a and h is li leas
than c.
3. S,.lvu '^^ =-^+J-«.
8j+1 a;+7
4. Factor JU-' - 14x-^ - 2ix.
5. Tlio exi.roHs train fr,„n Sarnia to Lf.ndon travels 'A2 miles per
hour aiu runchos London in 2 lio.irs lo.s.s time than the mixed train
at 10 uules per liour. Find the distance.
C. Multiply x>> + jc^+x* + xHl by x'^-l.
7. Express in 4 factors, 3(6.fH5x')'^-10(Gx» + 5ic)-8.
8. Divide (a' -hcf + 27 b^c^ by a' + 2bc.
V. ittauce ^4_9a.;,+os)^,_39^^ig to Its lowest terms.
10. Simplify ?-<flld^+«(«='r''_^>J.
b'ib+ax)
ly
id
the
EXERCISE XIV. •
l^iH ^i'f. ",f'^' ""^ ^«-^' + i7.;^-128a;^-14^ + 9and24xH22a;»-
3a/- 2^'"'^ ^'^ f'lctoring L.C.M. of 9a^-3^x\ 'ia'-^ax+x\ 2a^+
3. A herd of cattle cost $720, but two were stolen and the
average cost per head was then $4 more than at first. Find the
number.
4. Divide ce» - (a + h - c)x' + (ah -be- ca)x + abc hy x + c.
5. Factor x^ - y^ - .r(:r'^ - y^) + y(x - y).
x"-2-^ x^-4z~21 . 2a; -10
G. Simplify ^:-;;;x^
^ •' a;--49 x"
+SX+15 • x+7
a-?2/>-3o) ^^ ^''*'*'^""^' *^*^ quotient of a3-(26-3r)' divided by
8. Simplify -2^^^^±^^ j^nd test result by putting a;= 1.
!). Di V ide 1 + c"* + y' - 3y;: by 1 + ;/ + z.
10. Solve •6^--75-l-8x + l-2 = l-5x-y-5.
48
EXERCISES IN ALGEBRA.
EXERCISE XV.
1 _i
1. Find by factoring the value of-** — 5.
2. Factor (x""-^ a -iy-a\rr.
3. Factor 2(a=' + a'b + air) - (a« - ¥).
4. Show that {j' + yy + {ii + -.y + {z + xf + 2{x+y) (x + z) + 2(x+y)
(y + z) + 2( !/ + z) (x + z) = 4(x + ■;/ + z^.
(r:+z - 2m)' - (z+m - lyY
5. Simplify
(>;i +y+z)' - (m-YV - 52)-
ti. Show that / ., ,?tx /t.— TTx is always a proi)er fraction.
(a-+b') {c- + d-) '' ^ 1
7. Divide f - 2f + 1 by i/ - 21/ + 1.
8. Factor x^ + x{m + n +^) + p(wi + ^0-
9. Find three numbers whose sum is 51, and of which the
greatest exceeds the least by 0, and the third is one-lialf of the
other two.
10. Show that 6 + rf is a factor of (a + 6 + c + dY - (a - b + c - (Vf.
!
EXERCISE XVI.
1. Show that x{y+zy^ + y(x+zy^ + z{x + yy^ — 'kxyz — {y + z) (z + x)
(x + y).
o a 1 ^2,1 29
2. Solve ~ + ^- = ^-
3. A man bought tea at 78c. per lb, and ^ as many again lbs. of
coffee at 30c. ; he soil tea at 96c. and coffee at 27c., and gained
$12.60. How many lbs. of each did he buy ?
4. Shuwthat(a + i)^-(^ + ;)^=(a6-l)(^-5).
i). Snnphfy --'„,_;=
«). If a + b + c = show that o' + /;^ + c^ = 3a/jc.
7. Factor x^ - -»-''' - '/ + 1 ; rt^(^> - '0 + b\e -a) + c^a - b).
u o 1 4,7 37
8. Solve -,H — 77, ~-~rrr 7 :\'
9. Reduce to its lowe.st terms
a- \ a+b" +b+c" +c+'iab+'iac+2bc
a" -h" ~ c^ - 26c.
-__._.»
10 Simplify J«±''Hli«^) _ ^'Xi-m^Xi «-) .
MISCKLLANEOUS EXEHCISES.
49
EXERCISE XVII.
1. Solve ^s?+i=-4i^+4:t^.
o 15 o 45
2. If a^ + })' = 1 = c^ + d'^ show that (ac - hcCf + {ad + hcf = 1.
•■'■ Simplify „-4^+,^^^+..^^-
4. Divide {x^ - 2yzf - 27i/z^ by x" - 5yz.
5. Find the value of <t and b when x'^ - ax + 12 will divide x^ -
6.«^ + 75^:-108 without a remainder.
6. Simplify {x + 1) {x + 2) {x + 3) -(.«;- 1) (.« - 2) (x - 3).
7. Show that 2(a2 + 62 + c' - ah ~ ac - be) = (b - cf + (r - af + (a
-by.
8. Find two consecutive numbers such that the half and fifth
part of the less may be equal to the third and fourth parts of the
greater.
9. Show that (b - c)3 + ((• - af + (a - bf - 3(6 -c)(c- a) {a -b) = 0.
10. Show that a^ + b'^ + (f - 3abc ~ a(a'^ - be) - b{b^ - ca) - c(c^ - ab)
EXERCISE XVIII.
1. If x=b + c-2a, y = c + a-2b, z = a + b-2c prove x^ + y'^+z'^+
2xy + 2xz + 2yz = 0.
2 0- ,.- b"-c- , o"-a~ , a"-b^
. Simplify -j-i \ r •
^ ^ b-\-c c+a ' a+b
3.
4.
2a;
ix^
8x'
l~x \-\-x 1+a;^ \-\-x* l+a;»
x-'ia x+a ^-+Sa-
x+a x-'a x--a-
5. Prove that (x + )/) {x - //) + (i/ + ,:) (i, _ ^) + (~ + x) (z-x)= 0.
6. Prove that (f (/> - c) + b{c - n) + c(a - b) =- 0.
7. Show that (a + by + ^c{a + by + ScXa + b) + c^^(h + cf + ?u((h +
cf + 3a\b + c) + iv: ^
8. Simplify {a + b + cf - (n + bf ~(b + cf - (c + af + a'^ + b'+v''.
9. Divide 17<)0 yds. into two ])arts such that half of one part
with 2(R) yards may bo double the other part.
10. If p=^a-[-b-\-c, q = ab + ac + bc, r = abc, prove pq - r^{b + c)
{c + a) (a + b).
50
EXERCISES IN ALGEBRA.
EXERCISE XIX.
1. Simplify (3.« - 2)^ + (3^ - 2) {\ix - 4) + (Sx - 2) {^x - 5).
2. If x^ - yz = a, tf -xz = h, z^ -xy=^c, show tliut .o' + y^ + z^ - ?>xyz
= ax + by + cz.
3. Required the number whose ^, I, \ parts together are as
much greater than 223 as ^-, ,|, {- of it"are less than the same.
4. Add together the squares of ax + hy and ay - hx and subtract
the sum from the product of (a^ +(/''') and i}j^-\-x^).
K TP i. 64a'' 21b-'
5. Factor ^,^ -- ^^ .
6. Simplify ^ (7x^ + 4a; - 3) (7x'^ - 4x - 3) ^ 4- -! (7x - 3) (7x + 3) }- .
7. Solve
CC-l . X-
x-3
34
51
102
= 0.
8. Solve X — '-
9, Show that (a^ - be) + (b'^ - m) + (c'^ - «6) is not changed if a, b,
c is eacli increased or diminished by the same quantity.
10. Divide 32xH243 by 2x + 3 by factoring.
EXERCISE XX.
1. How many lbs. tea at 3Gc. and 60c. must be mixed to make
200 lbs. worth $80.40 ?
2. Write down the product of (l4:.« - 17) (14.t; + 17).
3. If (x + 1)2 = X, find value of llx^^ + 8x' + 8x - 2.
4. Factor 36x^ - 97xhf + SGi/.
5. The perimeter of a square field is 588 yds. and of another
672 yds. Find the [)erimeter of another ecpial in area to both.
6. What nmst be added to 6x-^-x^ + x^-x + l to make it an even
multiple oi x'^-x + l and .« - 1 ?
7. What value of a will make 3x'^-7x^ + 2ax'-llx + a exactly
divisible by x^ - :»; + 1 ?
8. Find the value of — -*- when a; = l.
9. Factor Oa^ - >jnb + 2ac - 2(>/>- + CAbc - 48el
10. If <t^ + ((6 + 529 is a perfect scjuare, find 6.
MISCELLANEOUS EXEUCISE8.
51
EXERCISE XXI.
1. If a, />, c, d, e, represent the first 5 numbers and a' = 0, find
the numerical value of x"((i + h + c + d + e) -aHb~c)-b'Hc-d)- c'^
{d-e)-d\i'.-x).
2. lix-\-y=^a, y + -==h, z + x = c,a'^ + ¥+c'^ = 0,Hh(m t\vdtxy + xz +
yz = ^(ah + ac + be).
3. Find H.C.F. of a^ + 6a* + lla'^ + 5a'' -'3a -2 and a^ + 3<i^-
6a^-5a'-3a-2,
4. If a; -2)/ is one factor of .)i:P-2xhj-'ixy^+Si/, find the others.
5. If Ax* + 12xhj + ax^tf + Qx\f + y* is the square of 2x^ + 3xy + ktf,
find a and k.
ft. What three linear expressions divided into a;^-7ic + 10 will
each give a remainder of 4 ?
7. If a + ,- = 1 and c + - = 1, show that &+- = !.
a, ' c
8. If x=19, find value of ^2±^ + ^^->^-^V^.
9. Find the equal factors of 9x* - 6x^ + 43x^ - Ux + 49.
10. Show that the sum of every fraction and its reciprocal is
equal to or greater than two, and that ^H f-jH l-r+->t>.
EXERCISE XXII.
1. Is (a' + c'^) (b'^ + d'^) greater or less than {ah + cdy when ad — be?
2. If m = 2x^- 16ic + 14, ?t=a;^-5x — 14, factor the sum of m + n.
3. Show that {Qx"^ + 4xy + y'^y - {3x^ + 4:xy + y'^)'^ is equal to 4a;'*
{2x + ijy.
4. Divide(^! + ":-2)'l,y 'i-^.
5. Find the value of x tliat will make a'^ + 6 A + 8x'c'^ + 10c* equal
to the cube of a; + 2c.
6. Factor mx^ - 129a; - 9797 ; 27a;'^ + 192a; - 4067.
7. Find the remainder when r)a'^ - 8x'^ + 8a; + 7 iw divided by
5a- - 3.
52
EXERCISES IN ALr.EHRA.
8. Show that the difference between -^ +— — 4.-^?^ an*l
, , r , s- . ^, , m-q^ m-r^ m-s '*"^
^ jn-8 ^^ same whatever m may be.
m-q m-r in -8
?k ^' ^' ^ '^^^ ^^*^ *^*^ '^ society. B gives twice as much as A
and {$20 more, C as much as A and B togetlier. What did each
give ?
10. Tf «-4, b-5, by what must ax^+bx + 1 be multiplied to trive
EXERCISE XXIII.
1. Tf (ly + hx = a, by - ax = h, tlien x'^ + j/' == 1.
2. Find the val ue of a^ - b^ - (a - by when a + 26 = 13 ; 2a + b= 32.
.3. Solve -*--- .% . = -2a_
x-a 7(x-a) x+7a
4. Solve ^^y-='^.
d(c+dx) d c+dx
5. A tradesman after spending $100 a year, increases the re-
manider of his property by J of itself, and at the end of 3 years liis
original capital is doubled. What had he at first ?
6. Show that x^+xf + z'-xy-xz-ijziB not changed by addint^
the same quantity to x, y, z. °
7. Factor x^ + Sx"^ - 79ic + 70.
8. Of the fractions 1^^ and |!^^, which is greater, wlien a is
greater than x. ^
<>. If .,; + ^, + . _ 0, prove that x(x' - yz) + y(y^ - ,r~) + z(z' - xy) = 0.
10. If x' + «2 = 2(xy + yz + nz- y^ - z"^), prove x=^y = z = u.
EXERCISE XXIV.
1. In the expression :);3--2a-.2 + 3,c- 4, substitute a -2 for x and
arrange in descending powers of a.
-.«/)
2. If X - i/ - 1, tlien (x^ - y^f = x^ - / + ,:y.
3. Prove that 2(x^ + y^ + z^- 'Axyz) ~ x + y + z^ {y - zf + {z - xf + (a:
4. If X - y = 2((, show tliat x"^ - imx + 9a- ■=((/- af.
MISCELLANEOUS EXERCISES.
53
5. A person leaves A for B at 3J miles per hr. ; 40 minnfes later
another leaves B for A at 4| miles per hr. , he goes h mile more
than half way when he meets the first traveller. How far from
AtoB?
1 _ J_ 2 - -
6. Simplify :7-7^ X =
x-'+l
:.--'- ' 1+'-..
X X-
7. Prove that jx'^i="£'
8. Show without division that x-a is a factor of x^ — (a-m)
r'^ + (f-am)x-af.
9. li X- a and x-h are each factors of a;'* + x + 1, then a"' + 6'' = 2.
10. If <t + /> + c=0, show that a^-b^=hc-<<c.
EXERCISE XXV.
1. If x + n and x + h are each factors of .r'^ + my^ + h then
m
a + h
2. Find the value of x which will make x^ - 2x + 3 a f actor of
x''-x2 + 5x-21.
3. Simplify (f - ^^ - l + l - ^)Ho-h){}>-c)(c-a).
, „ . 2x-7a , x-Sa x-7a 2x-9a
4. Solve -- -r- + ---^:. = ; - v, _ + •
x-'ia x-^a x-Sa x-5a
o. Find the value of a for which the following fraction admits
, ,. x'^-ax^ + lQx-a-i
,.f reduction: ^i (a+f^+as'^^a -'7 '
l + ^+l
6. Simplify ^-^ ^ •
X" " 3
7. Find by factoring what algebraical expressicm multi]ilied l)y
itself gives 25x* - 2()x^ - i')x' + 4.i- + 1.
„ c , 4;r-ll 2a;-7 ,11 r,x-U 3a--ll
8. Solve ~^- 6~+96^ 18 ~"""9
9. Prove algebraically that the sum of the squares of any four
consecutive odd (or even) integers diminished by 20 is a square
integer.
10. Show that x' + i divided by {x. + lf gives remainder 7(.''4-l).
54
EXEKCISKS IN ALGEBRA.
EXERCISE XXVI.
. } ' -^V,"^ ^^^^ ^'^^"^ "^ -^ ^^^^^ w^ 11 "lake x^ + 3a:>'^ + 4a'x - {ia^ eii ual
to {X + ay. ^
2. If two numbers differ by d, show that the difference of their
squares is d tniies their sum.
o jfb''+c^-a'^ An-b+c)(a+b-c)
2bc
•^ fo^i'\i ^^^^^^"^ (-^-y) (^ + y)-{x + yy whan Sx + 2y = 45 and
oy + zx — 15.
• ?■ ?9^® L.C.M. of two quantities is a* - 5a^h' + 4h* and the H C F
IS a' - h\ one quantity is a' - 2a'b - ah' + 2b\ lind the other.
6. The H.C.F. is m-7, the L.C.M. m^-10m' + llm + 70, one
expression is m' - 5m - 14, find the other.
7. Factor x^ + 25^ + 289.
8. li x'^-3x + 2 = 0, show that x* - lO.-j^ + 35x^ _ 50^ + 24 = 0.
lby2-(i + i+J;.
10 Show that (x'+6xy+4:y')'+(x' + 2xy+4y'f is divisible by
EXERCISE XXVII.
1. Show that x* + y* + z* - 2xY - 2xh'' - 2yh'' is divisible by each
of the four expressions x±y±z.
9. Divide ^' + ?^li + £zi
2. Simplify
5Ga;''-28a:"-42a;+14
42x2 -28a; -14.
3. Show that the value of the difference between (x + -)' and
(.'•--) is independent of x.
4. Show that any four consecutive odd (or even) numbers plus
16 may each be i)ut in the form of two equal factors.
5. From a + b-c take hi -hb- f c.
6. Show that ^ = x when x is any number.
7. Find the value of ^^'{ when a = 3, b--=2.
I
MiSOKLLANEOUa EXERCISES.
55
8. Find co-tillicioiit of x' in yX^x-Vx^'-^-y?, etc.)''*.
9. Of two sciu.'ire fields one exceeds the other by 100 acres and
its side is 400 yds. kniger. Find the length of the side of the
smaller Held.
10. Solve a{x - a) + ?>(x- - 6) + c{x - c) = lah + 2ac + 26c.
EXERCISE XXVIII.
1. Find the value ul c if x^ + 2x^ -10x^ + 3cx + 2c is exactly
divisible by x^ - ox + S.
2. From 16 (^'^ + '^^) take 32 {'^-J^-'-^).
3. Factor a;'-(f -^)
x
4. Find the co-efficient of x* in the product of
-, , a; , x^ , x^ , X* 1 -, , X , x- , x" , x*
1 + 2 +4- +8- +r6^'y 1 + 2 +4- +8- + Fg '
5. Write down the cube of l + x + x\
X x^
6. Write down the square of 1 + - + t- •
7. The depth of a cistern at one end is twice that at the other ;
water to the depth of 18 inches is frozen and the water below at
deep end is three times that at the other. What was the original
depth at the deeper end ?
8. Multiply l + ^a+^6 by l-|a+^6.
9. Prove ^^^ - %;}'^-''^' + ^'' " "^;t f " '^'^ = ^^^' ' ^^^V + ^^f-
10. Simplify a + h- (2a - 36) - (5a + 76) - (26 - 13a).
EXERCISE XXIX.
1. Solve
x-a , x-b , x-c
he
ac
ab
2 2 2
- + - 4- -
a^ b^ c
2. Snnphfy-^^, +^^3^.
3. What is the diflerence between 3a + m and 7am when a = 5
and m = ^1
, „. ,.„ a;<'+^ .i''+= x-^'+o
4. Smiplify „ X — ^^r- x
^.2C
>,26
,.2a
X*" X'
.5, Multiply 1 - Ax + ix^ by 1 + .^x - ^x^.
6. Tf $1000 be put out, part at 4% and part at 5%, and if the
yearly income is $73, how much is out at 4% ?
rui
r)(
KXEKCISKS IN ALGEnRA.
7. N\ rifu (litwii tilt! culm (if ';
/ /(
H. Show timt tli(! product of jiiiy f<iur oonsecutivo numbers in-
ereasod \>y I is ;i porfuct. s(pijire.
y. IVovo tliut ((mH- /'// + '•,:)-' -(/»,!• + (•(/ + '<•:)'■' i.s divisible by (u + h)
X + (h -f- '•).'/ + ('■ + ")'•» 'i"'^ '"-l^*' '>y "(•'^ " '-) + ''( J/ - •*-') + '-'{^ - y)-
10. Simplify '.+ '+JL+^^:;^'+^-'')'^^-'^.
' -' a -h ' b-c c-a 2(a - h) (h - c) (c - a)
EXERCISE XXX.
1 Simplify ^^L-'^^lzi^-^l'r , (a>/ -^a-p+(3a;-2//)^
^' •> 5(i/-a;) ^ y+a;
2. Prove (<(^ + h'' + r') {x' + y^ + z') = (aij - hx)' + (ex- azy+ (bz-
cajY + {ax + />;/ + <'2:)''^.
3. Add aw - cl — Ini and en ~ al — hw, and from the sum take
— cm - hi - an, and divide the result l)y b -c - a.
4. Prove that if one quantity measures two otliers it will also
n^asure the ditierence of any multiples of these two (quantities.
5. Factor (,r- + ;/ + ;; + a)' - {x - // - :; + af.
6. Factor 20a;' + 12rt.';H25/w2 + ir)r«?>x.
7. Find remainder without actual divisit)n of (or* - 3,i;^ + 4x* -
2.c + l)-^0*•^-.r+l).
8. Extract square root of \) - 24.^; + oH.r''' - iUVK^+VlSh-* - 140,c° +
lOOc*.
9. Simplify (,-«)(,s ~-l)H^r;.) when 2« = a + 6 + c.
10. Factor 7.'^'' - • «);/^ - x;/ + 19^; + 3% - 3G.
EXERCISE XXXI.
1. Apply Horner's method to find value of o.«'^ + 497,r* + 2()0a'^
+ 19Gj;^ - 218x - 2000, when x= - 99.
2. Show without expansion that {\ ■'t ■<' -\- x'^f - {\ - x + xA^ - i\x
(a,-' + .''' + l)-8.i--0.
3. Show that a\b-{-c)-}}\c + a)-irc'{a-irb)->rabi' is divisible by
a-h + e.
4. What value of « and b will make x'^ + 2ax + b^ the square of
5. A man is three times as old as his son, but 10 years ago he
was 5 times as old. Find son's age.
