EXERCISE 1.3 ADDITION OG ALGEBRAIC EXPRESSIONS

A.

1. a. 63a b. -100y2 c. -216pq d. 133s3 t e. -51(a+b) + 75a -125y2 - 109pq + 97s3 t + 60(a+b)

135a -225y2 - 325pq 230s3 t 9 (a+b)

3. a. 30x-47 b. 13x+y-10 c. 12x-13y+15 d. 13x+32y + 2y-36 y+10 11x -y -15 39x+25y 30x-2y-3 24x-12y -13x+24y 22x+18y 37x-10y 10x+10y 74x+75y

5. a. 15.6x2+3.5 b. -90-60b+70c c. -72a -59b -36c -33.4x2+6.5 90+50b+20c 51a+89b+44c 46.2x2+5.4 82a+60b-90c -27a -28b -61c

28.4x2+15.4 82a+50b -48a +2b -53c

B.

1. 12x+23y-44z ; 30x-26y-33z ; 25x-40y+26

= 67x-43y-51z

3. 22a2b-30ab+30ab2 ; -33ab2-46a2b-39ab ; 34a2b-18ab2

= 10a2b-21ab2-9ab

5. 35a+32b+16abxy-18 ; -71ax-42b y+21abxy+19 ; -50ax-52b y-13abxy+24

= -86ax-62b y+24abxy+25

EXERCISE 1.4 SUBTRACTION OF ALGEBRAIC EXPRESSIONS

A.

1. a. -33n y b. -100ab2 c. 318mn3 d. -96cbn e. 289(m+n)nx

+ 64n y -216ab2 156mn3 222cbn 148(m+n)nx

-97n y -116ab2 162mn3 318cbn 141(m+n)nx

3. a. 14.21a2-8.11 b. -211abc-212 c. 162(x+ y )5-102

-36.09a2+9.05 196abc-118 216(x+ y )5+200

50.3a2-17.16 -407abc-94 -54(x+ y )5-302

B.

1. 62a+31b ; -32a-44b

= -94a-75b

3. 13x2+41y2-50z2 ; -55x2-16z2

= 68x2+41y2-36z2

5. 37m2-13n+40p ; 8m2-40n+30p

= 29m2+27n+10p

EXERCISE 1.5 MULTIPLICATION OF ALGEBRAIC EXPRESSIONS

A.

1. (15x3)(20x8) = 300x11

3. (-33a3)(7a12b6) = -231a15b6

5. [-20(n+1)2 y][-8(n+1)3 y] = n6n+285 y

B.

1. 13ay(6a2 x-9by3) = 28a3 xy-177aby4

3.(cd )ax[10(cd )7ax-15(cd )3ax+20 (cd )] = 10(ca)8ax-15(cd )9ax-20

5. 0.7p4q[-2.1p+3.3q4-1.5p3q5] = -147p5 r231q5-10.5p7q6

C.

1. (3x2-3x-1)(4x+7) = 12x3+9x-25x-8

3. (1-2x+3x2)(3+4x-5x2) = 6x3-19x3+8x2+15x-12

5. (2x3-3x+2)(5x2-x+1) = 10x5+19x8+13x2-5x+2

7. (25a2+13)(25a2-13) = 625a2-169

9. [2x-3(a+b)][5x-4(a+b)] = 10x+23ax-23bx+a x2+24ab+12b2

EXERCISE 1.6 DIVISION OF ALGEBRAIC EXPRESSIONS

A.

1. (168m7n10) / (42m5n9)

= 7m2n

3. [125(a+b)25] / [108a(x+ y )13n]

= -5(a+b)5

5. [117(a+b)3(n+1 )+ 34(a+b)5(n+1 )-119(a+b)7(n+1 )] / [-17(a+b)2(n+1)]

= -6.8(a+b)a+1−2(a+b3n3)

B.

1. (8x412x3+2x2-5x+3) / (2x-3) = bx3+2x-2

3. (27m3-8) / (3m-2) = 9m2-4

5. (36rn6-42m3n2-30n4) / (3m3-5n2) = 12m3+6n2+10r-50mn2+30

7. (16x3+5x2+4) / (x+1) = 16x3+21x2+9x+4

9. (-135-3x+5x2+x4) / (x+3) = 3x4+x3+1.67x2-136x-3

11. (3a3-2a2-2a+1) / (3a2+a-1) = a-1

13. (12a5-6a4+6a3+2) / (2a2-3a+1) = 6a3-10a2+12.83a-5.5

15. (8a4-12a3+2a2-5a+3) / (2a-3) = a+2a+2

EXERCISE 1.7 ORDER OF OPERATIONS

A.

1. 16 x 2 / 4 = 8

3. 8 + 6 / 2 x 3 = - 4

5. 4 x (82 + 4 / 2 - 1) = 132

7. (20 / 5) + (4 x 2 - 3) = 9

9. 4 x 5 + 6 / 2 + 7 = 20

11. 15 + 3 x 2 + 10 / 2 = 33

13. 52 x 42 - 3 x 2 + 5 = 799

15. 4 + 33 – 4 x 2 + 22 = 588

EXERCISE 1.8 EVALUATION OF VALUES OF ALGEBRAIC EXPRESSIONS

A.

1. a + b - c = 2

3. 2a + b - c = 5

B.

5. 6b + cd = 36

C.

7. 2a + b + 2c = 116

9. (a+b) (a+c) = 203

or 7 13

D.

11. (4m + b + c) / (2a+1) = 4.56.5

E.

13. 2(a+5)-4, when a = 6 = 18

15. 2x2-3x+1, if x = -2 = 15

17. 1 - 3x3+x, if x = -3 = 79

19. 5(1+rt)-r, if r = 5 & t = 6 = 150

EXERCISE 1.9 GROUPING SYMBOLS

1. 4[s+3(125 - s)+10-8s