MISOELLANKOUS EXKUC'ISES.
67
♦I. Kxprcss ,r-f*J(>i/' + 70;.H 52//;: + 4.I-I/ + (),*•;. as the Hum «.f three
S({uaroN.
7. What ((uantity will divido without reiiiain»li!r into .»•* -
2ax^ + {ii' - h')x' + 2al)'x - aV and ;/;* - (a' + h'^)x^ + o-h' (
8. Factor x* + 12,r'' + HOx^ + 84..- + 33.
1). Express .»•* — 28,*''^ +• \(\ as the product of two (juadratic factors.
10. Prove that (3.x'^ - ox + 8)- - (2x^ -x + 4)'^ is divisible by (..■ - 2)».
EXERCISE XXXII.
1. How much tea at BOc. per lb. must be mixed with 100 lbs. at
87Ac. per lb. that the mixture may be worth «)2ic. per lb. i
2. Show that (dx-^y)'^- {7x + 3yf is a multiple of 2u;- 7;/ and
Idx — If.
3. Extract sq. root of 9x* - A2x-^ + IWx^ - 154.x; + 121.
4. Find the conditions that mr^ + nic''^ + 32x + 15 may be divisible
by 2x - 3 and 3x + 1 f(jr all values of x.
5. Find the value of (m-n) ('m + n)-{m + n) (m + n) when 3w
+ 2n ^ 45 and 3/t + 2m = 15.
6. Factor 112«H 138a6 - 1356'^.
7. Show that (5.f- - 'Sx + 2)- - (2x' + 3x - 1)'^ = 3(7.''- + 1 ) (x - 1 f.
9. Find the conditions that x* ~ px!* -\- qx^ - rx + n may be exactly
divisible by a; -a.
10. Divide
2 - in
by
'2+m , 2.-m
2+?ft •' 2-)/( 2+/rt
EXERCISE XXXIII.
1. Determine the value of p and </ when the expression 4j/'*-
12i/-\-py^-{-<iy + lQ is a complete square.
2. Show that ^^"^^-^^ = ^^ .
(c - a)" - (a - 6)' ?*+c-2a
3. The product of any three consecutive integers being found,'
and also that of any other three, tlie difference of the products will
be a nmltiple of the ditlerence of the middle integers of the two sets.
4. Show that ax^-'rhx'^ + cx + d is divisible by .*'- + //^ if ad = bc.
5. If x + a be the H.C.F. of x'^+px + q and x'^ + mx + 71, then
a- n
a^- •
5ft
Kxi'ju'fsRs IV .\F,(;i.;uuA.
(». If x' + 2u .-'.ih' is <liviail)lo hy x-o, provo a^h or -h,
7. Wtml v.'iluo of ^> will make '.\x-Ai\ niensuro of J8u;'^-/>.. +28?
8. Rosolvo into 5 factors .*■"'- (JHo.'JO.
9. Factor 9.*;'' + 48.1-2 -i- r)2.i; + 1(».
10. l^.ictor2.i;H1b;i/ + 12//' + 7j- + l.V + 3;j».
EXERCISE XXXIV.
1. If x + 1/ + :; = 0, show that ^^^^ + ^i'L-j?!! . i(£!Lzl!) _ n
y-z ^ z-x ^ x-y ~"-
2. A rrango (.« + ,/ + ,)a , + (,• + ,, _ ,),, ^ + (.,. + .. _ „),, , + (v + ^ - .'■)«.
Ill thruo teriiiH involving x, y, z respectively with co-otticieuts a,!
«2» «;j> 'h-
3 Find tlie condition that x' - Sb'^x + 2c-' may be divisil)le by x - a
whatever })e the value of ic.
4. a;*- 4u:3 + 6a;2-4x + l is a multiple of x'^-ax + l, Hnd a.
5. Prove that the product of K.C.f. and L.C.M. of any two
quantities is ecjual to the product of the quantities.
r/ f; ^ih'^'i'^^.'V*^''''.?^ expanding(a;2 + ^.j/+,/.)3 + (.^2_ ^. 2)34.
G{x^+\/){x* + xY+i/). -' J I '
7. If 9./:*-30.«3y + axV-10.ty + ?/* is a i)t!rfect sfpiare, find a.
8. Find value of 3.«5 + 54^ » + 50.^3 _ j 9 ^.2 _ 35^ _ ^g ^^j^^^^ _^ ^ _ j^
9. Solve i^^i-l-i?^±l=5?-t.
18 13a; - 16 9
10. If lr:-cy=p, cx-az = q, aii-hx = r, then <ip + b<j-\-cr=0.
EXERCISE XXXV.
1. Find value of 2x* - 510r' - 513.«2 + 25(u; - 1024 when a; = 256.
2 Prove by fact.ms that (7a; - 3)H (7x - 3) ^^ + 2) + 7x-(x - 4) - 3
( it' - 4) = ( < .1; - 3j (9x - 5).
3. H x = a + 2b + 3e, y = b + 2c + 3a, z = c + 2^ +3b, show x + y + z
= b{a + b + c). .
4. Facte )r 2y-^ ~ hxy + 2a;^ -ay -ax- a\
5. Reduce to lowest terms — i'iL'-i«*-i2 __.
2a;-'-l(ijr'-!-'24;r+'288
0. What number addel to fj + 1; + ;'!; + ^^ will make it a perfect;
scjuare? -^ " '"' '
MIHCKLLANEOUi) EXKliCIUES.
59
7. If .rj* + px^ + qx'' + r.r + s be a perfect square, show that r' = p»s.
8. Find tvvti o(|ual fuctors of
9. Show that the sum of the cubes of any three consecutive
numbers is divisible by 3 times the second of tliym.
10. Simplify ~ •
X - )/ x-\-y
EXERCISE XXXVI.
1. Solve 1 + 1-1 + 1 = 78-
Z. Write down the .s(juaro of ~^Z\ z '•' "^s simi)lyst form.
.'i. Find a (juantity which when multi[)lied into .*■•'' - 4j"^ + 5*- - 2
will make it a perfect square.
4. * ind the sunplest form of n — \ -r •
5. By what quantities must x^-x^-jc + 1 be multiplied to make
it a complete cul)e.
G. Find the values f)f {-, + - ) and "^ " when x = 6, u = 4 ;
will the values be the same if |/ = t) and x = 4: 1
7. Find the (juotieut of -r - '- by ^ + V+'-^-
8. Prove that the ditt'eronce of the s(}uares of any two consecu-
tive even (or odd) numbers is 4 times the intermediate number.
9. Find value of :>••' - Si/ + 29z^ + ISxyz when 2j/ - .t + 3^. and z = 5.
10. Multiply {(i + b)'^ + {a-by^ by {a + b)(a-b\ and divide result
by <i^ + a-b + ub'^ + b^.
EXERCISE XXXVII.
1. If (t - /) = 1, show that (a - bf (a + b)- = a^ -b^ + ab.
2. If a^ + cf + l-O, show that a^-l.
3. What value of m will make 6x*-2x^ + 2mx^ + 2x + m exactly
divisible by x'' - a; + 1 ?
4. If x + y-z — 0, show th&t x'^ + yz=y^+xz.
I
f!
00 EXKUC18U.S KV ALUUUUA.
r». Prove thixt .»"4-»/ is divisihlo by .'•*4-l/* and write tho result.
7. If X is pi.Hitive, sliow that ^^"1 + -— - 2 is positive.
8. Which is greater ^-, + |, or j+ ^, when .*■ aiul ;/ are positive/
9. Find two linear and one (lu.idratic factor of (/^ -!/-)'-(«/'' -
10. A courier ])as3ing throutch a certain place (I') g()es 2^ miles
per hour ; 4 hours later another passes through, ,i,'oin<,' 'Mi nules per
fxour. How far will tho second travel from (I') before he overtakes
the tirstV
EXERCISE XXXVIII.
1. If ac = 6-, show that {a + 2h + c) = ^""f •
2. Show that a-h, h - r, c-a cajinot be all ])ositive or all
negative.
3. Show that a^ + // is greater than a%-^ab'^ unle.ss a = /i.
4. Prove a + h = -^^ + ^~'
X , a
c . a
c
5. Find by factoring for what values of x are - + ^«i»d - 4-
equal to each other.
6. Ii2a = y + z, 2b = z + x, 2c = .»• + !/, find the value . - a* + b* + c*
- 2a^b'^ - 2aV'2 - 2b'^c^ in terms of /, y, z.
7. If axi + bhi-r^ = 0, (n/ + h^x - c^ = 0, and x + u-e = 0, then
b^ = ac.
8. If ic^ = x-2, show that xP= -x + (i.
9. If a;'' = j: + l, show that .T» = 5,r + 3.
10. A sum of nvmey is divided among A, B, C ; A is to have
1?120 less than half, B ^0 less than J of it, C $32 more than i of it,
what did each i*eceive ?
EXERCISE XXXIX.
1. Find by factoring the value of a in order that x = 2 may be a
, ,. n ox , X ^'iaxi-2
solution ot : + „- = — .vir— •
a - 1 2a dax
2. If — H — ^ + — =0, show that x = y = z.
x-y y-z t-x ' *
maCBLLANEOUS FXERCIflRH.
61
3. Prove that if any two <|uantitio.s be added together and their
«uni divided by tlm sum uf thoir reeinnxials, the quotient will be
ei|iiHl ti> tlie product of tlie two <iuantitieH.
4. If (f -/>(/,/> -^-7/-. (■^/•,s find II -{-}}■{- ('-^ pM, sliow thnt (11 ^hy^
'}. If .i'-f-|/4;-H). show tlmt jix''-yz)-\->j if - xzj +z{z- - juj) =0.
(5. Prove (.»-/*)(./ + /> -'•) + (/'-'•) (/>+i'-a) = (a-r) (n + e-h).
7. If .»;^ = fT'' + 6'^, }/ = c'*-^fl\ which is greator, ..■(/ or <ir + hd, ad
being not e(|ual to be ?
8. Which is greater, ii^ + l or n'-'+u, ;i being not = I /
9. Show that - +^~^ -1 and ^ + -' -2, a and /> being both
jiositive.
i/» T« ^-c 0-0 a-6 .1
10. If x= a , {/ = -(r' '^^'V *^^'^'" ^l/2 + .'H-(/ + 3-0-
EXERCISE XL.
1. If a, h. A- be positive integers, ascertain whether (a + A;)' —
(b+lif is divisil)le by a - b.
2. Factor 4(a + />)* + %, - />)« - 12{a^ - b-)-.
3. Factor 12.f2 - 31u;;/ + 20i/-' + 29r2 - 387;: + Uzi
4. A person bought 80 lbs. tea, some at oOc. and some at 75c.
He finds by selling all at 7oc. his gain would be $2.50 more than by
adding 12ic. per lb. to the price of each. How much of each did
he buy ?
5. Prove that any trinomial is a complete s(|uare if the square
of the middle term is ecjual to 4 times the product of the first and
last terms.
(). If 16a* + 48,»;*j/ + (/xV + 24.xi/* + 4/ be a perfect square, find q.
7. Prove that (\-2x + 3x^ - ix\ etc.) (1 + 2;r + 3.«- + 4.f' + ) = (1 + x^
+ x* + , etc.)'.
8. U 6- = a + ^ , show that a* + --2 = s'Ys^ - 4).
9. Find the equal factors of x^ + 4x(/ + 4i/2 - 4^ - 8(/ + 4,
10. Divide a\c -b) + b\a - c) + c\b - a) by a - 6.
62
KXKHCISKS IN AUJKBllA.
*
MISCELLANEOUS EXERCISES.
A.
1. Divide the product of 12«''-lla-.W juid 28a2-86a + 6G by
2hf--5a-44.
2. Divide (2'x^ + 3x - If - (x'^ + ix + by by the })roduct of (3x + 4)
(.. + 2).
3. Iix+ii = 2a, .>;-i/ = 2ft, prove that x* -23xhf + v*::={7a'^ -3b^)
4. Find the value of x* - 47xhf + y* in terms of p and (/ when
X + \i= p and X- y = q.
5. Show tl it the square of x + l exactly divides (r' + J^' + 4)' -
(x^-2x + 'Af.
6. Prove that (x + zf + S(x + zfif + 3(x + z)y'^ + }/ = {x+ iif + 3(x +
yyz + 3{x + vyj + ::\
7. Find the quotient by factoring of 9a- -|- 6ab + b" - 4*;'^ - 4cc? - <P
di vided by 3a + b — 2c ~ d.
8. Factor x'^ - 2mx + m'^ — w^.
9. Factor (a - b) (a^ - c') -(a- c) (a^ - 6").
= 2{a + b + cy+ai + b^ + c\
10. Show that
(a+b)'''-c-' (?> +c)'' - «••' (c+g)"'- ;*''
(i+fr-fl />-fc-a c+a-6
U. Divide (/H J,)' -8 by (x -i)\
12. Resolve into 4 factors (x'''-3x)2-2(ic''- 3.«) -8.
13. If x = b + (' -a, y^c + a-b, z = a + b-c, Und the value of
y^ + i/ + z'- + 2xy -f- 2xz + 2yz in terms of a, b, c.
14. Divide (4w + ^bdy - (4(u? + 8/)c')'' by {a + 26) (c - 1/).
15. Divide, {x^ - 3xhjy - {3xy' - fy by {x - yf.
10. Find the difference between the squares oi 3503 and 3497.
17. Find the algebraical expression which, divided by x'^ + x-l^
gives 2x^ - Gx^ + 8u; - 14 as quotient and 22.* - 14 as remainder.
• 18. Prove thtit ,— — 4~- — r. = 1 when m
(TO-c)(m— <i)
a+b-c-d
MISCELLANEOUS EXERCISES.
63
19. Simplify
(x -y)(y- z)+(y -z)(z-x)+(z- x) (x - y)
x(,z-x) + y{x -y)+ z{y - z)
20. Find h.CM.oix^-if, x^ + f, x"" + xhf + y*.
21. Find the fraction in its lowest terms which is the sq. root of
l+4x-2x--12x^+9x*
22. Simplify
l-4x+Gx--ix'>+x*
x'*+x^y"+X'y+y^
x*-y*
23. Find by factoring the sq. root of (.7:^-3^; + 2) {x^-^x + 3)
(x^-5x + 6).
24. Find the co-efficient of x* in (x + a)^ x{x- fif.
25. Show that ac^ - {d^ + h)c^ + IP' is divisible by ac - 6.
26. Factor x-''-9xH 11* + 21.
27. If X - -^1, show that x^--^ = 4.
a; ' x^
28. One lb. tea and three lbs. sugar cost 75c., but if sugar were
to rise 50% and tea 10%, the cost would be 87^c. Find price of tea
and sugar per lb.
29. Find co-efficient of x in Ot;-|-2) (.x -6) (x - 10) (x + 14).
30. Find the first four terms of {[i- i/+>f - [P + )'.
1 1
31. Simplify
32. Solve
+
18
6a- 18 6a-M8 o- + a* + S\
bx-Gi 2a;-ll 4a;-55 x-&
a;- 13 x-6 x-W x-1
33. Resolve into 5 factors u' -I- x* - Kir* - 16.
34. Multiply (3 + X - 2x'0' - (3 - x + 2x'f hy (3 -{■ x + 2x'f - (3 - x-
2xy.
35. Divide the ])roduct of 2.<'^ -f- .*• «J and Hx'^-bx + i by 3x^ +
5x - 2.
36. Show that (2a' - 3) (.'' + 4) exactly divides the difierence of the
squares of 3jr-j-8x - 25 and .»'--|-3,'' - 13.
37. If x+ij^ta and .'.- !/-=/«, then H]{x* - 7 x- if + i/) = {5m' - n^)
(bn^ - m').
38. Find tlie value <>f x* - '2x^ij + 2xif - i/ when x=-(i+J>. ;/=-a - 6.
39. I f <( + 'M- '• - 0, sliow that (2(« - bf + {2b ~ cf + {2c - af = 3(2(t -
b) ^2b - <•) , 2f - a).
64
EXERCISES IN ALGEBRA.
4G. Factor x' + 3x^ - 13a; - 15.
41. What number must be added to x\x-\-2) + 7 in order that it
may be divisible by ac + 4 ?
42. Divide x'^ - xy + {%i/ by j; - 1«/.
43. Find the remainder when 5x* -7x^ + 3x^ - x + S is divided by
x-4.
44. Factor x-'^ - 2xy - '32'Sy\
45. A boy spent h of his money for marbles, J of the remainder
for oranges, and I of what then remained for a book, and had 120.
left ; what had he at first ?
46. Divide the square root of ia' - I2ab -Obc + 4<iti + %'' + c'^ by
2a -3c.
47. Factor x^ + bxy- 30]/''' + x- 4'i/.
48. Tf X -^ =1, prove that x^ + \^=S.
49. By what (juantity must 3.f'^-4xt/ + 5/ be multiplied to give
18x* + 19a;2|/'^ + 12.r ;/ + 35 1/* ?
50. Multiply {x - I) by {x - ^^) •
51. Show that (a + b + ef + a' + b' + c^ = (a + by + (6 + cf + (c + a)\
52. Factor dx' - 24xij - 9^'^ + Hiy'.
53. Factor 99x2 4.ci/-143!/'.
54. Solve 19x - 21(/ - 100, 21x - I9y = 140.
55. Find the s(i. root of (x - yY - 2(x' + y'') (x - yy + 2{x* + y*).
50. Without actual division prove that x^-7x* + 17x^ - 22a3''' +
25x - 18 is divisible by x - 2 without a remainder.
57. Find the remainder when xHUx^-o-f'^ + Ox + K^ is divided
by X 2.
58. What must be added to 5(t*-7"^ + to make it exactly
divisible by a + 3 ?
59. If « - 5x - 3{/ - 2;:, b-=r>ij -'3z 2x, c = 5z-3x-2y, then a +
6+o--=0.
bO. Solve 4=0; -^ — ^'
MISCELLANEOUS EXERCISK8.
65
61. Write down the quotient without actual division of 8x' +
8y^ + z^ - 12xyz by 2x + 2y + z.
62. Factor a^ - 26^ - Gc^ + ab-ac + Ihc.
63. Find the value of 3(x + ]/ + z)" - (x^ + y^ + ^s) ^hen x = 3, i/ =
-5,2-7.
64. Factor 4xV + 4(a + 6)a']/ + (a + 6y.
65. Factor (:c-3(/)3- ( J/- 3xf.
66. Factor {x' + 7x + 0) (x^ + 7x + 12) - 280.
67. Solve
123; _2_ ^ _2_
a;--9'^x + 3 a;-3
a'-xy\ . ja''-xy\^
„ ,., / a--xv\ I , a--xy\ , ,a^-xy\^
68. Simplify ( y - ^') (x + ^) + (-^r^ )
69. Solve ax + & (/ = c, a'^'x + 6'^ j/ = c\
70. Divide a* - 6* by a. - h, and from the result write down the
quotient of {a + hy - 16c* by a + b- 2c.
71. Multiply a^ + 256'^ + 40^ + oab - 2ae + 106c by a-5b + 2c.
72. Factor x^yV - xh - yh + 1.
73. Divide (a + 26 - 3c + df - (2a + 6 + 3c - df by a + 6.
74. Solve 15x + 17i/ = 79, I7x + 15(/ = 81.
^^ „ 1 4x4-18 , 3x-2 l O.c+28
75. Solve 2-^q^ + -j:f:3=-2Fr8"
76. Factor x^ - (/^ - 3x - j/ + 2.
77. Show without actual division that (6x2 - 4x + 2)' -(4.r^ + 6x
- 10)^ is exactly divisible by x - 2 or 2x - 0.
78. Show that (1 - xf is a factor of 1 - x - x^ + x*.
79. If 2{a'^ + b'') = {a + by, show that a = b.
80. Show that m(m + n) (w + 2h) (m + Sn) + n* is a perfect square.
81. Find a number such that if § of it be subtracted from 20, and
/,- of the remainder from ^ of the original number, 12 times the
second remainder will be h of the original number.
82. Factor iix' - iJif 2Qz' + 22yz ■\- Ixz - 5xj/.
66
EXERCISES IN ALGEBRA.
83. Which factor a; — | or x+I is likely to belong to cc'-^^ -
f- + |, and why?
84. Find by factoring the H.C.F. of a;'-8a;2+19x- 14 and x*-
7x3 + 8ic'^ + 28x-48.
85. Write doAvn the co-efficient of x in (x^- 21.c - 1.3) (x' - 2x - 1).
86. Write down the co-efficient of x* in 1 — 2x + 'ix" - 8x^ + 16x*
multiplied by 1 -f- 2x + 4x'^ + 8x^ + 16a;*.
87. Write down without dividing the quotient of x*-5xh/ + 4y*
by x'^ - 3xy + 2i/l
88. Factor ah{a + h) + bc{h + c) + ca(a + c) + 2abc.
89. A number consists of two digits and another is formed by
reversing the digits. If the sum of the two numbers is 99 and tlie
difference 45, find the digits.
90. Factor x* - x^ - Sx^ -|- 7x + 7.
91. Solve x--|-2j/-}-3z=4, £c-f-3i/-f-2 = 4z, aj-f23-)-3 = 4t/.
92. Find value of r-^ when x — —T~i-
93. Solve 49x- -f- 37i/ = 1230, 37a; + 49 ;/ = 1350.
94. Simplify u;"+*+'=xcc«+^-<'x.t''-''+<^xcc*+''-'»,
95. If a = y + z-2x, b — z+x-2y, c = x + y~2z, find the value of
b^ + c^ + 2bc-'a\
96. Find the remainder when a''-9a"* + llrt^-7 is divided 1)y
a -4.
97. Solve - + f = 3
a b
f +? = 5
1/ c
? + '- = 4.
a c
98. A starts frt)m London and travels If miles per hour, B starts
8 liours after in the same direction at Ig miles per hour ; how far
will he travel before he overtakes A ?
99. Factor (x2 + r)^- 8^ Y.
100. Factor6.f2_i3^y^6y2^5^_5^^^1^
MISCELLANEOUS EXERCISES.
67
MISCELLANEOUS EXERCISES.
B.
1. Factor (he + ca + ahf - (6%^ + c V + a%'').
2. What is the least multiplier tliat will make .«•* - bj^ + 5a: - 1 a
multiple of x'^ - 4a:; + 3 ?
3. Solve '¥' = ^^±^ + -?^?-
4. 44a*-83a3-74a'^ + 89a + 50 is the product of two factors, and
one of them is lld^ - 7a - 8, what is the other I
5. A man has two farms rented at |7^ i)er acre and his total
rent is ^3,375. If the rent of the tirst was reduced to $6|, and of
the 2nd to $5.00 per acre, his rent would be $2,500. How many
acres in each farm ?
6. Factor xS^ 3a; + 1 + 2 X- !/ + 1 4- 3 ;/ + yK
7. Factor x* + 2a:ht^ - x^ + a^ + i\x - 9.
8. Find co-efficient of .^ in (.»; - 1) (;/• - 2) (x - 3) (:c - 4) (x - 5).
9. Resolve (x - 1) {x - 3) - (,c - 1)^ into factors.
10. Factor {a? + 6'' + 1 + ah + a + 6)'^ - {ah + a + h)\
11. Extract the sq. root oi x^' - G..^" + 13./:^ - 14,- « + 10a-* - 4a;H 1.
12. Find H.C.F. of 3a;3- 13x'^ + 23.(-21 and iSj? + x' - Ux + 21
and what value of x will make both vaui.sh i
13. Divide l+^by 1-- •
14. Divide -i3|-i by -— -|-/- giving the ([uotient in its simplest
form.
15. Find the remainder when the divisor is .'-1-1 and the divi-
dend is the product of {x + 3) {x -f 4) \;x- 7).
1 1
16. Solve
^ + :^
a;+2 ' a;-MO x-f4 ' x-\-S
17. Extract the sq. root of (Ga^ -|- a - 2) (3f r ' 7'< - G) (2(/^ - 7« -I- 3).
18. Divide 20 into two parts so that the square of the greater
shall exceed the s(juare of the less by 80.
19. Find H. C.F. of 6x* -h 26.i» -|- lox^ - 1 Gx - 10 and 30.^'* + 13Gx'' -f
95a:2 _ 79^; _ (35^
68
EXERCISES IN ALGEBRA.
20. Resolve x* - 4x" + bx' - 20 into 3 factors.
21. A dealer adds 20% to the cost of an article to make the
selling price, but he gives a customer 10% discount from the
selling price, and then has a profit of 7oc. Find the cost price.
22. Find the value of ^-Jt+'^rll'- '•'"' .
6a;-
5x - 5
1
23. Solve ^'--^^Zl
9x+(i l2a;+8 12
24. Solve 3x - 2y = 13, 3// -2z- 16, 3z - 2x = 9.
25. Extract the square root of O.i-^- 12x'^ + 22.r* + xH12cr + 4.
26. Divide — + -- })v 1- •
l-a^l+a J' l-a l + a
27. Express (x^ - 3x^y - {3x - ly as the product of 5 integral
factors.
28. Simplify ^"•'-^"'^+^'-^» lZJ.
1 •' 7n' -2711" -4m +8
29. Fhid L.C.M. of o^ + 6a + 5 undo'*- a.
30. Factor 2abc + a\b + c) + b'^c + a) + c\a + 6).
31. A boy i)lucks from a tree a certain number of plums, an-
other ^ as many. They both have 5 times as many as a third takes.
All have 84. How many has each ?
32. Write down the quotient without actual division of x* - 4:x^u^
+ 4/ - x^ - (ixij - 9,/ by x' -x- 2(/' - '3ij.
33. Solve x-y + z = 5,3x + 4 1/ - 5,: = 13, x + ^ +
z
i ' 3"
14.
34. Solve '-z^zl + -^lz^-^^ _^-^^--^5 , ^-*^-n
35. Factor 200;^^ - llj; - 42.
3(). Find the remainder after dividing x* - 3.r + 7 by x-2.
;{7. Div'uo (4x- - 4// + 7-:)' - (3.*; - 10;/ - 7^)" by 7(.'' - 2y).
3S. Write down the square root of (.«•'■' - 7)'^ + 24.''(,>'^ - 7) + 144a;*.
39. Siuq,lify ('-'-,+ "- + l) (-«^^-)^ _ill(iLl-'i> .
40. Multiply x-^ + 2ax + 2bx + a' + 2ab + li' by x + a + b.
41. A roll of cloth was bought at 66c per yd., ami another n>!l
'lb yds. longer at 60c. per yd., the two together coat $241.80. How
Uiany yds. in each roll \
MISCELLANEOUS EXERCISES.
69
42. Find the value of ^-I^l^;^^!^! when x = 2y.
43. Solve 3.f - 1/ + 2z = 11, 3;/ - 2 + 2a; -=\), 3z-x + 2y = 16.
44. Express in words the following algebraical expression : y{y-
l){>l-2){y-:i) + l = {y'-'3y+lf.
2(x+'l)
x+6
40. OUnpmy ^_^^2)(x+.^) (x+5) (.c-I)'^(x'-l) (j;+2)
46. Solve Ki^'^' + 5) + K'^'- - '">) = K-^-*' + + K^-*' " !)•
47. Solve (/ - 3(.^- + 1), 4.<; - (/ + 1.
48. The product of two numbers is 75, and the quotient of the
sum by the ditference is 4 times the quotient of the difference by
the sum ; find the numbers.
49. Snnphfy ,y_,„^3+ ^,^,,,.3 +„,.^^-
50. Show that (x + // + 2 + ((^' - {x -y-z + a)'' = 4(.c + a) (y + z).
51. Show that -^ ab{t> - a) +ac{a - c) + hc.(c -b)\- ^{h- c) = (a - b)
(c- - a).
ah+'2a" - 'M)" - 4l><- - ac - c"
52. Show by factoring tliat
53. Show that
•2a + Sb+e
i'ld'-Tlab- 12^y - + (ic- Ibc - c-
= a-b-c.
4a — 36 — G.
3a+46+c
54. Factor x^ - %•' ij - 3 </'' + 4yz - 2^.
55. Simplify (4.«' + 5;/ + 2)^ - l\{ix + 5y + zfz + 3(4x- + 6y + z)z' - z\
56. Find L.C.M. of u{x + l), b{x + l)(x-l), <ix'' + 2x-S), d{x-' +
4x + S).
__ ,,1 X (I , X-h X-C .) /I ,1 , 1 \
57. Solve .^. + ~- + -^= 2 (- +,-+- ).
68. A man bought sheep for $528, and having lost 10, and sold
20 that were diseased at $1.20 per head less than cost, he disposed
of the rest for .S4<)4, thereby realizing his outlay. How many did
he buy '(
59. Factor 42br + ix' - iiW - dc".
60. Simplify j:^,^-^:^- =^t-
61. Find the value of
i +
a-b
l + ba
a{a - b)
l + oA
70
EXEHCISES IN ALGEBRA.
62. Solve a(x + y) + h{r - y) = 2a , y(a + 1) - x(a ~h) = 2h.
63. Show that the sum of the cubes of 2x-3y and 2x + 7y is
divisible by 4{x-\-y).
64. Factor (a' + a- 2y + {2a^ + ,( + '^).»' + a^ - 1 .
65. Solve ,. , +-S , ,. ^IH-
ix + a 2x+^n
66. At an examination A got 3 marks above 50% and B got 6
less than .'{3.'j ■' , and A's marks were twice B's, iiow many had each ?
67. If the dividend be .'li*-4..'H8.>''^-7u-+ ' ind remainder 34.''
-30 and quotient o:*-'' + 5.«' + 17, hnd the divisor.
68. Divide (pq + m'f - (pa + qr)- by (p - /•) (q - s).
69. FindH.C.F. of 3.«,"' + 5r'(/ + 9a'V + ^'/^ + 6'/* and 2,>* + ox'^y +
5a;V - 3uy - 9(/'.
70. Sh<.w that U^'r^^Y - ^7 1 <J!±21S}^Iizm!Lz3) I ^ .'^'zl)! .
^ 12 Z ' 1. 432 ' -256
71. Simplify (a; - af - (.»• - />)2 - {a - b) (a +h- 3x).
72. Prove that (« + 5)'^ -(<» +2>'' = 9(a + 5) (a + 2; + 27.
73. Solve :/; - 11(/-1, Illy - 9x = 99.
74 Simnlifv ("'-")Oi-a) (w-bU,i-h) (m-j^(n-c)
(r-h)
75. Express in factors ij.C.M. of 1 - 8.*' + 17.'-- + 2.i ' - 24.*'* and
l-2'x-13.i'' + 38.."^-24.f^
7Vk Find H.C.F. of (ix^ - llx'^ - 37:^' - 20 and 2j- - 4x' - mc - 7.
r.f- o- IT '■'•i'lX* - V.iX" + l
78. The sum of two numbers is 57«)0 and the ditterence is ^ of
the greater. Find the numbers.
hrn ui i.\ i. 12a;+10a , n7a + 28x io i
79. Show that ^^^^ + \m+2x ^^'^ when ..• = 3«.
80. Factor 40./-' + ()!.*•// 84//''.
81. Find by factoring tlie H.C.F. of 2r<-' - 21?/- - 45o''' + <»/) + 62/)r +
(U- and .'!«- - 21//- ~ 45(.-' - 2((h + 62/jc + dar.
82. Divide (22..' + :•.</ - zy - a7x - 2y + 3zy by ox + 5// - 4^.
8:5. Find the remainder after dividing .*■■'- 0.1-' + 7,c - 9 1^- x + 3.
MISCELLANKOUS EXKKC18ES.
71
84. Factor 45.r" - Ti^ - 'MOH.
85. Divide the sciuare root of ..•^ + 18.r« + 117jr* + 324x« + 324 by
80. Find L.C.M. of 8»'3 + 27, lHx* + .%x2 + 81 and 6x^-r)x-6.
87 . U •ix-2y = x + 2y - 1 , show that x^ + \f = 2xy + 4.
88. Write down the product of {x - 2>y - oz) (2x - S^y+z).
89. Factor8y'' + 18x!/-5i/'^-2.''-38i/-21.
90. Divide a- + (2ac - W^yx" + chi* by a - hx + cx\
91. How much greater is the co-efficient of x in the product of
(x + 1) {x + 2) (x + ii) (.0 + 4) than in that of (x + 2) {x + 3) (x + 4)?
92. Find the vahie of 25a'^ + (a + 4/>)^ when 3a + 26 = 7, a + b^2.
93. Find the value of ;*•* - llx» - lla-^ - 13a: + 11 when x = 12.
94. Factor 56x-2 + S6xy - 20y' + 28x - 10.
95. By what must a* + a^h + a'^b^ + ah^ + b*' be multiplied that the
product may be <i^ -b^'i
m. Show that (x' + Gxy + ^y^f + (x^ + 2xy + 4 j/^)^ is exactly divisible
by x- + 4x2/ + 4j/^.
97. Solve 5ic + 2y-l = 3x-i/ + 14-ic + 19|/ + 6.
98. Write down the quotient of the sum of the cubes of a + 6 and
c + d, by a + b + c + d.
99. Solve 4x '- 6y - 3 = 7.'»- + 2 j/ - 4 - 3y - 2x + 24.
100. A farm was rented, ])art at S5.00 and part at $8.00 per acre
for $t)80, but if the rates had been interchanged the amount would
have been ^620. How many acres in the farm i
KXKUOiSEa IS ALOKBKA.
MISCELx^ANBOUS EXERCISES.
r.
1. Write down the (luotieut with- ul aotiml division of {x + if)^-\-
3(x + ijYz + Six + y)z^ + ;;» by {x + »ip + 2(.i; + ijc. + ::'.
2. Find tliu fa-.tors of the quotient of 84./ ♦ - 5rw;»i/ - a^O^V -
G0.r|/* + l05;/* divided by 1y^-\-'Avji-'.\>,K
2x-\
3. .Solve ~-~ + 4a; = 12+ ^
.5 4 O
4. Simplify ii'^ + b'-\-c'^ -'■^ah-be-%ir. +,t{a Arh + c)-{h - cf and
divide result by a — h.
5. Factor l^{x' + >/)" + 40(.<;'^ + if)z^ + 25:«.
6. Without actual division find the remainder when r^-Sj-^ + O
is divided by .*; - 5.
7. Write down quotient and remaintier of ~- •
8. Simplify -±l}JlLLA±L .
* ^ (./;*+'••) (x«+«) {j;''+'')
9. Divide (x* - 1 )- 5(x - 1) l)y (.»•- 1)1
10. Divide(./j + l)(x + 2)(i*^ + 3) + 6})y u; + 4.
11. Solve -V,— .5 + roTj^ = /'? iv., , / ■ , \., •
12. Find the remainder when 2x^ -2x* + 3x^ - 7y^-hbx-S is di-
vided by .*' + 2.
13. Find the .S(iuare root of G7.c2 + 9x*-70x-3O.rH49.
14. Ex[«uid in consecutive powers of x the expression (1 -x + x'^)
(1 -.!;' + ;*•«).
15. What expression must be added to j."'' + 11j:"' +21 {x + l) that
it may be exactly divisible by x + (i 'i
16. Find all the factors of (d^ - 3a)'' - 2a^ + Ga - 8.
17. Multiply x(4x + 3//) - {x + 2yy by 44^; + y)-y{ilx- '3y) and
divide the product })y (3.*; - 4y) {-ix - 3*/).
18. Find co-eilicient of x^ in (x + 2) (x — 3) (x - J) (x -i- 8) (x - 9).
19. Factor (1 + x)'(l + y^) - (1 4- #(1 + x^).
M18(!ELLANIiOUS KXKJ<Cl8Ka.
73
20. Simplify ^^, ^ ,. ,27»:Tri *
21. ^nuplify ^— ,y,-^^^j), - ^«.,
22. Solve 7'+ 0|/- 7 1, 9.*--7!/ = 17.
23. Muj! 1 [.ly .,' + 4ar'» + 6e-' + 4j + I by j"^ - 3.i-"^ + .3x - 1.
24. Tf 3/) = .r + i/-HM .sh()wthat(/>-.r)34.,/>- «/)' + (/>-::/ = 3(/>-a:)
(p l/)(/'-4
25. Factor 6.i' + ir).' +1).
2b. Express an a HiiiLfle fraction ^■i^2x-'i^ x^^'^x^h'^ x*V*ix^l '
27 . Factor {x - 3) (x + 1 )" + (x - 3) (.r + 2)^
2H. Divide .;•' -\-y^ + x* + x^ + x"^ + I hy ..^ + ./;* + u"' + j-'^ + .»• + 1.
29. The batiki I discount on a sum of money at 5 ))er annum
is equal to the true di.sc.unt on a sum $50 larger. Find the sum.
30. Factor 2xy + 7' + «// + 21.
31. Factor x*-\- 4(;c - 1).
32. Factor (x 4- ;/ ' + 2xi/(± -x-y)-\.
33. Di vide '.r}'' + ^-^ + 11 .y .#•* + x + 1 .
34. Find the co-etticient <.f ..♦ in {\-x-¥x^-xy.
35. Write down the remainder of 2a* + 3a' + 4a'' + 5a + 6 divided
by a -3.
36. Show that
a + h-c ff-h+c A(l}-c)-
4(6 -P)
«_6+c (t + h-c a^-(h-c)- a+b e
37. Expand in powers of x, (1 - xY(l + x*').
38. bimphfy - ,._7^y-gj^ " "
39. Find co-e .cii-nt of .'•* in (1 +.»)'.
40. Fin«l the remainder when x^ + fjx^ + qx'^ + rx + s is divided by
X — a.
41. Factor {x!" + x)' + 4(x'^ + x) - 12.
42. Reduce to its lowest terms -TTa^. -3a"*.£^0a^J^2^ "
43. Factor a''- 14a26^-h6\
KXEltCISK8 IN ALOEBHA.
44. Ruhf rart (,r 5)'' from (r 3) (.t - I) (..• 4 .3) (x 4- 1).
45, Show by factoring that '^ .>' + if)- + 2(x+ ij) (z+ti)- b{z + 11)* is
divisiblo l)y r+y- . n and >\rit« down tpiotiynt.
40. Show witliout dividing that (l+x + .'i.'-'H3x')''' + (l -.»+3x'-
;ir')» iH divisiblo by l+.V.
47. Employ dotacbed co-efticients to divide jr'^ - lijc* + x* + x" - 3x
+ 1 by ..-fl.
4«. S( ,1 vo (n + j'Y 4- (h + J-)' + (<- + xf = IX<( + r) (/) + x) («• + x).
4J>. Solve
2a?i'+2j;4-:i
j-i 1
50. Find the rt'inaindor without division when x^ -7x^(i-^fixa* +
15(r' is divided l)y x + tid.
51. Find the continued product of a' + d + l, a'^ + a-l and a* -
lV + ,i'^ + 1.
52. Simplify
x*+a-x" -b-x^ -a-b- . .r» «*
a;*+a!'j"'-4-«*
^•'+a-'
C.J w If (2j-2 + .5j-+2)(j-'-3a!=-J-4-3)
o.i. Snn])lify , „ , ., „ , ,; ,„,,, .. , •
54. I f (<r 4- /' ') (.'■'••' + if) - (-?.'• + '>!/)- prove that f = | •
55. Sliow that (4.'' + 7;/)* - (.V 4- H»/)* is divisible by 7x + 15i/ or a; - y.
5fi. Resolve Ki.r' -81.»'3- l«j't4-81 into five factors.
57. If x = h + ('- 2(1, ii = r. + (i-2b, z^a + h -2c, find the value of
'j^+if + z^-3xiiz.
58. Show that ( « + h)^ + {a + cf + (u + dy + ( J + c)'' + {b + df + (0 + 1^)^
= (a + ?> 4- + (/y^ + 2(< (' + b' + c' + (/-)•
59. Prove that (o + 'Ab)^ + {b 4r )•' + 8((t + [ib) {b - 4c) (a + 46 - 4c) =
{((+4b-4cf.
CO. Solve 5x + 2y + 'Sz= 13, 3x + 7y-- = 2, :*• - 2i/ + ^ = 5.
01. rV man is able to pay his creditors 25c. in the $ ; but if his
assets were 5 times as much and his debts rj of what they are, he
would have a balance of ^1,400. How much does he owe !■
62. Write the co-efhcient of x^ and x^ in the product of Sa-^-Sa^^
+ 5x'^-llx4-13anda;5 + 9a;* + 7x3-lla:^-8x + 2.
63. If 'x+- = y, find the value of a^ + ^, •
MlStKLLANi:ors KXIlKiHBlH.
75
64. Find the (liffHreiico of tlio H<iimreH of the hi','hoHt find lowest
of any three conhecutivct uuuibera in terms of the middle number.
65. Simplify
■X
'^-7+i
<;<5. Factor Km' Uhf ~2x^ + 2xY-'dxy(S ■ x*).
07. !?»(j1vo
17
3
2
68. Solve ^-__^_-^ = ^^,-„— •
6t). Simplify {l - ^:,+J^,] I I + ,U'^hy
70. Divide (...'' - p-) (x* +p'x' + /»*) 1 .y (^- - p) {x' + px +p').
71. Uu.solve 64.';'';/'-' - (/ - t>r)«M' + 4/^ into factors.
72 Find the ju-oduct of (r"' 4- .'{.''^ 4- r))' - (/■^ + [h-'' -nf and {x' + x
7:1 Find the co-eHiciunt of .»■■» in the product of l-2x + x' and
l + '2.i+.V- + 4.»;'' + 5..-'.
74. A 's money, twice B's and V> times C's = $190
lis " " C's " 3 " A's= 175
A's " 3
Bs= 176.
How much had each?
75. Extract tlio square root of x^ + {l + x'^) (1 +x)\
76. Showthat(a' + o)(,r + ft)(.*- + c) = (a:-u)(.K-6)(x-c) + 2-J (a +
77. Multiply]/ - f ^y'l + l'
78. Find the sq. root of (2.c + l) (2x + 3) (2x + b) (2x + 7) + 16.
79. Solve -2^rr-+— 3i^-i 45":^"
80. If x* + 8*' -aa-^- 168.»; + 441 is an exact square, find a.
81. Divide (oc + hdy - {ad + bcf by (a - 6) {c - d).
82. Show that a' + a?) + 6-- a(a-c) + be = (a + 6) (?J + c).
•ix--<lx+5 4a;=- 5x4-7
I
83. Solve
3x-4
4«-6
7()
KXEHOISKS IN AUilCHRA.
84. A number consists of two digits. If the left hand digit be
incri ised l)y 5 and the right diniinislied l)y 5 the original number
will be doubled. If these digits be transposed the result will be 1
of the original numl)er. Find the number.
85. l*rove that half the sinn of the squares of
x+a J a; -a 8n-x- , ■.
80. Simplify -_--^ - +~-^---- •
87. If 9x* + 24:X-'ij + ny'y'-Kfii^ + t/ be a perfect square, find n.
88. Find what value of d wi'l make <i- b + e-d{-bc -ad + bd-
m' + a'^ + //^ + c^ vanish when 4a -t- .,6 = (;, 3a + 4/> + l = 0, b + c + 4 = 0.
89. Factor 108,/^ - 1383,i-// + 4277 ;r.
90. Factor 4(;«' + 2)* - 37^'(^ + 2)-^ + S)x\ .
91. Find the H.C.F. of 21a;^ + :-;8..+r) and 129.»-'' + 221a- + 10.
92. Prove that (a; + 3)^ - (x + 2)^ is ecjual to 3.r-^ + 15a! + 19.
93. Divide by factoring a%b + e) + b%c + a) + c\a + b) + 2abc by ab
+ (ic + b'- + bc.
94. Find value of 3.<;»- 160.k* + 344.c^ + 700..- - 1910r + 1200 when
. — f;
51.
95. Simplify ■{ {a + %f + 2{n + 36) (a - b) + (« - bf [- ■',<i-b\- \
9o, Simplify
a^ -X"
«••' - x^
97. Write down the [ji-oduet of iy^ + llh''' - 4j- -\d by Ox'-lSx'^
-4.f + 15,
93. Find H.C.F. of 1 - .'• + ij + z- j-y + j/;;; -xz- xyz and 1-x-y-
z + xii + yz-i-^.z-xij::.
99. Factor 12«'^ - 132rt/; - 143ac - 1566c - 1446- - 12c''.
,, 12.C+97 .
100. Solve ^:^7. =G,
■ 1^ '
4J-+81
10m ■
15y - 17
MISrELLANKOUS EXEHCISES.
77
i
ab
MISCELLANEOUS EXERCISES.
D.
1. Ups = qr, show that ^1- + ¥>-:^p>-><=.{p:+ , )\
2. Find the relation that must exist among a, /^ «■ so that ,'■* +
(*,»''' + 6x'^ + (:,'' + l may be a complete scjuare as regards ,/•.
3. Find the value of x that H.."'' 3(l'''^ + o<'».' -39 may be a
complete cube.
4. Find the relarion between /> and c that y^ + '^<t.>''^ + l),>--{ r. may
be a perfect cube for all values of .»•.
6. Show that (a^ + h'^ + c~) {x'' + y' + ~P') - {ax + Jni + czf - (1r, - cyf +
(ex — azY + {a [I - hxy.
G. Under what C(mditi(ms is - + 7 + - = , . . > when a. h, <; are
not each eijual to zero{
H ti 1 13a; -10 , 4a;+» "(a; -2) lax -28
7. Solve .-^^— +----— ^ =mr6^-
8. Find 4 values of a for which (m-^ -fare -35 is resolvable into
factors of the Hrst degree in x wliose co-efficients are integral
numbers. State how many more could be found by your method.
9. Find the values of c and d so that x^-\- I2.t'^ + 8,0^ + 0,'' + '^ may
be the sijuare of an expression in the form «»f x' -\-i)x-\-(i.
10. Find the value of p and q if '^x'' -V2x*if + 'M,-h/ ^Aj^y* -
pj'if-{-<[[l^ is a perfect s(piare.
11. li x = a-\-d, y=^b + d, z = e + d, then x'^ + y'^ + r.^ -xy -xz—yz =
a'^ + b'^ + c'^ - «/' - w - be
12. Show that if any integer be put for x in tlie expression x*^ -
4.';^-|-14,i^*-32x'' + 49,r--60x + 3() the result will be a square number.
13. If (6 + c)x = 'f, (c + rt)(/ = 6, {a+b)z = c, prove xz+ifz + X)i + 2xiiz
-1-0.
14. Write down the quotient of x^ -3x^y + '3x*if ~ x^i/^ - x'^ -S-
6x^ -12x hy x^ - x -xii-2.
15. Extract the s([. n^ot of (.'/ + -) - 4( j/ — ) •
IG. Prove the following identity :
+
+
'8
P:XEU0ISES in Al.liEliRA.
..
17. pA"t-raot thfi sij. root of (m - I) (m - 3) (m - 5) (m -7)4- 16.
IX. Resolve 9rr(.,-» + 1.2a/>'0 (4^^ + 2430.^) into four factors.
19. Sliow tliat (h(. -I- /(, + /' + <i)"- = {m + nf + {m + pr + (m + q^ +
{n+pf + in + qY + ip + qY - 2(m''+H' + p- + q'^).
'20. Solve - -f ,- +- - h{a + b + cf h{ ,"- + 'V + A") '
a.r bx rx -^ ' - ^ hex acx abx '
be ar "'' ^" '•
nh
^(( ' b
22. Fmd value of (- -,,, + , J in + yr,- ,Zl) '
2'A. Fiiul sq. root of (■'■-' + "'-' 2.'') ( 1 + J-J 4- 1 .
24. Simplify 2 (- -}- ^^ + ,. ) ,^. -^ -„y- •
25. Show that {x. + yY + {x - nY - 2(.r-^ - ,fY^l6x'y\
2G. Find the s(i. root of ,.•* - .r'-hr" + 4r 2+- •
27. From the sum of the extremes of ^^ ; ^^y^ ; ^^ ; ^:p^„take
the sum of the means.
28. What (juaniity must be added to 9.r* -a.'3 + 4.V^ - 14.C + 25 to
make it a complete s(|iiare ;'
29. Factor {>t'^ + 4;( + 4 ).*•'' + (2<r + a - •))..■ + <>'^ 8a + 2.
30. If 2.)'-' - lO.'-^j/ + 25.o''ir - Rxi/ + '20;/^ is divisible by x'' - 'Sxy +
'iy'^ without a remainder, lind Ji.
31. Factor (3a''^ - 5« - 2).>;- + (^a^ + m + 2)r + - 1' + 2a.
32. Factor .*;« - / + 2x!i(x* + .<-hi' + ;/*).
33. Find the value of 4..'> + 9..'' -r),«,-- + 23..+() if 2,.;-^ = 3x-4.
34. Show that 2{d- + lr + r'-\-id, + nr-\-hc)-{ab + ac-{-bc)^{a^b^
•''' "*" a-^b+c
35. Show that {2x + 3i/y^ + i2ii + 3,"}'^ + (2,: + 3.i;)-' + 2(2.r + 3(/)(2«/ + 2>z)
+ 2(2(/ + 3;:) (2;: -f 3./,') + 2(2^ + 3.-; ) {2x + 3i/) - 25(^ + ;/ + zY.
36. Simplify
-^^^ ^"'^^ 2-3x + cT^ + 10+^=^-
MISCELLANEOUS EXERCISES.
79
38. Find the value of a in the expression lOOcc^ + SOx fa so that
one of the factors of the expression may be 4 times the other and
the sum of the factors is 25x + 10.
21 fi SI
39. Extract the sq. root of IQx^ - 9Qx + 216 - ^ + j-, •
40. Solve
X--2.X x--bx+\ x^+1x~% a;M--c-12
a;^-3a;+2 aj'-^-fi.c+S x" + x-l'2 x^-\Q
41. Find an expression containing no higher power of x than the
first which added to x* + ^x^-\-12x^ + ^x+\ will make it a complete
square.
42. Find the value of a when the fraction ^.t^..^^^^^^,,,. 3^^ ^ admits
of reduction, and reduce it.
43. Obtain an expression which will divide both 4a^''^ + a.'« - 10 and
4a;3 + bx^ -3x- 15, if ^ = 2a + 1 = 7.
44. Find what values of ta will make 3mx"'' + (6m- 12)x + 8 a
perfect S(|uare.
x*+2x '^+x--7x-S _ x*+6x'>+2x"-16x-4 ^
j?^s+3d:+5 "" x'^+lx+lO.
45. Solve
46. fennphfy ^^^zT)^:^Ti'
47. If {a + hy + {b + cy-t:<' + <lY=^4(ah + be + cd) then a=-^h = c = d.
48. Multiply and arrange in descending powers of x the expres-
sion {\+X + X^ (1 + J^ + X*) {1-X + x').
49. Solve ig{x + 4) - ^^ ~^) = - " ^^ ' ^•«) " i(^ + ^)-
50. Factor x^ -\-ia-b~- i)x^ - (« ~b + ab)x + ab.
51. Find the co-efticient of x^ in (j -f 1)'.
52. Find the co-efficient of ..• and x^ in (1 +.»+j;0'' + (l -x + x^)'.
53. When a^ + b-==c\ find the value of {a + h + c) {h + c-a) (a +
c- b)(<i + h-c).
20X-+13X-21 I2x'^+x-m 22a!---H03x-f91
54. Snnpliry y,^;j^iQx--i\ ^ ■i4x--i7x-m l0a;''+49a;+49 '
55. Find the factors of .i-^'" - (/•*" when m and n are positive
integers.
56. Find the conditions that x^ + bx'' -ax + b may be divisible by
both X - 2 and x - 5.
80
EXEROiSES IN ALGEBRA.
57. Find the conditions that x'^ + 73c^ + ax + h may be divisible by
x + 3 and x-2 for all values of x.
58. If h — — = r h rr~ » prove a = b.
a-x a+x b-x h+x^ '■
2
a*
o9. If H = -.prove -+ -
x-y x-z a-'i y z
60. Determine which is the jj;reater fraction, ,"^ or t— when
° ' 4 + m 4-H
m is any positive number, and ?t any positive number less than 4.
61 . What must be the value of x in order that -—^^r— . ., =11
when a = 67 ?
a-+70aa;+3a:-
62. The first two terms of a certain jjerfect square are 64.*'^ - 64.i"\
and the last two terms 14a; +f^. Find the square root of the
expression.
6."}. What number must be added to the product of any four con-
secutive odd numbers to make it a complete s(juare i
64. If ah + hc + ca = 0, prove that (a + !> + cf = <i^ + />=' + c^ - 'Sabc.
65. Show that x*^+i/ + {x + yy is divisible by j;'- + ,'■;/+*/- without
a remainder.
66. What is the least integral multiplier that will nrike 17^;'* -
68a;*i/ + 102xY-^8icY + 17a;?/* a complete cube ?
67. Show that the product of any four consecutive integers
increased by unity is a perfect square.
68. If |(6-r)+ t{r-~a)+'j{a-b):^0, prove i (,-,/)+ |(.,-3)
+ l(y-x)^0.
69. It — 26;r" + "^^~" ' + •'«// =" ^•' '*'^''"' ^^^^ (" +^- '■) (^ +
c-/))(fe + o-rt) = 0.
2ac
70. If (rt^ - be) (b'' - ac) {c' - «6) = 0, show that
rt""''t^' c"^ a"b^c-
X'-y- xy
71. If-^ •/ =-^"and ,
a-o 2 b-c
y--z^ yz
,, Z--X- zx
, then —
J- ' c-a 1/
72. If 3((<- 4- 6- + <-')= (" + '' + <•)% prove <t-/j^c'.
7ii. Find the co-etticient t»f .*•* in multiplying 1 +2.Af;i/;^ + 4u;'-|-
5a;* by 1 - 2.'' + :3^- 4.*« + 5a;*.
MISCELLANEOUS KXERCISKS.
81
n
74. What is the least factor that will make x^ - 11.k^ + 40x-48 a
complete square ?
75. What is the least expression used as a multrplier that will
make x^-bx^ + bx— 1 a multiple of 5»'2-8u'4-7 ;
76. If x^ = 3ic - 4, prove that ;<;« + 45x = 44.
77. Prove that the product of the sum of the s(piares of any two
quantities and the sum of the squares of any other two quantities is
always equal to the sum of the squares of two quantities.
78. Prove that if the sum of two numliors be multiplied by the
sum of their reciprocals the product is not less than 4.
70, If x^j/ =z(x + y- ,-)^ prove (.r - zf = yz.
80. If X + !/ + = - ryz = 2, prove (1 - ,':)'•' = (1 - x//) (1 - xz).
81. If b(hx'^ + a^y) = a{aif + h'^x), \:>Tove bx + ay =nb find ay = hx.
82. If (a + h- c - d)x = cd - ab, prove (a + x) (b + x) = (c + .*•) (d f x).
83. If (a'^~bc)x + (b--ca)ii + (c''-ab)z^O and ;>■ + ;/ f,--0, prove
ax-\-by + rz = 0.
FA. If (2a - 3i/)v -= (^ - xf and (2a - oz)z = {x- yY then x + y + z = a
and(2a-3..>c = (i/--)l
85. If a = ^, 6 = ;;-^ r=^, then prove (ti«)(;t:)(S) = 1-
I
on
86. Show that (1 + x) (1 + ;.-') (1 +/*)... -to n factors =
87. Is x^ - xy + y"^ a factor of (x - yf - xy{x - ;/) (.»■' + !/2^ ?
88. Prove (x - yf - x^> + y^ =-- 5.i-i/(x' - .'•;/ + ,'/-).
89. li{x-y)z^ = c?,{y-z)x' = o\ {x-z)y^ = ¥ and (x-y)(y - z)(z - x)
= 3abc, prove ct^ + b^ + c^- 3abc = 0.
90. If x{l + y) = l and ]/(l + ;;) = ;:, prove - .: - i + x + 2x' + 4x^ + etc.
91. The H.C.F. of two ex})res:nons is a 7. and the L.C.M. a^ -
lOrt'^ + lla + 70, and one of the quantities is a'~12(r + 3o, what is
the other ?
92. Simplify -
x"*- V'
.»yn+l
x'-^y
93. Prove that if the sum of three (juantities is zero, then the
sum of their cubes is equal to three times their product.
82
EXKRCISKS IN ALGEBRA.
94. If the sum of the cubes of three quantities be equal to three
times their jiroduct, then the sum of the quantities is zero.
95. Prove that a'^ — hc + h'^-nc + r'^-ab is not changed by sub-
tracting the same quantity from each a, b, c.
96. Show that the vahie of x^ + 1/ + z^ - 3xy:: is not changed if
X' - yz, y^ - xz, z^ - xy be substituted for x, y, z respectively.
97. Find the value of ;»; that will make both of the following
equal to zero, icHSxH I2x-Ui and y^ - 13x + 12.
98. If a + 6 + c = 0, show that \, „ + ■ , ■ :. h _ ,. =0.
h-c
c-a
a-b
99. Determine the values of }> «iid q that will make 4»/* - }2)/ + py^
+ qy + 16 a perfect square for all values '>f .»•.
100. If ax'^ + bx + c becomes 8, 22, 42 respectively when x = 2, 3, 4,
tind its value when x= - |.
101. Fuid the value of x that wi)! make x* + 6x^ + i^h'- + I'Sx 1 a
perfect square.
102. Detennine numerical values for A, B, C, 1), so that 2.*"'' -
13.r2 + 26.>; - 14 - A{x - 1) (x - 2) {x - 5) + £(x-\) (x-2) + C(x - 1) +
D, may be an identity.
103. If x + a is a connmm factor of x'^ + px + (( and x^ + lx + )ii, show
that'"-^
= tt.
l-P
104. Find H.C.F. of 1
?((
- >H^ + vv" and 1 - m* — m^ - m?.
105. For what numerical values oip can the fraction -^f^A^^Z^t "
be reduced to lower terms ?
106. Show that xy + xz + yz is a factor of {x? - yz) {y' - xz) + iz^ - xy)
(t/' - .xz) + {z^ - xy) {x^ - yr.) and tind the other quadratic factor.
ANSWERS,
ANSWERS.
EXERCISE I.
ADDITION.
Page 1. (1.) 84a + 12/) -7c. (2.) 7ax'' + 21ax ~2hy-3i/ + o.
(3.) ,V* + rti^ft + |A'^. (4:.)2a+2b + 2d. (h.)'p + ,i + s.
(6. ) 3x2 - 5^3. (7. ) 9(1 - 7h + 4c. (8. ) 70(a6c + a%'c-).
(9.) 6x-3y + lbz. (10.) 2^ •
EXERCISE II.
SUBTRACTION.
(1.) 2a-2j: + 18. (2.) (t'-^- 14^/ + 2(y'' + 4;:'' + m.
(3. ) 16« - 14<^ + 14 CI/- - V3\/a'- h\
(4.) i)--17(/-22)'/ + 99.
Page 2. (5. ) ((^^ + "■■'& + 9a/>'^ - 2/>». (6. ) x^ - \xy.
(7. ) 12(f + \0h - 22c. (8. ) x^ - Q.c'y + llxi/' - 6i/'.
(9.)aH« + TV (10.) -V'.
EXERCISE III.
ADDITION AND SUBTRACTION.
(1.) -4« + 27c^ (2.) 2 + 20X + 29A
(3. ) bx - 3fxyz + 11 1/ + 9z. (4. ) x? -a^+ if + b'.
(5.) ' (6.) c\ (7.) 0." (8) 6'^
(9.) ?5'' + 26. (10.) (r + /)'^ + cl
EXERCISE IV.
Page 3. (1.) a-46 + 3c. (2.) 3a-2;r + c+l. (3.) 3c. (4.) c.
(5.)5^-.V. {H.)2a~b-il. (7.) -46. (8.; 0.
(9.) 6/> + b(/-<>r. (10.) a.
[851
I
86
Page 4.
EXEROISES IN AI.OKBRA.
BXERCISB V.
MULTIPLICATION.
A.
(1 . ) ^"^ - 3.r'i/ + '^.>•^f - >/ + J-' - 2..;!/ + J/'.
(2. ) r.r'' I/-' -f z' + 'Xniz. (.'i. ) 1 - ^"^ ■ 1 2..
(4.) :V2yHiA (r>.) <(*-2a^//' + M + 4(i/>r'-n*.
((i.) ..« - 57^-H2<5(i/^- 1. (7.) 1 -X-. (8.) (x'-a«)^
(9.) x2"' + 2x"»i/'» + iy'^". (10.) a''-^^
B.
..4
(1,) ^„^i, + r, + df. (2.) ,/« -4a* + l«a^-16
(3. ) 27x-=* + Sif + z' - 18.'!/.:. (4. ) 1 + \i' + ■:'' -
(5. ) ax, - a'x*. ( 0. ) a>' - 2a^/>* + />"•
(7. ) u:* - \/ - s* + 41/".^ - 6jy-;:;' + 4i/r'. (8. ) j-^"
(9.) d'-{mx-nx''f. (10.) a;«-^-
:>i/2.
C.
(1.) x'-'+!/". (2.) (6 -a).
(3.) (.'•■' + «/>)' - («•« + /^•'•)'- (") 0.
(4.) a-" - 2;>'V(/ + n^if - !/^ (5.) :>:' - - (<t - hf.
(6.) 3-3 - (a 4- /) + c).';^ + ("'> + '*'' + '>'')^ - "^'■•
(7.) Co-etticient of ,'• above m(ab + a<' + hr.) :.S>iS + 8y. -
2 + 3 X -2 = 2.
(8.) ,c^-(l+2 + 3).t'^ + (2 + 3+tJ).r-0-x='-6x'^ + llx-6.
(9.) x^ - 15x2 + 71x - 105. aO.) 1408.
EXERCISE VI.
DIVISION.
A.
(1.) 4a2 - 3a + 5. (2.) a'' - 2ax + x\
(S.)x*~x^ + x''-x+L (4.) ..• + /). (5.) .r'-ax + b.
Page 5. (7.) 8x-3!/. (H.) x' + iSx' + llx + (l (9.) .r^ + j/».
(10.) (x + ay^-0e + */)/>-f6''.
ANMWKKS.
B.
8;
(1.) a + 6 + 2r. 2.) .r* + Sx^ + 2.
(,S.) Hint (I'-^ + h^ c^ + Snhr is (livJHihle by <t + /».■,
{,,f + {'2"\^ - ('^ ' ' 18..(/;; is diviHi})k' by j-' + '2y - A~.
Rem)' ^.h'^-2xy + 3xz + iiyz.
(4.i (o + l') -;j(c4-rf). (6.) (l+X. (fi.) (x + b)(.r-\-r).
(7.) '2<r - 3M/> + 4fc2. (H. ^ rt* - a»+ 1 . (*.».) x - .n/ + */*.
(10. 'ivitlu in the (irdiimry way, and since renuiindcr in
zero, « = 2.
C.
(1.) a 20, 6 = 85. (2.) m-ofi.
(.1.) Apidv |)i'i u 7-^ . ett ., then - „ ' , „ . ■ > - -
4,,.'^ + 8r (/ + 7 !/' - 6x2 + 'Mfz + 9^'.
(4. ) .'^ - .'•* + 1 .
(5.) Aj»j»ly ditterencc of 8<iuaies. Result, «-- 6.
H>.)x~Ik (7.) 6 + c-a.
Paged. (8.) 8.'''''-22«a-4-15a-. {U.) 7x + ^z. {W.)u+b + c.
EXERCISE VII.
MISCELLANEors EXERCISE.
(1. ) a' 4- 4<t + 4. (2) x' - 7.<'"'//'- + >/. (3. ) hx -y-z.
(4. ) J'* - 1 0.,'^ + 9. (5. ) 4x' - Hx + 7.
(8.) 2an.
(0.) A-u + 2u^-4a». (7.) i+?f-?
(9.) a = 7. (10.)^^-Y^ + 7j •
(11.) Write 1st (9.. - - 6ax + a') - {x^ + 4<ix +4:a^\ 2nd
(9..'^ + Cidx + f<0 + (x' + 4:ax + 4a'). Sum unci differ-
ence, etc.
(12.) .r + 1 4- p, • (Ki) x'' + 3x + l.
(14.) Perform the divisicm and remainder nuist be zero,
etc., r^^a(i+pb-2ab~a'^p + a^, s — bq-b'^-apbi-
a'h.
(15. ) X* + x' + 1 + -\ + 7. • (16. ) - 26. (17. ) a« - 6«
Page 7. (18.) a---- -1, 6^0. (19.) j, + |. + ;i -^,-
(20.) u = 5, 6-3. (21). X - i • (22.) m = 2.
^1
<l^
"^.^^
.a>.
IMAGE EVALUATION
TEST TARGET (MT-3)
m
f/
/.
/
1.0
;siM iiM
:!f 1^ 12.0
i.l
y5 lllll-L4_ mil 1.6
6" —
V^
'<^>
W
^
^m oS
O
fjy^//;'/.
J
Photographic
Sciences
Corporation
4h
<^
^<b
\
23 WEST MAIN STREET
WEBSTER, NY. 14580
(716) 872-4503
^
o^
88
EXERCISES IN ALGEBRA.
(23.) f^- 6, ^*-ll, .'= -6.
24.) Divide in (n-diuHiy way and remainder must be
zero ". n^(('^ - iio'^ = a'^ or n'^ = ?< + 1 .
(26. ) a* + 2rt* f Aa^ - Sa^ - 16a - 32.
(26. ) a;« - 3x^ + O.r^ - 27;>^' + 81 .^^ - 243.r + 729.
(27.) ((=-1. (28.) 10. (29.) ;i,-^ + 2a; + 3.
(30.) 2.*+ 3.
EXERCISE VIII.
HORNER' S METHOD OF DIVISION.
A.
(1.) a' + 2a+A. (2.) 5x' + llu; + lL
(3.) .c3-3x'^ + 3x + l. (4.) 3x*-2x-3-x + 5.
(5. ) 2x' + 3x - 1. (C.) x^ - xy + f. (7.) 2xH x + 1.
(8.) ir*-4*=^ + 6..-^-4x' + l.
(10.) 3x3 - 2.i;2 - 5x - 3.
(9.) x^- 29^3 + 47^2 -25.
Page 8. (1.) -7.
(5.) 101.
. (10.) -7617. (11.) 0. (12.) 0.
B.
(2.) 15. (3.)
(6.) 20. (7.)
-205. (4.) -G400.
1. (8.) 1. (9.) 943.
EXERCISE IX.
Ii;VOLUTION.
(1.) a2 + 2a?> + 6^ 4^(2 + i2a^ + 9//' ;
(2 .) d^x^ + 2obx + })' ; 225x2 ^ 420.x (/ + 19())/ ;
^a^ 8 , 1^\ i= 4. 2 . ^\
SSx--"^ ^ '^ <da- ' y'- ^ x'^
(3. ) i^ - %ih + ¥ ; a'^ - 4a6 + 46^ ;
4x^ - 24x1/ + 36*/^ ^'-2+'S-
91/ ^^
(4.) 49x* - 70xV + 25y* ; |. - 2 +;
.2>
■9 "" 6 "^16' 16
6.1- -'j/2 16y«
5 ' 25
(5 . ) a'^ + 6H c2 + 2tt6 + 2(tc + 26c ;
4x2 + 9,j^2 ^ i(5.i + I2x J/ + ICfs + 24i/2 ;
l + 2x + 3x2 + 2x3 + x*.
)e
ANSWERS.
89
(6. ) Ida' + 25^2 ^ 3^,.-. ^ 4,)^.^,, ^ ^g^^^. ^ ^^^^^ _
4+9+16+3+4+6' ^+^ + l2" + 3+|
(7. ) r/-' + h'' + r^ + 2(«/) - 2ar - 2hc ;
«2 + 62 + c-^_2a6 + 2ac-26c;
a'^ + //^ + '-' - 2(t/> - 2ac + 2hc.
(8.) x^ + |)/Hl-.«v + 2x-j/;
a;* - 2(tx-3 + ((t^ - 2b)x' + 2ahx + //^.
/Q \ X* 2.r'' , 4a;2 , ,
(^•) -9+-3-+-3-+^+l;
9.t-*-2.r3- 171^.2 + 2^ + 9; !!i! + »!il_2
9/4= 4w-
(10. ) a V + ?,2,/ + ^4 + 2ff A- + 2a6,n/ + 2hc'y ;
a;" H-' »= •>« o- <^..
^-' + ^'4.1= I !? 4. 22 ,2?/
?/
EXERCISE X.
Page 9. (1.) .T-H3a'V + 3,«yH?/^;x''-3xV+3a;i/-j/3.,.3 + ^3 + ,3 +
3x^,/ + 3^^'^ 4 'd.nf + ^xz^ + 3j/s^ + 3(/'^;^ + Qxyz ; a;^ +
i/« - z^ + Zx-hj - ^x^z + ^xy' + a.-;^^ + 3^^v2 _ 3 ,•,„ _ g^^^
(2-)mH^ + 3m+|; m^*
— T- — 3m + - ;
(3.) (t3 _ ^,3 ^ ^3 _ 3^^7^ ^ 3^^^, _!_ 3^^«^ _^ 3^^^,, _^ 2^^^ _ ^^^^ _
6a6e ; 0} - W - c^ - 3(f -^t + 3a6'' - 3a'^(; + ^ac^ - 36^c -
36c^ + Gabc ; 1 + 3a; + 0*'^ + 1x} + 6a;* + 3^-^ + .r«
(4.) 4(a + i)^ (5.) a\ (6.) (7.)
(8.) 2(rt-c)(6-rf). (9.) 2(1 + 3a*). (10.) 27.r'.
EXERCISE XI.
MISCELLANEOUS EXERCISE.
(!•) (2.) 0. (3.)
(4.) x2 + i/ + c'^ + 2.i;|/ + 2.r~ + 2i/,t. (5.) 2('l + 3a;*)
(6.) {a+hf.
(7.) Factor expression u-" - 8j/''' - 27/ - 18.ri/.:; and o..e
factor \fi x-2y- 'Sz, which is ecmal to zero. • a:' -
8i/3-27z' = 18.np.
(8.) =0. (9.) 8x3. • (10.) 0. (11.) a\
(12.) 8(x'^ + |/^)l
90
EXKRCIHii'S IN ALGEBRA.
FACTORING.
EXERCISE XII.
Page 10. (1.) (a + b + cd)x; (a+p) (x + y + z).
(2.) {a-b)(x-y){x + y); (l-a)(l-6).
(3. ) 3h\:^a* + lla'b - 4/>^) ; (1 - x') (1 + x^ +p + q).
(4. ) 3ac%oh''c + 4rt^fe - 7c') ; (2a' - I) (x" - 1).
(5".) (4x+ii)(a + b); {n-l)(a + b).
(6.) {2x + 2f)(a + b); (x-3)(x~y).
(7. ) (ax - b) (ex + d) ; (u-^ - a') {x' + ax + a').
(S.)(l-a + b)(l+p + q).
(9.) (l-b)(a-b + c); (a^-l) (a + 1).
(10.) (a + b-c)(d-e+f).
EXERCISE XIII.
COMPLETE SQUARES.
(l.)(a + 4^)^ (a + 7by. (2.) (a + 18)'; (x~5ay.
(3.) (xy-Sf; b'^lx-oy)\ (4.) (mV + l)'^; (4rx' + 2)\
(o.) (a-^Y; (1-A.xy. (^.) (^x"" + 2^jy ; (C^-lf.
(7.) {xy-'-Qf; (a + b + c+d)\
(8.) {x + y + zY; (Ax-' + ^yy.
(9.) (3a -26+40)2. (lo.) (3x + 2y-z)\
Page 11.
B.
(1. ) (^x!" - 4i/s)^ ; (a - 6 + cf. (2.) 0.
(3.) (2a'^-3& + 4c)2.
(4.) (.--- + — + ^j .
Note. — Question shovild be — tt— •
(5.) (26 + 3c-l)^ (6.) {x- l + l)'-
(7.) i^ + q + r-sf. (8.) g--7;36t/'.
(9.) r2a'^-3a + 4)'^ (10.) (a' + b"" - c^f.
ANSWEHS.
91
C.
(1.) -{ 2{a + h) + '^(c + d) y^ (2.) {2a-h + cy.
(4.) Multiply second expression by 2 and add to first, etc.
(5.)a«-aS-i7a* + "''^'- 1"^"'
■4« + ,
(6.) (3a + 2)2 (a - 3)'^ (2a. - 1)1 (7.) (f + 3 - f )'
(8.) Multiply out and re-;irrange, etc.
(9.) (.e'-2xy + yy. (10.; (2-^-£)^
EXERCISE XIV.
DIFFERENCE OF SQUARES.
0\2
(1.) (2x-Sy) (2x + 3y) ; (I2x-17y) (12x + 17y) ;
(4x'^-l) (4,*;''' + l).
(2.) (2a - 6 - c) (2a - 6 + c) ; (4x'+ */ - z) (4x + y + z) :
(3r/i + 2)1 +p) (3m + 2m - j?).
(3. ) (200) (198) ; (x - y+ ;;) (x + y- z) ;
(a-36 + c) (a-6-c).
(4. ) (x' + 1/2 + z^ + 2xz) {x" + y' + z^- 2xz) ;
(a-b + x + y) (a -b-x- y).
Page 12. (5.) (b + c-a + d) {b + c + a-d) (a + d-b + c) {a + d +
b-c); (a + b + r) (u + b- c) {c + a - b) (c -a + b).
(6.) (x«-i/) {x" + y-) ; 16(l + a-) (1 -;.•) ; 4(a + c) (b + d).
(7. ) (x' + y' + z") (x-^ + 1/2 + z' - 2xy - 2xz - 2yz).
(8.) lo(x-2y)(x + 2y); 3(9x' - 4.y') {9x' + 4y^) ;
{l-2ab'')(l + 2ab^).
(9.) (3a-56 + 4c)(-a + 6-4c);
(a' + a-b' + b){a' + a + b'-b).
{10.) {x + 2z)(x-2y).
B.
(1.) 7x-5y + z. (2.) (x-' + yy-zK
(3. ) (a^ + b' + c' + d') (a' + U' -c'~ d').
(4.) Factor dividend. (o.) 840.
Apply difference of squares to 6, 7, 8, 9 and 10.
02
EXEUCISE.-^ IN AUiEBRA.
EXERCISE XV.
EXTENDED APPLICATION OF (x±yy AND x^-tf.
A.
(!.)(;*'- "^'Xy + 2i/2) (:r2 + 2xy + 2y^) ;
(x' - 3x- 3) (x^ + Sx - 3) ; (x' -x + 1) (x^ + x + 1).
(2. ) (lOx^ + 4x + l) (K).*'^ - 4j- + 1) ;
(x' + 3..' + 7){x'-3x + 7y, (a' + 3ah- b^) (a' - 3a/> - 6^).
(3.) (2a2 - 5a& - 3b') {2a' + Mb - 3b'') ; (3x^ - a;j/ + i/)
(3j;''^ + i^'.V + ?/'0 ; (i«''^ - 4xiy - /) (x"^ + 4xy - y').
(4. ) (m' + Amn — n') (m^ - 4»/i?i - 71^^) ; (x*-x^ + l)
(x* + a;2 + 1 ) ; (c'^ - ac + a^) (c'^ + a/- + f t^).
(6. ) (a' - 4((l> + 8/>'^) (a' + 4<(b + Sb'') ; (26a^ - 5a + 1)
{2oa' + 5(t + 1) ; (o- - bab + 3b'') {a' + 5ab + 36^).
(G.) {3<(''-ab + ^) (3a' + ab + ^-^) ;
(2x^-1 -3) (2x^+1 -3).
Page 13. (7.)(.^-f + ^^)(x^+f4-;^);
(x'^-5x + 25)(x''' + 5x + 25).
(8. ) (4a' - 3<ib - b'') {4a' + 3ab - b') ;
(x^"* - 4x''"(/'» + 8)/'*") (x'-^'" + 4x"»)/'" + Sy""") ;
(x'^-3x + i)(x'^ + 3x + l).
(9. ) (4m' - 2mn - 3n') (4?n2 + 2mn - 3n'').
(10.) -{ (,t + yy'-3z(x + y) + z' y -{ (x + y)' + 3z(x -\- y)
+ z"-',
^ (a -f oY - c(a + b)-c'\- -{{a + b)' + (-(a + b}- c^) J> .
B.
(1 . ) (3a'' + 3a?; + 2b') (3r»2 _ 3a/> + 2¥) ;
(x' + x + 4)(x'-x + 4);
{4x' - 6xy + 9(r) (4x' + Gxy + 9?/').
■ (2.)(r'"'+i)(C+"''+3);
ne ~ r2 ■" 9 / Vl() "^ 12 ''" 9 / '
(3. ) (a' - j'-c'- 2br) (a^ -b^'-c'^- 2bc).
(4.) ^x^-2x(^i^-z) + 2{y^z)'y
^ x-^ + 2x(y + :-■) + 2(y + zf y ; (a' + 3^2) (3a' + b"^).
AXSWKRS.
9d
)■
-n
'■)•
n.
(5.) (J..-f.UM(l-U-')-
(6.) 4(r»'' + 5(t/> -2//0 (/>' + 5rf/, - 2a-).
(7.) ■\Aa'-ha{h-c) + 2{b-cy)- -^4a» + 5a(6-c) + 2
(8. ) -{ (^-^ - x// + ff - 3(..;» + ?/>) + (x+ yf y ^ (x^ - xy
(9.) {a' - 2ab -f 5//^ {5a' - 2ab + b'') ; (LV - x-i/ - 3v^)(2;r'-i
xij ~ 3if).
EXERCISE XVI.
TRINOMIALS.
A.
(1.) (x + 2) (x + 6) ; (x + 4) (,x + 5) ; (a; + 37) (a; + 10).
(2.) (.*' + 40)(.c- + 49); (x- 13) (.--14) ; (.,• - 25) (.• + C)
(3. ) (,. + 20) (,. - 4) ; (,: - 26) (a- - 62) ; (x - 40) {x + 3).
(4.) (5?; + 4) (3,.- + 1) ; (3a' + 2xj) {2x - 3*/) ;
(4c-7a)(4c + 3u).
(5.) (^• + i)(^--i); (a; + f)(cc-f); (a;-l)(x-}|).
(G. ; (,i- + 12) {x + 21) ; {x - 99) (.«■ + 7) ; {x - 48) (x + 11 ).
(7. ) (3.. - 7i/) (7x 2;/) ; (x^+ i) (r^ - |) ; (x- - |) (.
(8.) 13x(13//). (9.) \ 4(,f + 2)^'-x'^ ^ -^ (x + 2y
(10.) -^(a-^)*"-!! ^-^ (a-6)'»-33[..
+
+
^)-
■11M(«
B.
Page 14. (1. ) (8x - 9) (9^ - 8) ; (4.t - 5) (2x
9a;2 }..
-7).
f>2).
(2. ) (3.« - 4//) (8x + ^) ; (5./; - 1) (2x' - 3).
(3.) (15a; + 99)(a; + l); (4x- - 3) (3a; + 7).
(4.) {2a ^W) (3a - hb) ; (4;:; - 5x) (8,-; + 4ic).
(5.) Multiply by 4 times co-efficient of first term thus—
4 X 4132*^ - 4 X 413 X 606»j/ - 4 x 413 x 299*/^ then
add (606//)2-(600)/y^ .-. we have difference of
two .squares - 1^826^; - GOG;/)'' - {9t2%y)\ Factor
in ordinary way and divide result by 4 x 413, =
(59.« + 23|/) (7x-13w). Second part (17a; + 8y)
(12a; - 25)/).
i)t
EXRIUJIHKS In AI.fiEHRA.
(6.) (5..- + 151) (<>,.• 140) ; (i\.,+:>:i) (rxr+w)).
(7. ) (!.•;..• 4;»r>) ((;.,■ '.17 > ; (7,.-^ 1,;,-)) (Sr - 1«;9).
iH.) (<;,.• - 111) (7.. ■+ 107) ; (8.M-I(M0 (12y-01).
(J>.) (17.v + 2ir))(2.r-14;i). (10.) (7r/ -437) (4a + 191).
EXERCISE XVII.
POLYNOMIALS.
(L) (4a;-2i/)(5.>; + %-2)
(3.) (x-6y){7~2x-:iy).
(5.) (x-3y)(x + 2ij~4::).
(7. ) (3a: + 2!/ - 4,^) (2.r - 3j/ + 5^;)
(8. ) (ll(t -b- He) (5a - 06 + 2c).
(9.) (2a-3fe + 4c)(3a-7/>-c).
(10. ) (3m -n- br) {m + n + r).
(2.) (2a -56) (3a + 46 -3).
(4.) (r + 5;/)(3a- + 4// + 2).
(0.) (3x'-2//)(G.^--4*/ + 34
B.
(1.) (7a; + 6i/ + 8)(a;-y-2).
(2.). (5a;-5j/-22)(4a,- + // + 4).
(3. ) {Ax + 5 J/) (5a; - 4y + 7). (4. ) {x + 3*/) (x- - 4 ./ - 5).
(5.) (3x-2i/-2.:)(2..;-3;/ + 4;v).
Page 15. («;. ) (2a - 56 - 7c) (2r< + 36 + 3c).
(7. ) (5.«' + 4;/ - 6) (3x - 1y). (8). (5a - 46 - 2) (a - 36).
(9. ) (6x - 4 J/ + 3) (3a; - by). (10. ) (5a; + 4 */ - 6) (4.« + 3 </).
C.
(1.) (a-36)(4a + 76 + 4). (2.) (3a; + 4.y-8;:)(2.x-5y + 6^).
(3.) (2a- + //4-7;;)(.c + 2i/ + 3,;). (4.) (3.f + 5v)(8x--y + 4).
(5.) (3.v-4]/ + ,-;)(a; + i/ + ,':).
(G. ) (2a; + 3m - As) (a; + m + 3«). (7. ) Am + 3p- 9n.
(«•) . (9.) .
(10.) Factor in ordinary way (8x- - 3j/ + 0,~) {2x-by+8z).
Then ^ llie simi = ()ne (]nantity, ^ the diiierence
■{Xw otlier
(5.,.-4// + 7;)^-(3.*' + ^-~)^
AXSWKUS. 96
EXERCISE XVIII.
APPLICATION OF j'^±y\
(1. ) (a + h) (a^ - ah + fe') ; (a + x + y) (a' + 2ax + x^- ay -
^y + y'^) ; (m + w + jo + n) (m + nf - (m + n) {p + y) +
(?' + '/)'.
(2. ) 2(wi'-^ + n') (m^ + 5m2w2 + n«) ; (a^ + 1>') (a* - a»fc-^ + h%
(3.) (a* + />0 {a*" - o^)* + />') ; (a^ + />») («'" - a'Jf + />'") ;
(2a + a/;)(4a'-^-Of(/> + %0.
(4. ) (x« + y>) (x^' - jt^y^ + iy6) ; (Sx^ + 8 (/«) (25x" - 40:r' i/« +
64(^«); ^ (o -;>-<•) ;- ^ a2 + a6 + ac + 6H26c-|-c2 )>.
(5. ) (2a; - 4i/) (4^-2 + 8:/-// + IGy^) ; («« - 6") (a'« + a«6" +
6") ; (ic - (t + 6) (x'-^ - 2ax' + a» - 6x + a6 + 6*).
(6.) (7.) (9.) (10.) Use ^^^', etc.
Pagre 16, (8.) {x + af -h{x + a)-\-lA
x±y
EXERCISE XIX.
GENERAL EXERCISE IN FACTORING.
A.
(1.) (x + y){ax + ay-bc); (5jt> + 24) (3j3 - 1).
(2. ) 2a(26 - 2c) ; {a-b-c-2) {a + h + c).
(3. ) (2x + 3// + z) {X + 4*/ + 3:;) ; {x' + 4y'') (x' - y'').
(4.) (b-c)(x + ay; {2ai-2h + l) (a + h + 2).
(5. ) {x + y) (x - y) (u- + .r // + ,/) ;
(a - 1 - 6) ((('^ - 2a + 1 + tt6 - 6 + 6'^).
(6.) (a^ + 62) (c^ + d') ; (x'^ + 5;*^ + 4) (a;'^ + 5a; + 6).
(7. ) (..■ + !/) (a;2 + xy + >/) (x' - x,/ + r) ;
(a;'^ + l)(c«^ + *-!).
(8.) (a-l) (a'-a + l);
write exjiression x^ + 1 + 4a'''' + 5a; + 1 factor by i)arts ;
{x + 1) (a'-'' + 3;,' + 2) ; (x + 1 ) (x + l){x + 3).
(9.) (..' + 1 ) (.>' + 2) (.' + 3) ;(.'-- 1 ) (.,; - 2) (x - 4) :
(a'-2)(.r 3)(.,'-4).
(10.) (,. + l)(.,.-2)(,.' + 3); (a: -2) (a- -4) (.. + 5).
(.^-l)(.c; + 2)(.«;-3).
96
KXKUn.SKH I>f ALfJKHUA.
B.
(1.) (x + 2)(x + :\)(2x + l); (x + l)(x + 2)(3x + 2).
(2. ) (x + 1 ) (.'Ir- + 2.1- + r.) ; (x + 1 ) (x + ;{) (2;*; - 1).
(3. ) {2x 4- 1 ) (2x - 1 ) (x + 2) ; (M.r + 2) (3u; - 2) (x - 5).
(4. ) (x + 2) {x - []) {iix - 5) ; (a + 2by.
(5.) . {(I) (x-l)(x + i){x^-px + q).
(7. ) 3a/>c. (8. ) (x + 3) (..■ + fi) (.*•'' + lb- - 2).
(9.) (x - 1) (x + 1) (x'^ - 10). (10.) (x + l){x- 1) (x - 2).
EXERCISE XX.
H. C. F.
A.
Page 17. (i.) 2(a-x); a + h. (2.) x-a. (3.) x-7.
(4.) a; -12. (5) x^-2. (6.) x-2. (7.) 2x + 3.
(8.) 12a;'^-5. (9.)ft-l. (10.) a^(3a + 2).
>il
(1.) a; + G.
(4.) a; -3.
(7.) x + S;
(10.) x»(a;-l)(a;-2)».
B.
(2.) 7a' - 2y. (3.) cc' + 2x1/ - ■»/.
(5.) 3u;'^-2. (6.) x + y.
: x= -3. (8.) a = 6. (9.) x-l.
EXERCISE XXI.
L. C. M.
Page 18. (1.) ()x\3x-l) ; (2.) (x + 2) (x + 3) (3x + 2).
(3.) (:x-l)(x + l)(x + 2).
(4.) (x + ^) (x + 1) {x-2) (x + S).
(5.) (rt2-l)(rt2_9)(„4-5).
(0. ) (ic + 3) (x + 4)(x + 4) (x + 5).
(7.) (.«; - 1) (u- - 2) (.. - 3) (.. - 4). (8.) (aHl) (a«-l).
(9.) (uj^-i/OM-^^ + a-'y + ■?/*).
(10.) (.r-3)(.,--8)(x + 8)(.f + 9).
ANHWKH8.
97
EXERCISE XXII.
(JENERAL EXKUCISE li. C. F. AND L. 0. M.
(I.) H. C. F.=a-1; L. C. M. .»»-5aH7«'-a-2.
(2.) H. 0. F.-(.r+l)(.'' + 2);
L. C. M. (.r+1) (.r + 2) (.r + .'J) (.'• - 2).
Note — (J>ue.sti<)ii .shoiiid be w^ + .!•'', etc.
(8. ) (.r' + 5u; + (}) (x' + Ix + 8). (4. ) a -^ 12.
(5. ) .* 20. ((). ) a ^- 1 2, t -- 12. (7. ) <* = 1 0.
(8.) />-2. (9.)f= 114. (10.),,-^*). (II.) nhx\
Page 19. (12.) w-28. (l.'J.) 4ax ; %x and 2<ix ; 4l>x.
(14.) «— {); other expression i»" + 2. (lf>.)
(16.) . (17.) . (18.) x-' + dx + d.
(I».) . (2(».)
(21.) (.T + 1) (;i!4-2) (:*; + :i) (x - 7), .-. a- = 7 to make each
vanislj.
(22.) H. C ¥.=x~ 1, .'. .7—1 to make each vanish.
98
EXKKCI8B8 IN AI/JKHKA.
FRACTIONS.
EXERCISE XXIII.
Paere 20. (I.) "— • ?— — . (*/)''—■ "
(^.)a + h + r; ^-. (4.)^'= 1. (5.) f ;
fix X
/^ \ (a~iy . a-Zh
(8.)
2a + 4
(7.)
a;-3//
(9.)
(a-l)(a-2) "^"'^ (x-1) (:i-x)
EXERCISE XXIV.
(10.)
221 - .{Ox
14
(a!«-l)»
Pa«e21. (4.)|^.
(!•) 'tS^^^^ (2.) a (3.) ^^-^^
(5.)
(8.) 4a' -9x'^. (().)
(x - 2a)s
2ffl(a«+/>')2
3x+l
05.)
(a^-b'^y^
"^ • (7)0
(10.) X.
EXERCISE XXV.
av ab+ae+bc ^ .„ . a^+x^ 24xj/
x-3
0='-x8 ' 9x'^ -4i/2
2a-'f«f2+afe»-6i
a(a--b^)
(5.)
(6.)0. (7.)^:- (8.)1. (9.) a. (10.)^/;ii^,
EXERCISE XXVI.
Pagre 22. (1.) 1.
(o \ ^±3: . (2a+3) (4a+5) .„
^ •>'6-a' (3a+4")(5aT6)' V^'-' ^ 5 ^•
(4.)
8««
a*-x«
ax
X* -y«
(•^•)1- (C-)~|-- (7.)
X'-'j/-'
2a6
(8-) -x-i^^- (9-)i. (10.) 0.
ANHWKFiH.
99
■24
113
EQUATIONS.
EXERCISE XXVII.
A.
Page 23. (1.) a; = 7. (2.) x-16. (.'J.) a; =15. (4.) y = 2 .
(6.) a! = 3. («.).r = 8. (7.)a;-13. (8.) a: = 28!'
(9.) a; = l«. (10.) .1-10.
B.
(1.) x-3. (2.);r = 5. {•.\.)x:
(4.)r--107. (5.) ..•..7ij. («.) .r = 2f (7.) .r-H.
Page 24. (8.) x- - 3IJ (9.) .r:^7. (10.) u: = 8.
C.
(1.) a; = 20. (1-.) a^= -2. (3.) x = 4, (4.) a; = 8.
(5. ) u' - 4. ((i. ) T = 2. i7.)x = 3. (8. ) u' - 5.
(9.) cc = J. (10.) x = 8.
D.
(1.) ae«2. (2.) x = 2. (3.) ic==7. (4.) a;=4.
(5.) x = S.
Page 25. (fi.) x=7. (7) x^^^^^^^^. (8.) x= -2.
(9.) u;-l. (10.) x-= --H.
E.
(1.) a; =4. (2.) .«•
2A.
(3.) x = ^
cd-ab
ac
/ A \ Itt Ct" - I) -
a+ft-fl-rf
(6.) a; = i ; .
rtft
ah
a-h
(5.) j' = (tfec.
(7.)^=-/>. (8.) X:
m + »
*/i
(9.) x- = 4a. (10.) x-^'
F.
(1.) :c = ll. (2.) a;- -21. (;j.) x-24. (4.) ,-^5.
Page 26. (5.) x- = 7. («.)x-l3. (7.) .«-10. (8.) .<• H.
(9.) x = l (10.) ,'^ -2.
100
EXEIUJIKKS IN ALGEBRA.
(I
f
(3.)
(5.)
(7.)
(9.)
Page 27. (10.)
(13.)
(10.)
(18.)
(20.)
(23.)
Page 28. (24.)
(27.)
(30.)
(33.)
(35.)
(37.)
EXERCISE XXVIII.
PIU)BLEM8.
22 miles. (2.) $180 fcr liorse, $HX) for burrgy.
A §^93.50, B .1?280.r)0, C !5«1]22. (4.) 4^ liours"
!^3200. ((J.) 240 yds. long, 80 yds. wi(fe.
A $142.50, B $47.50. (8.) 15 at $38 and 8 ac $50.
$15.00.
$0. (11.) $800, $3200, $1000. (12.) 50 yds.
$1.60. (14.) $20000. (15.) $750.
90 head. (17.) 18, 22, 10, 40.
40 and 35 bags. (19.) A $2542, B $2422, C $2430.
08. (21.) $11100. (22.) 182 and 10.
A $048, B $472, C $410.
$18000. (25.) $2400, $1000. (2<5.) 11 horses.
41 a lbs. (28.) 09 and 81. (29.) 144 sq. yds
$1050, (31.) 84. (32.) 18x12 ft.
10 and 24. (34.) A $70, B $120, C $190.
10 vols. (30.) $(550, $750, $050, $450.
24000 men.
EXERCISE XXIX.
Pa^e 29. (1.) r, = 7 ; // = 2. (2.) c*' = 7 ; 7/=3.
(3.) ;. = 7; ?/-3. (4.) ^' = 3; ,/ = 5.
(5.) u' = 5; // = 1. (6).i. = 90; (/ = 72.
(7.)j- = U>; yr=24. (8.):»'-12; (/ = 12.
(«•) ■>' = !; II--1. (10.),r==18; 7/= 10.
Note- Question should be r + - =8.
3 5
(11.) :*=00; // = .30. (12.) .r = 12; y^S.
(13.) x^a + b; ,j.-a-h. (14.) x
iii"-n- n--m-
ati - hill
(17.) X-^-h; If = 2.
» - >n am - n
«,-i ; y
(15.) .• =
(1().) .;
a-l
am -tin '
_'''■ . .„ '"'
« + '> ' ^ "^ a^-b
(18.) .r^2; y^\. (ID.) .,...2; ,/-4; .^0.
(20.) ..^ -I -y^i- ::^0, (21.) .,.:{; „^\ .
(22.) .r^l; y.-.2; Z = 3. (23.) .:^3; .,=^5; z
-1.
A NSW Kits.
101
Paere 30. (24.) a-lO ; y = Q ; z = 0. (25.) x-^ ; y= J^ ; z = \.
(20.).r=-i; i/ = i; . = 1.
(27.) a- = 2; |/ = 1 ; .~=--4 ; p-3.
(28.) a- = 1(5; |/ = 7-7r); 2 = 5-5.
(29.) x^r, ; !/=3 ;~=4'
(30.) ^■ = — ; !/ = -2-; '~=^^-
= 8,
(4.)
(7.)
Page 31. (8.)
(10.)
(13.)
(15.)
(17.)
(20.)
Page 32. (23.)
(25.)
(32.)
(35.)
Page 33. (30.)
(37.)
(38.)
(41.)
EXERCISE XXX.
17 yds. ; 13 yds. (2.) | ton ; 1^ tons. (3.) 45.
GOc; l()c. (6.) $1.00; GOc. ((5.) lo ; 25.
A lOOc, BGOc.
$48 = cow, $96 = horse. (9.) 8 and 15.
$180 and $120. (11.) 3, 4, 5. (12.) 8, 12, 18.
A = $1.00, B-$1.12. (14.) 31 and 23.
John = $22, Tom = $2r). (16.) 41 and 7.
72c. and 40c. (18.) 35 and 65. (19.) 24.
3
(21.) 40 and 90. (22.) 3,
40 and 65. (24. ) A - $232, B = $332.
A-$31, B = $27. (26.) jV (27.) f^
fandf. (29.) 26. (30.) 75. (31.) 69.
7 1 and 4^ hours. (33.) $5000 each. (34.) 35.
72, 64, 56, 48.
$540 and $360.
30 and 50, and 70 and 20, or 60 and 20, and 40 and 50.
12 sheep, $40. (39.) 10, 22, 2C>. (40.) 2Js., 2s.
3, 5, 8. (42.) $3()0 and $600.
(43.) 80 and 120. ( : i) 48c. and 40c.
MISCELLANEOUS EXERCISES.
A.
Page 34. (1.) y^-4x + ll. (2.) 0. (3.) j^-6x+8.
(4.) (..-I) (..■-3). (5.) 16.n/. Hi.) ~4ab.
(7. ) 2.r. (8. ) ((/ + b) (a - 2). (9. ) x'' + Sx"" - 153.
{h).) »,^ \ h\ (11.) 43. (12.) .»'' + H;.-!/»7!/-.
(13.) al,{ii+2h) {2(1 +l>).
(14.) x{:ix + 2ij); 5/»(3(r-2/> + 3f). (15.) x-y.
102
EXERCISES IN ALGEBRA.
(16.)
(19.)
(22.)
Page 35. (25.)
(28.)
-47. (17 . ) a^ - b' - --" + i . (18. ) abc(^^ -3b + 2c»).
$70. (20.)m'-n^-3?riH(m-n). (21.) |^-
(8..' + 2:^) (x - 2). (28.) 10 - 8./;. (24.) x - 5.
(x + 4ii + o)(;x + y). (2().) 0. (27.) 12r»-146.
(4.r - 11) {7x - 8). (29. ) x = 7. (30.) 60 and 40.
28a. (32.) -2. (33.) 94.
(31.)
(34.) (ax -ay) (ax + ay). (35.)
(37.)
(41.)
(44.)
(46.)
(48.)
Page 36. (49.)
(53.)
(57.)
(60.)
(62.)
(64.)
(68.)
(71.)
Page 37. (73.)
(75.)
(1.)
(3.)
(4.)
(6.)
(10.)
(14.)
(17.)
(18.)
Page 38. C-O. )
(23.)
(26.)
(38.)
a + h
X- -4
(36.) x-2.
X*
(39.) ;^^ (40.)(x + 2)(.: + 3).
4.
1. (42.) ;r^ + 3.T + 5. (43.) 4.
1150 and .^120. (45.) {x - 1) (;i: - 4) {x + 1) (x + 4).
(47.) ic^ + t/^ + «'- 3x1/2.
3x + /y
Zx - y
(2cc-3i/ + z)'.
9. (50.) a(a'^ + &'0. (51.) 0. (52.) a' = 4.
a« - 1 . (.54.) X = 7. (55. ) d'h - .0. (56.) 74
-2i/. (58.) a- = 7. -"^ A„„i„^-.v^
(59.) App]y^-,56(x + j/)
x-y
(61.) rt = 65.
a;- -2a;+3
(4a; - 15^) (6.r + Qy). (63. ) 2(m2 + (f) {x^ + ]/').
14. (65.) a- = 7. (66.) -1. (67.) a">-l.
x = 7. (69.) 0- = 3. (70.) (a- -4) (a; -5) (.-^ + 11).
A - $160, B = $400. (72. ) (2.r - 11) (a- - 5).
X = 10| . (74. ) (/>i -n + k-I) (m -n-k + l).
(d - h) (rt + m + 1).
B.
(2.)
2a;''
46. ,^., ,
"^ '' X*+X' + l
x^ + (a + l> - i')x'- + {ah - ac - hc)x — abc.
x^ + 10.r^ - 47.'' - 504. (5. ) (^1'^ + c«9 + a^ + a^ + l.
^^;^. (7.) .-2^.
(50
(8.) h:ach$10. (9.)
a+x
u'-7. (11.) -2bc. (12.) u;- 3. (13.)^.
72. (15.) 4. (16.) 0. '^
Examine for complete square a^.
,,-•4- //-' + ,.-' _ 21,,. ^,fi,_ „c_ (^] t)_) 2xy{x'' + if).
1 >»f tiist, 3 of .second.
.'• + 3. (24.) x^-H.
16//»-27r'-36(/^(4(y-3;s)
(21.) 0. (22.)
(27.) xy.
4{a-+b-)
(a -6)- '
ANSW'KRS.
103
(28.)
(29.
(33.
(35.
(37.
(38.
(41.:
Page 39. (43.
(4t;.;
(49.;
(50.
(54.
(58.;
(GO.
(61.
(63.
Page 40. {m.
(67.;
(C8.;
(69.
(70.
(71.
(72.
(73.
Multiply first e(| nation by <i, second by h, third
by c, etc.
I (30.) x = h. (31.) 0. (32.) ^:^^'
(m + q + n- p) (in + r/ - /(. + p). (34 . ) ,*• + - •
(x + y + a) {X + II + h). (36.) 3 + 66.
v^ - bax + la'K
m=-510. (39.) x = 10rt. (40.)
(42.) w.
(44.)
x+l
(45.)
X'-x-XI
Second, ^^^—2^; + 2. \--. ,
. (47.) -30. (48.) l-rW.
x'~xy + y\ (ii) (x'-.r^,/ + y*){x'-jcy + y')
(x'^ + xy + y'^).
39. (51.)
1-//
(52.) x*-4.«;H16. (53.) -8.
^:- (P^-)'^- (5«.)8. (57.)
a- -46^
a
27y''
i,3
2a
(l-4a^)(H-a)
a(a - 1) (a + 1) (a + 3) (a - 6). (62.) ^"
3x+2
C. (64.)
5a;+l
(65.)
(74.)
(75.)
{x 2y + u) (x^ + 4y^ + x'^ + 2xy + 2)/s - xz).
(x + 2i/ + z) (x^ + 4y'' + z' - 2yz - xz - 2xy).
(2a + 36 - (•) W + 96'^ + 0^ + 3/n; + 2uc - Qnh).
(2a - 36 - (0 (4<t2 + {)6' + c- - 36c + 2ac + 6a6).
{x-\-2y- 1) (iT'-' + 4)/" - 2x[i + ;r + 2;/ +1).
(ic - 2|/ - 1) {x^ + 4(/- + X + 2.;,-*/ - 2y + 1).
(:.' + l)(..' + l)(./:~^2)0r-3).'
Hint.— If there is a binomial factor the co-efticient
of X is unity, and the second term nmst be ±
one of the factors of 105. Using the remainder
theorem, the expression vanishes when 1, 3, -^ 5 or
- 7 is put for X, :. factors = (a- — l){x- 3) (;x + 5)
(x + 7).
Write {x-'-7xy + 22{x'-7x) + 120, etc., {x-2)
(x-3)(.,-- 4)(.i--5).
(.*■ -S){x- 5) (x + i){x + 8). (76.)
104 EXERCISES IN ALGfiBRA.
{11 .) it' +h'' - c:'- - (IK
(78.) Write oxpmssioii {ir -hcf ~ {a' -he) (h'-ac) (e^-
oJ>) <.r (rr - hr) ^ {a' ~ be)' ~ {Ir - ac) {c' ~ ab) \- +
(f - '"•) ■' (/'' - ^<cf - (<^^ - ab) {a' - be) \- +(c' - ab)
^ (,'i _ al>y _ (,t^ _ be) (// - ,,r) \- = a(a' - hr) -> (a» +
b^ + c^ - Sabc) + b{b' - ,((•) („•■* + h^ 4- o3 - 3,j6c) + c
(c" ~(ib) )-, etc.
(79.) Write expression a* + lOx^ + 2bx' - 8(x' + 5x) -
33=^(x' + 5x - 11) (x-' + ^x + 'A). (80.)
(81.) (r<-^-//0(a-4/r). . (82.) m.
(83.) (2<r-.,y^(a-.-)2. (84.) ^•=-.110.
(86.) Note.- Question sliouldljoay-^ + G?/"-;, etc.,.-. factors
(x + 5m + ^hi) (x + m - 37().
(80.) a,'« + ^; + i. (H1.)Q(a^b)(a' + ah + b').
(88-) . (89.) a;'^ + «l (90.) li=6h
EXERCISE I.
EXAMINATION PAPERS.
Page 41. (1.) 96. (2.) k (3.) 4*. (4.) 12. (5.) 21.
(0. ) 3.r'' - 2x' - 5.*; - 2. (7. ) (m - n)' -(p- qf.
(8.) a« + a*/>^ + />8.
(9. ) {a - 36) (a + 26) ; (<«' -i- 6'' - 5((6) (a^ + 6^ + 5a6) ;
(5x + 4]/)(3,« + 4|/).
(10.) Apply principle, difference of square.
EXERCISE II.
\x
(l-)-^- (2.) (l + a-6 + c)(l-a + 6 + c).
(3. ) a-''^ + '\f -^-z^- xy -xz- yz. (4. ) ,
^'- ) ^y ' (^0 (^^'^ + ^fO- - («(^ + &C)^
(7.) (}ix + 2>j){2x-3if); (:>c' + y*)(x^-xy + ,/);
(2a + 36 + l) (a + 2/> + 2). (8.) -1. (9.) |+|
(10.) (a-b)(b-c){c-a).
EXERCISE III.
Paere 42. a.) Ih.
(2.) Reduce each fraction to a mixed number, hence
x = 2. (3.) A - $840, B = |G00, C - $840.
ANSWRnS.
105
(4.) Put in form of fraction and eanci;!, —x^-xy + y-.
(b.) 2x'-'3xy + 7if. (Vk) x- + :h-+\.
(7. ) 4(./^ - xy + >f) (:>•' + .«•»/ + I/-') ; {h - c + o) (b-c-n)',
{2a~x)(n+2x).
(8.) x^9. (9.) u;-lll. (10.)
EXERCISE IV.
(1.) X. (2.) a;'^>--l)(x + 2)0r + 3). (3.) c = l, (ii) No.
(4.) . (5.)ay^+l + ^..-
(6.) Factor denominator and cancel, 1.
(7.) Apply ''^;J;;J', etc., 2a* + 10a'b'' + 2h\ (8.) x-20.
(9.) a; = 7. (10.) Factor dividend, (x + y-zf.
EXERCISE V.
Page 43. (1.) Apply ''-l^\ etc. (2.) -2b.
(3.) 74. (4.) 13.
a;-' +?/•■'
(5.) (a + by-c*.
(6.) Apply
(7.) -14.
, etc., {x + ay^-b(x + a) + b^.
(8.) (ic-l)(.c + 2)(.«-3).
(9.) 4(x'H]/Hs^). (10.) 8.
EXERCISE VI.
(1.) a= -4. (2.) x^ + 4.x^^-lQ,x.
(3.) Write expression (a;'^ - ;c)"'' - (2)\ apply * ~'^
x-y
, etc.,
(ic2 - « - 2) (x* - 2^3 + 3x'^ - 2.*; + 4).
(4. ) (9.K^ - 5) (4.x- + 3). (5.) a;3 + 24u'i/(a' + 8j/) + 512i/.
(6.) x^-~^' (7.) 2(a4-6 + c).
(8.) 8 first-class.
EXERCISE VII.
Page 44. (1.) 4. (2.) (a + />)(a-c).
(3.) Apply principle difference of .sojiares, 7x + y + 2.
(4.) 2{a-b). (5.) a = 4. (0.) 1. (7.) x-5.
(8.) Reduce fractious tu mixed numbers, u; = 3.
(9.) 15 and 20. (10.) x = 7.
106
EXERCISES JN ALGEBRA.
EXERCISE VIII.
(1.) x = 5.
(2.) z(y-x) (y + j-~z), (.'»;- tt) (x - M ; (x - 4a - 46)
(x + a + b). ^ a/ ^ f
(3.) 0. (4.) G0(x-l)(.,- + l)(a;-2).
(8.) Multiply and <i + l=0, :. «=-l, b + a + l = •
(^•) Apply ^, etc. ilO.)^{iy-{iY.
EXERCISE IX.
Page 45. (1.) a-^ + u^^ + i + I +i . (2.) a«-/A (3.)
(4.) a- = 17. (5. ) Apply ~±^l^, etc. , x' - 4xhjz + 7yh^.
(a -by
(6 5j-\' and .38jV past 4. (8.) ^ ^.
(9.) ^•«-.o« + 2.r'-'-2. (10.) u.-^ + (a + 6 + c)x'^ + (afc + rtc +
bc)x + ahc, x^ - (w. + u +p)x'^ + (mn + mp + np)x -
mnp.
EXERCISE X.
(1.) -(4a6 + 4ac). (2.) (x + iy.
(3.) x^-x+l + l+^^. (4.) Barley oOc, wheat 65c.
(5.)x-4. (Q.) a-^ + ab + b'. (7.) x-
(8.) ,;= «''^ . (9.) ,„ 1
^ ^ ah-ac + bc ^ -'
af(i* - d)
n
a- c
(10.) x = 2.
EXERCISE XI.
Page 46. (1.) a + x.
(2.) Write the expression —^ x
l + x-'^ +;*;*, etc. ^"-^
1 - X" 1 + x« 1 - «*«
1+.
1
-X'
(3.) x = 95. (4.) a + b + e. (5.) .!•= - 6, |/ = 11,;2= -6.
(0.) (x^-m) (x + n). (7.) ^r + 27. (8.)
(9.)
a+ab+l
(10.)
A NSW K us.
107
I -4b)
= 0,
EXERCISE XII.
(1. ) Factors are (x + 1)'^ (x - 2), .•. other factor x-2.
(2.) . (3.) . (4.) =400 gallons.
(5.) ^^ (C.) 2x^-7x-3.
(7 . ) (^- + 2 + 2a - '(/) (x + 2 - 2a + y). (8.) y = 1.
(9.) !8!7r)2.
(10.) Write expression x* + Mx"^ + 729 + 6x(x'' + 27) -
27.'y^ - {x' + 21 f + ^x{x' + 27) + 9x^ - 3(u-^ = (.^^ +
27 + 3x)'' - (6.>3)^ = {x' + 9x + 27) (x* - 3x + 27).
'i/'2l
f ac +
EXERCISE XIII.
Page 47. (1.) 4(2x^ - 1) (2x^ - 3x - 1). (2.)
(4.) u;(3x + 4)(x-6). (5.) G4 miles.
(7. ) (6x + 1) (3x + 2) (3x + 4) (2x - 1).
(8.) a*-bnH)c. + l^lf-c\
(9.)
4a:-'-15a;+13
(10.)
. (3.) ar = 3.
(6.) «">-!.
65c.
EXERCISE XIV.
(1.) 2x^ - 5x + 1. (2.) 9(a + 2x) {a - 2x) (2a - x)\
(3. ) 20 cattle. (4.) (x - a) (x - l^. (5.) (i/ + y) (x - y).
(6.) h. (7.) Ai)ply^4:«''^*«-;«' + «(2'^-3c) + (26-3c)='.
/a \ 2+3a;
L+5a;
(9.) H-/- + ,2_,^_,.
^2.
(10.) 3ai
5ff'
Page 48.
= -6.
(1.) «-
^•
EXERCISE XV.
(2.) (x + l)(x-l)(x + a + l)(x + a-l).
m~y
(3.) (a + ?>)(<^-' + a/> + n (4.) . (6.) ^^
(0.) The denominator =^{ac + h(iy + {ad-hc>)\ which is
greater than the numerator, etc.
(7.) ■!/* + 2//=' + 3!y'^ + 2/; + l. (8.) {x+p) {^ + m + n).
(9.) 14,17,20. (10.) Apply difference of squares, etc.
108
EXKUC'ISKS IX AL(JRimA.
III
(1-)
coffee.
EXERCISE XVI.
(2.) 10. (3.) 90 lbs. tea, 120 lbs.
(4.) Expression a'^ - h'^ +-..- .- =
1 (a" - bi) (a^b^) , b^-a*
a-b"
u-b*
(!'^ab')("al') = (l-'0 i''^-i)
(5. ) -%zr ' (t>. ) Factor last expression, etc.
(7.) (u; + l)(..--l)r!/+l)(i/-l), (a~b){a-c){h-c).
(8.)u. = l. (9.)"^~K-T- (10-) 0.
EXERCISE XVII.
Page 49. (1.) x = 4. (2.) (d' + b-') {c:' + d') = l, etc. (3.) 0.
(4.) 'j:*-x'yz + 7in''.
(5.) Divide, and remainder equals zero, .'. a = 7, b = Hj.
(().) 12.t;'^ + 12.
(7.) Left side = a' + <t'' + b^ + h' + c" + c' - 2ab - 2uc - 2bc =
(<(' -2<d> + b') + {a' - 2ac + c + (b' - 2bc + e') -
{n-by + ia-cy-{b-c)\ etc.
(8.) 5 and (5. (9.) Factor as a^ + b^ + c^-'3abc, etc.
(10.)
EXERCISE XVIII.
(1.) Add = "»and..' + // + ;:==0, .-. (x+!/ + ;:)2 = 0, etc.
(2.)0. Ql)^- (^.)^- (5.)
(6.) . (7.) . (8.) 0.
(9.) 1328 yards and 432 yards. (10.)
EXERCISE XIX.
Page 50. (1.) (3,f-2)(9..'-ll). (2.) . (3.) 280.
(4.) (a^ - .«^) (// - >/). (5. ) [j-tJ 1-25- + -5- + — ; •
(6.) x^-1. (7.) x = l. (8.) x = 6.
(9.) Let m be added to each a, 6, c, .". (a + my-(b + m)
(c + m) + (b + my - (c + m) (a + m) + (c + my -
{a + w) {b + m). Multiply out and add, etc.
(10. ) Write ^"^J^±f^, etc. , 16x-* - 24.x^ + mx" - 64a; + 81.
ANSWKHS.
109
EXERCISE XX.
(1.) 140 lbs., 60 11)8. (2.) VMix'-2S9. (3.) 1.
(4. ) (3.; - 2,1) (3.>- + 2y) (2x - :hj) (2,' + 3,/).
(5.) 993 yds. nearly. ((}.) - ll.i--' + 17x - 12.
(7. ) « = 7. (8. ) 4. (t). ) (2rf - bb + (k) (3a + 4h - He).
(10.) /> = 4(i.
EXERCISE XXI.
Pagre 51. (1.) -()().
(2.) Square each =« and add, .•. 2(x^ + y^ + z^ + xy +
xz + yz) = a' + h' + ('■' - 0, hence, etc.
(3.) a2 + 3a + 2. (4.) (./' + 2i/) (j;-2./).
(5.) a-13, A = l. (G.) ^-i, x-2, i; + 3.
(7.) a == 1 - -, .-. cH ^ =1^ hence etc.
(8.) 23 J. {^.) {3x'-x + iy.
(10. ) If a > 6, then a - 6 > 0, .-. a'^^ - ftz > 2a6 or
5 + ->2, etc.
>yi.
EXERCISE XXII.
(1.) Equal. (2) 3x(a;-7).
(3.) Apply difference of squares. (4.) (--*)'.
(6.) (7a;-101) (8..+97) ; (3.- + 49) (9x-83). (7.) 10.
Page 52. (8.) Write „-^^ ,^,, etc
(9.) A IGO, B $140, C $200. (10.) 2x + 3.
EXERCISE XXIII.
(1.) Square each, add and factor. (2.'i 4oG0.
(3.) :f = 7c<. (4.)x>=|. (5.) .$1480. (6.)
(7.) Write x^-l-^ 8,.;^ - 79./- + 70 + 1 ^ {x - 1) (a- - 6)
(.*•■ + 14). (8.) The latter.
(9.) Factor expression,aud one factor is equal to zero, etc.
i:
110 RXERCISES IN ALOKDUA,
(10,) 'rraiiHpMst!, jiihI x'^ - 2.»'i/ 4- y' ^- .'/■' - 2}/,~ + z^ + z'^ -
'2ii:: + 11- --^ (x - J/)-' + ((/ - zy + (r; + a)'. Sinco the
s(|naro <»f fmy(|U!mtity is positive, .". each (expres-
sion ia jjositivi!, Jind cniinoi ho zero iinleHH each
(juaMtity is zero, .". j;-i/ = 0, y~z = 0, z~u = Oy
:, x = y=z=n.
EXERCISE XXIV.
(1.) '<'-8a'' + 2.3rt 20. (2.) . (3.) . (4.)
Page 53. (6.) 21) miles. ((J.)-- (7.) . (8.)
(9.) x' -((i + hyx + dh^x^ + x + l. Since co-etticients of
like i)ower3 are equal, .". a + b= — 1 and a6 = l, .'.
(«» + ?>» -2.
(10.) a + b= -c, multiply by (t — }>, etc.
EXERCISE XXV.
(1.)
13a
(7.) 5^-^ -2^-1. (8.) x = 10.
(5.)a = 8. (6.)-:-.
(9.) Let 05 — 3, X - 1, x + 1, x + 3, be the numbers, etc.
(10.)
EXERCISE XXVI.
Page 64. (1.) x = 10a. (2.) Let iC = one, x + d the other, etc.
(3-) -%tf-- +1 = ^^ + 1' ^'tc. (4.) 216.
(6.) a^ + 2a'b-ab''-2h\ (6.) m^- 12m + 35.
(7. ) (x' - 3x + 17) (x' + 3,c + 17).
(8.) a;'^-3x + 2 is a factor and =0, .'. expression =0.
(9.) Write (l - 1+1-1+1-1-1)^2-
(^ + l+^)'^t«-'=l- (10-)
EXERCISE XXVII.
4a;-+2a;-l
(1.) Factor the expression. (2.) ^
(3.) . (4.) . (5.) -ia + f/>-^c.
(6.) . (7.) ];;..
Page 55. (8.) G. (9.) 405 yards. (10.) x = a + b + c.
ANHWKKH.
Hi
EXERCISE XXVIII.
(l.)a (2.)20x-32;/. (3.) (.•- J )(.*•- ^ )
(4.) i\. (6.) l + 3x + (M;'' + 7j'' + 0.'* + 3a;Hx«.
(6.) l+.r + ]!.fH.^.>'" +?,'•♦. (7.) Oft.
(S.) l-ia' + 'ib + lh\ (9.) . (10.)7a-56.
EXERCISE XXIX.
(1.) x = a + b + c. (2.) 2(a* + <W + h*). (3.) 291.
(4.) 1. (5.) 1 - ix - kx'+hp-' - i^*- (<■'•) i^^OO.
Pa^e56. (7.) ^^ - 3^ (J - J) •
(8.) Let X, x + 1, x + 2, x + S be the numbers, etc.
(9.) Apply diuorence of (squares. (10.) 0.
(3.) l-m-n.
EXERCISE XXX.
(1.) 2Ga;'-48j-i/ + 2();A (2.)
(4.) . (5.) 4(.«- + a)(i/ + s).
(6.) (4x'' + 6bx){r>x + V>a). (7.) x.
(8.) 3-4x + 7^-'-10a:». (9.) -2.
(10.) (7a; + 6i/-9)(x-f/ + 4).
EXERCISE XXXI.
(1. ) - 20. (2. ) (1 + .*• + x'Y -(1-X + xy - 6x{x* + x^ + l)
is a cube, etc.
(3.)
(4.)
b" - m-
2(>« - a)
(5.) 20 years.
Page 57. (6. ) (^' + 2|/ + 3~~)'^ + (4 ./ + 5.-)'^ + (6zy.
(7. ) {x - a) {x' - b-'). (8. ) {x' + 6x + 1 1) (x'^ + 6x + 3).
(9. ) (x^ - 6x + 4) (x' + ().>• + 4).
(10.) Apply difference of squares.
EXERCISE XXXII.
(1.) 200 lbs. (2.) Apply difference of squares.
(3.) 3y^-7.«' + ll.
(4.) Divide the expressiim by product of (2a; - 3)
(3x + l), and the co-efficients of like powers in
the remainder must be e(][ual ; m = C, 71= -37.
112
EXEKCI8KS I\ ALOKMIiA.
.Hi
i
(5.) 2 in. (6.) (8,t + 15/)) {Ha - 9h).
(7 .) ViicUiV ]iih hnud Hide, etc. (8.) (.r + z).
(9. ) a* -pa' + qa' - m + .s ^ 0. (10.) ^ ^'^ •
EXERCISE XXXIII.
(1.) ;) = 26, q^ -24. (2.) Apply ^]-^^ otc.
(•U . (4.) . (5.)
Page 58. (<J.) . (7.) p-\'^.
(8.) Write (./•)'•■' -(2)'". Henco (..•" i- 2") (r^ 2*) (.-«« + 2»)
(.• + 2)(,r-2).
({>. ) Write 9./"'' + .•^Cu;^ + 1 2..'^ + 48.r + 4;i- + 1(5 - 9/^(..- + 4) +
12.r(ie + 4) + 4(..- + 4) = (.'J^i; + 2f (.*: + 4).
(10.) (2x + 3i/ + ;j) (x + 4i/ + 3^).
1
EXERCISE XXXIV.
0)
(2.) (rt , 4- a. 4- a .,, - 114)3: + (^i + o ... - a., + a^ )y + (a j - a^
(3.) a-''-3tt62 + 2c3 0. (4.) ^*-2. (5.)
(6.) A cube. (7.) ''=31. (H.) -I. (9.) .r = 4.
(10.) Multiply Hrst =n by a, second by 6, third by c.
EXERCISE XXXV.
(1.) -1024. (2.) . (.3.)
(4.)(2j/-x + a)(i/-2x-r0. 0->.) .2^-^48 •
(6.) 2(? + ^)p + i')-
Page 59. (7.) Let .r2+wcc+i/s = aq. root of expression. Square
ami e'jiate co « flicieuts, 2m— p 2mp/6' = r, .*.
y/^' - /-.
20:2/
(8-) (:-:+:-:+! +r- (»•)
(10.) Multiply each term by x^-y'\ etc.,
EXERCISE XXXVI.
(1.) ,.-10.
(2.) Factor, simplify ; .square remainder =a'' + 6' + c' -
2ab + 2ac -26c.
ANHWKHH.
113
(3.) Factor ftiul othorJactor ri«iuire(l ==a;-2.
H.) KxpresHion = ^^—^--^ = ----^^ •
(5.) Factors arc (r - I)' ('• + !), •. wc muHt multiply hy
(r -l)(r + l)-. (0.) 2,^uml7«}.
(7.)5-r («•)
(9.) Write cxprossiou j-^ (.i: + :k)' + 25k'' + tM.« + .'k),
etc. , = 2'J* - 260. ( 10. ) 2(a - b).
EXERCISE XXXVII.
(1.) Sinco <(-/» = !, ..(a- />)» = !, .. (a - W^t + '')'=«' +
2ab + h' = "'"''- + ah = a« - /»» + a/* .
(2. ) Multiply by a - 1 , etc. (3. ) w = 2.
(4.) a; + i/ = 2, multiply Wy .*•- ij, etc.
Page 60. (5.)
((•>.) Since :.■" + !/' iw divisible by ..• + !/,.•. C'' *)" + (//*)' i«
divisible by ..• * + >J K lle8\dt, x* - x hj ^ 4- »/ *.
(7.) a;^ + a;'^ + 2.i+l -2u3'^—2ic>0,.'. expression is positive,
(8.) The former.
(9. ) x(x + y + z) ( j;' + if + z' - xy - x.z - yz).
(10.) 35 miies.
EXERCISE XXXVIII.
(1.) (ir- = b'^; add 2h<' + e'^ to oiich and divide by <■, etc.
(2.) If a - /) is ])ositive, .-. a > />, also b ^- c, :. a > e, hence
<•- (f is negative. (3.) . (4.)
(5.) r or ''/• (6.) - xi/^(a- + v + 4
(7.) Subtract 2nd from 1st and factor, etc.
{8.) x' = x-2,:.x* = (.'• - 2y. 1 1 ence x^ = x(x - 2)'^ = -
3ic* + 2x= -3(x-2) + 2x=-x + ^.
(9.) . (10.) A $648, B $472, C $416.
EXERCISE XXXIX.
(1.) 2 or i. (2.) See hint, question 10, Exercise xxiii.
Page 61. (3.)
(4. ) {<t + by + {b + rf + 2ac - 2ac = {c + af + 2h{a + b + c) -
2ac = , etc. (5.) Multiply out and factor, etc.
114
EXERCISES IN AUiKHKA.
0>-) . (7.) ;'•'/ iH greater.
(H. ) }(,« + 1> <„2 + n or ,r - /i + 1 > <>i or (n - ] )2^0, etc.
(10.) aw/;:; = ^--^:i:^-''-)-^''> (>. + ,, i. -A ^ (^>-^)(^ -^)(«-^)
.'. , etc.
EXERCISE XL.
(1.) Apply ^^^^ etc. (2.) •{ 2{a + hf-'^{n-hf y.
(3.) (3:r, - 4?/ + 2;;;) (4..- - 5,/ + 7"). (4.) 50 lbs., .'30 Djs.
(5-) . ((i.) (/ = r)2.
(l + a;Hx*, etc.) (! + »;''' + .«♦, etc.) = (l + .*y^ +
u'* + , etc.)'^.
(8.) . (9.) (»- + 2?/-2)l (10.) (a- .0 (0-6).
[jf
MISCELLANEOUS EXERCISES.
A.
Page 62. (1. ) (4a - 6) (4a - 9).
(2.) Factor l)y difference of squares, (.7^ + 1) (.v-3).
(3. ) x=- a + h, y = (,-b, expression = -{ (.<■ + y}' + 3xy \.
■{(x- yy - 3x11 \- = (4a^ + 3a' - 'Sh') (46^ - 3a'^ +
3/>0, etc. (4.) ,V i^P' - f) {5'l' - r).
(5.) Apply — _'-, .". expression is divisible by (x^ +
a;2 + 4) - (.,.••' - 2x + 3) - .f 2 + 2x + 1, etc.
(C.) Each side of =//acube. (7.) 'So +h + 2(' + d.
(8. ) (x - m - „) (x - w + „). (9. ) (a - h) (h - c) (r - a).
(10.) Divide numerator of each by denominator, etc.
(11.) Dividend is divisible by (.rH^-2). Apply prin-
ciple^:f;.-H2.«H^+J, + 6.
(12.) C;- + l)(.^-l)(.r-2)(,.--4).
(13.) Add the equations, etc., (a + 64-c)^
(14.) A[»j)ly difFereufe of sciuares, l(\(a^2l>) (c + d). ■
( 1 .5. ) (x + (i) (x' - 4x,i + if'). (10. ) 420( )().
(17.) '^x^-4x\ (18.)
ANSWERS.
115
Page 63. (19. ) 1. (20. ) {x' - if) (x* + xhf + /),
(21.)^+/J- (22.)
(24.)
(20.)
(23.) (x -.-«)(;. -2) (,.-3).
0. (25.)
Factors of 21 are ±1, ±3, ±7. Honce (r + l)
Page 64.
Page 65.
(28.)
(30.)
(32.)
(33.)
(34.)
(35.)
(36.)
(38.)
(40.)
(43.)
(46.)
(4?)
(51.)
(53.)
(55.)
(58.)
(61.)
(62.)
(64.)
(66.)
(68.)
(70.)
(71.)
(73.)
(74.)
(75.)
(76.)
(78.)
(x - 3) {x - 7). (27.) Cube x- i = 1, etc.
Tea 62^ cts., sugar 6 J cts (29.) 512.
,f-2y^+3f-4f. (31.) /!|^,-
Reduce fractions to uuxecl numbers and eciuate
remainders, etc., x = iO.
(x - 2) (x + 2) (x-^ + 4) (x + 1) (:«2 - x + 1).
Apply difference of squares, 144,r'^(l - 4.*;''').
{2x - 3) (2x - 1).
Fact(;r by difference of squares.
mih\ (39.)
(x + l){x-3)(x + 5). (41.) 25.
884. (44.) (x-19//)(* + 17i/).
2a-3h+c
(37.)
(42.).-f.
(45.) 45 cents.
(47.)(.'; + 9;/ + l)(..--4;/). (48.)
8x 1
2a -3c
Gx' + 8xy + 7y\ (50.) x^
. (52. ) (3x - 4 */ - 3;;) {3x - 4 j/ + 3^).
(llx + 13|/) (dx - 111/). (54. ) X - 13, y = 7.
x" + y\ (56.) . (57.) 125.
600. (59.) Add =- us, etc. (60.) x = b, y = 2.
Ax^ + 4|/ + ^i''^ - 4.f 1/ - 2xz -2yz.
{a + 2b '- 3c) (a -b + 2c). (63. ) 1.30.
(2x11 + « + '>y- (<5S-) 4(.'- - !/) (7x2 _ 2ot/ + 7//"0.
(.,. _: 1) (,,. + 8) (.«2 + 7.,. + 26). (67. ) X = 1.
a
(69.)
c(c - b)
c(c- a)
'' b{!l>-a)
a{a - by
a3 + «-7> + nl,-' + J>\ <r' + b^ + 3a^b + 3(1^^ + 2a-c + 2/A^ +
4<d)c + 4nc' + 4:hc' + 8c\
a^ - 125// + 8r' + 30a/>c. (72.) {xJ^z - 1) {ifz - 1).
Apply "~ ^" ; I'csult 3(6 - n - 6(,' + 2(i).
;»-3, // = 2.
Reduce to mixed numbers, etc., ,'—8.
(..■ + ;/ -1) (,.■-;/ 2).
. (71>.) (80.)
(77.) Apply ^:;f, etc.
no
EXERCISES IN ALGEBRA.
n
I
(81 . ) 24, (S2. ) (3x + 2»/ - 4^) (2x - 3i/ + 5z).
Page 66. (83.) The second. (84.) x-2. (85.) 47. (86.) 16.
(87. ) x' + 'Sxy + 2,/. (88.) (</, + h){h + c) (c + a).
(89.) 7 and 2.
(90.) (f-x-l)(x'-7). (91.) x=-l, y^l, z = l.
(92.) ^J. (93.) a; = 10, ,/-20. (94.) a;2«+'^''+^
(95. ) Factors are (h + c,~a)(b + c + «), but b + c + if. = 0, .-.
expression =0.
(96.) 14249. (97.) x=:a,y = 2h,z = 3c. (98.) 70 miles.
(99. ) (x - yY (x* + if + 2xhj + 2xif f Gx'Y)'
(100. ) {2x - 3i/ + 1) (3..; - 2y + 1). '
MISCELLANEOUS EXERCISES.
B.
Page 67. (1. ) 2abc(a +b + c). (2.) x - 3. (3.) x=5.
(4.) 4«2 _ oa - 7. (5.) 200 acres, 250 acres.
(G.) (x + y + l)(x + y + 2).
(7. ) (cr^ + a' + X - 3) (.«2 + a2 _ a; + 3). (g.) 85.
(9.) (l-.T)(;«2_3a; + 4). (10.) (a + b + iy{a^ + b^ + l).
(11.) x«- 3.^* + 2^2-1. (12.) n.C.F.-3a:-7, .-. x = 2l
(13.) /-^ . (14.) .^''-y^ - (15.) 48.
(16. ) x=-(\. (17.) (2r( - 1) (.3a + 2) (« - 3).
(18.) 12 and 8. (19.) 6x'^ + 2..--5.
Page 68. (20. ) (x-2)(x + 2) (x^ + 5). (21.) $9.37^
(22. ) ^^^, . (23.) X = 2k (24. ) cc =-- 3, 1/ = - 2, . = 5.
(25. ) 3x' - 2x" + 3a; + 2. (20.) ^ •
(27. ) (x - 1) (x - 1) (x - 1) (.K + 1) (x2 - 4x + 1).
(28.) ;^^ (29.)a(«-t-5)(a''=-l).
(30. ) ((( + b) (a -h (■) (b + c). (31.) .30, 40, 14.
(32. ) x' + x- 2if + 3y. (33. ) x - 6 , ;/ = 10, 2 = 9.
(34.) Reduce to mixed numbers and etjuate remaining
fractions, x = 7. (35.) (5x' + <))(4,»; - 7). (36.) 17.
(37. ) .'■ + 6 (/ + 14^. (38. ) x' + 12.(; - 7. (39. ) ^- •
(40.) {x + a + by. (41.) 180 yds., 205 yds.
ANSWERS.
117
Page 69. (42.)
8(1/ + 2)
8.V-]
(43.) x = 2, .v = 3, 2 = 4.
(44.) The ymxluct of any four consecutive numbers in-
creased by unity is a perfect sc^uare.
(^^■) (.---^kx^^' (4fi.)- = 5. (47.).: = 4,!/ = 15.
(48.) 15 ajul 5. (49.) 7. (50.) . (51.)
(52.) . (53.) . (54.) (;>' + !, -;:)(x-3y + ::).
(55.) (4.i'+5!/)l (5«.) ahrd(x-l){.,- + l){x + 'A).
(57. ) ,.: = a -f h + r. (58.) 120 sheep.
(5'.». ) (2rt + 7h - 3<') (2rt - 7h + 3c). (00.) 0. (61.) />.
Page 70. (02.) ,.=1, i/ = l. (03.)
(04.) -; {a+2)x + a-l \- ■{ (a -}),, + <, + 1 )-.
(65.) .*•=3r^ (66.) A = 48. (07.) .'■- - 3,/; + 2.
(68. ) {p + /•) (q + .s). (69. ) x" + 2x11 + 'W'-
(70.) . (71.) (a.-&)x. (72.)
(73.) :>^ = 100, ;/ = 9. (74.) 1.
(75. ) (1 - 5,.'. + Ox--) (1 - 3./; - 4.^0 (1 + 3-'- " 4.-^;')-
(76.) ,>;+ 1. (77.) —^fr' ("^•) ^"^^^^ ^"'^ 2304.
(79.) . (80.) (5.v + 12;/)(8,>:-7»/).
(81.) H. C. F. =(a-Sb + or).
(82.) Apply difference of squares, 39x + |/ + 22!.
(83.) -102.
Page 71. (84.) (9,*; + 71) (5.; -48). (85.) x' + ii.
(86.) (3,»'4-2)(64,.;«-729). (87.)
(88. ) 2.i_'2 + 9if - 5.:'-' + 1 2//~ - 9xz - 9.ni.
(89. ) (4.r -ij-7) (2..' + 5;/ + 3). (90. ) a + hx + cx\
(91.) 24. (92.) 226. (93.) -1.
(94.) (4,.' + 4j/ + 2)(14,^--5i/).
(95.) a -/^ (96.) . (97.) u; = 6, i/-=l.
(98.) {<t+hy-{a + h){c + d)^-{c + d)\
(99.) .1-3, «/= -1. (100.) 100 acres.
Page 7
MISCELLANEOUS EXERCISES.
C.
(l.) ,.• + ,/ + •:. (2.) (4,.' + 5(/)(3..'-7!/). (3.) .r = 3.
(4.) 2.< <•. (5.) (4y- + 4;r + 5::7'. (0.) 3005.
(7 . ) .f'" + .'•'■'// + .'"ir + .'• ' !/'' + .'•"!/* + x^i/ + .!•*/ + xhf + x'Y
+ u;j/*+;/", remainder 2]/".
118
EXERCISES IN ALflKBRA.
(" I'
(8.
(11.
(14.
(16.
(18.
Page 73. (20.
(22.
(24.
(26.
(27.
(28.
(31.
(32.
(33.
(36.
(38.
(40.
» (42.
Page 74. (44.
(46.
(48.
(49.
(50.
(53.
(55.
(56.
(57.
(60.
(62.
Page 75. (64.
(66.
(68.
(71.
yi-\-h+c^ (<).) :,-3 + 2.-«2 + 33; + 4. (10.) ,r' + 2x + 3.
a- = l. (12.) -166. (13.) 3,f2-5:« + 7.
1 - X + x' - ..•••' + ,,■* - x^ + .*•" - ,.•' + x\ (15. ) 2x + 21.
(ti - 1) (a + 1) (a - 2) (a - 4). (17.) x"" - -/'.
-63. (10.) 3(1 + ;*:) (1 + ;/^ (x - (l) (I - xy).
X-+X+1 ' v'^'-'' a:2-l\ x"-i /■
x = b,y = A. (23.) (..;2-l)«(x + l).
. (25.) 3(.*: + l)(2x + 3).
3a;'' + 15:r2_-i^^315
ix+3)(^5j(^+7j'
(x-;i){2x + 'A)(x' + 3x + 3).
x'-x + l, (2».) $1000. (30.) (2,1 + 7) (x + 3).
(.•-])2(^H2,.,- + 3).
(x + y - 1 ) (u;- + }f + X + y-\-l).
x^-x-' + x}-:t'* + j,^-x + l. (34.) 3. (35.) 300.
. (37. ) 1 - 3.C + 3,1 '- - .«3 + X* - 3xs + 3x« - x^
(.i- + 8//)(./- x). (.39.) 35.
(t^ + a'> + ,i\ + ar + ,s. (41.) (x' + .x + 6) (x'+x - 2).
X +1^ • (43- ) (<''^ - ^<th + h') (a' + ^ab + b').
-16. (45.) 3(.: + //) + 5(-+»0.
Apply ~S- • (47. ) X* - 4x3 + o,*9 - 4a: + 1.
;«:
Reduce to mixed numbers, etc., x = 2.
- 37a'^ (51.) ftS - 2^" + a* + 4^3 - 1. (52.) *^^' •
Apply difference of squares.
(x - 1) (,,.-' + .T + ]) (2x - 3) (2x4-3) (4x'' + 9).
Add =>(.s, etc., 0. (58.) . (59.)
X - 3, */ = - 1, ;^ = 0. (61.) $2400.
156 and 13. (63.) ]/(//^-3).
4 times. (65.) ^±-^ •
(2x -f- y) (x - 2//) (2 - ;r) (4 + 2x + x'^. (67.) x = 51.
.-:4. (69.) 1. (70.) (x + ;>)(x'^-^x + ^r).
(// + 2x) (// 2x) (2x + 1 ) (2x - 1) (4x'^ - 2x + 1) (4x^ +
2x + l). (72.) 80x^(x'^-9). (73.) -6.
ANSWERS.
119
(74.) A JB'ia/B $.".(), 0$;;5. (75.) 1+^- + :*•"'• (7<"».)
(77.) "1^;/ • (78) 4.i;H1(m:+11. (7!>.) ^'--l-
(80.) rr-2a (81.) {a + h)(c + ,l). (82.) . (8.'}.) a; = 3.
Page 76. (84.) 45. (85.) . (8«i.) j ■
(87.) n- 10. (88.) (^-5. (89.) {S)x - 47!/) (12x - Ol./).
(00.) 4(1 -x) (1 + 2,1;) (a; + 4) (3.r + 4). (01.) 3x + 5.
(92.) . (93.) a + (!. (94.) 27. (95.) 4(<(•^-//0'^
(96.) 1. (97.) 36u;«-217:«* + 40().t;''-225. (98.)l-x.
(99. ) (12a 4- 1 2/> + <■) (a - 12?> - 12^0. (100.) x = 2|, (/ = 3^.
MISCELLANEOUS EXERCISES.
D.
Page 77. (1.) • (2.) a = c and 4?) = c2 + 8. (3.) a; = G.
(4.) ?/' = 27cl (5.)
(().) a + /> or 64-c or c + « is eciiml to zero. (7.) ;'' = 6.
(8.) Factor in ordinary way. The product of G and 35
= 210, and the difference of the factors of 210
will be co-efiicient of 2nd term, or equal a ; for
example 1 x 210 is one pair, and .'. 2nd term
would be 209 and expression 6.x2 + 209x-35.
The other co-efficients of x would be 103, 67,
37, 29, 23, 11, 1.
(9. ) c = - 1G8, d = 190. (10. ) /) = 20, r^ = 25.
(11.) .»••- = «' + 'lad + (^2, f = , etc.
(12.) S(i. root =x3 + 2a;'H5a;-6, etc. (13.)
(14.) Write expressioji (.»•- - x\})^ - (.' + 2)1 Apply prin-
cii)le of dift'erence of two cubes, x> - 2j'Sj-\-x^f-\-
x^ - xhj + 3./;2 - 2..' J/ + 4u; + 4.
(15.) !/-,^ -2. (1(1.)
Page 78. (17.) »*(-■- 8w + 11.
(18. ) (:j( ( + 26) (3,t - 26) (,.■ ^ 3a) {x" + ^ax + 9a2).
abc
x-y + xy
V
(21.) a- =
(24.)
a+b+c
abc
(19). . (20.) x^
(22.) Hi\f- C-^'M
(28. ) 24. (2t>. ) -; (« + 2)..' + a-l\- ■{ {a + 2)x + a-2)-.
120
EXERCISES IN ALGEBRA.
(.'iO.
(.'i2.
:j;3.
('
(.'{4.
Page 79. (:w.
(40.
(42.
(45.
(40.
(47.
(48.
(50.
(53.
(5().
Page 80. (57.
(5l>.
(60.
(02.
(04.
(00.
(08.
(09.
(70.
(71.
(72.
Page 81. (74.
(75.
(77.
K^'M. CM.) pu' + .r + a) (a.r - 2,r + ,r-f2).
( .'•'•■' + 2,»';/ - //-') (.>■- + .r,i + ,r) (x- ^ j-tj + ,f).
Divide l)y Ilonior's inetlKxl nnd the reiruiiiulur will
bu tho v.-iliio, divisor ^2x^-3;c + 4, answer -=10.
. (35.) . (;}(}.) 4. (.37.) .*:-H.
a -10. (39.) 4;*;- 12+^. .
Reduce to mixed miml)urs aud ecjuate remainders
ill lowest terms, .*; = 2i. (41.) O.f + 3.
" =^ ' ! (7TT? • (^'^•> -^-^^ - '-♦■ (•^^•) "^ = <^ "^^^ if-
Reduce to mixed numl)ers, etc., .♦: = 10.
.>•'-.>■ + 1.
See i)age 52, Ex. xxiii, (juestion 10, etc.
.,.« + LV' + 3.r-* + 2..'''' + J. (49.) .,•= -4.
(.»• + '( ) (.*■ - b) (x - 1 ). (5 1 . ) 35. (52. ) 0,0.
4a-lr. (54.) 1. (55.) {.<'"- >j") {.,■-'" + .»■'" if + i/*').
Divide by (x - 2) (x - 5) and remainder is zero, .-.
a = 74, />-120.
r, = 0, 6--36. (58.)
Reduce lst = ?i, xz + xij = 2y::, divide by ;*•//,:, etc.
The former. ((il.) x= -].
8x'~4x-l. (03.) 10.
{a + h + rf = a^ + h'^+r'^ + :\(<,h + („■ + ,■<() (,t j-h + c)-
'6abc, :. {(t + b + if^a^ + h^ + r^-;i,,hcHUn;v((b +
bc + ca^i). (05.)
P]xpres.sion = 17.«(.'' - //)*, .■. to make a complete cube
we must nuiltiply by 289,i-(.«' //)l (07.)
Multiply by xyz, re-arrange terms and divide by
abc, etc.
Write expression ^J:i^-1 + '•^+«=r'^+ i +
ihc
a" + b" -(■-
2ah ^""^^^ simplify, etc.
Reduce to form it^b"^ + aV^ + bV^--=(d)r*, + etc., and
divide by (r'6'V', etc.
~,^— =a-^ y^ =b~e; . . , etc.
See page 52, Ex. xxiii, question 10. (73.) 3.
Factor expression, ;. factor recpiired is .*; -3.
J- -7. (70.)
ii(' + b'){r- + iP) = ,etc. (78.)
ANSWERS.
121
. 38
(71K ) x^y = z(x - ,^)■^ + 2 j/^sc -z)-\-'^ \i\ :. z{x - zf = xhj - 2yz
{x-z)~zi/in-{.r - zy=-ijz.
(80.) Multiply l»y .»• uiul mU unity to (\'i<;h .side ; .'.
( I .*•)", etc.
(81 . ) /> V - <t',j'' = <(hh- - (iVnj, etc.
(82.) {a + b)x + (ih=-, etc.
{&3.)x=-{ii + z), .: -{,j + z) {a'-bc) + {h'-m)y + (c'-
ah)z=^), :. {l>'-ca-((' + bc)ij = {a'-b<' -c' + ah)z,
etc.
(84.) Subtrnct 2nd from 1st, divide by y-z, etc.
(85.) l + '^^l + l^^ and l-. = l-^.-.j:«=^,etc.
(8(5.) Write (\^) (;:::),etc.
(87. ) x^ - xij + / = 0, .•. (x - \if = - xij, :. expression =
x^y'\x - ;/) - Xji{x - y)x[i — 0, .". x'^ — x\j + \f is a
factor. ' (88.) ' .
(89.) Add the etiualities, etc., but left hand will be
(■'' - .'/) {{I " ■') (' ~ ■'■) which ='6abc, etc.
(90. ) - ,: - ^2ox ) \\\n^\\ divided out gives l + x + 'Ix' + , etc.
(91.) .r- 5a -14. (92.) _^. (98.)
Page 82. (94.)
(95.) Let m be quantity subtracted, and instead of a, 6, r,
write a - m, b - m,c - m ; .'. ex])ression = {a - m)'^ -
(b - m) (c - m) +{b- m)'^ - {a - in) {c - m) + {c-
m)'^ - (a - m) {b - tn). JSiniplify, etc.
• (90.) . (97.) x = l.
(98.) Expression ^a{b'^ + bc + c^) + ,etc., ^(a + b + c){bc +
ca + a ■>), .-. = 0. (9!>. ) p = 25, q= - 24.
(100.) -f. (101.) x^5.
(102.) A=2, B = 3, 0=1, D=l.
(103.) Divide each by x + a, and remainders =0. Sub-
tract, .-. a{l - p) = in - q, etc. (104.) 1 - »<•' - //(*.
(105.) If reduced, x + 1 <>i' •'' + 2 must be a factor, :.x= - 1
or - 2, and hence jl» = 3 or ^.
(10(>. ) {.I If + x:: + ijz - .c^ - (/'^ - z^) is the other factor.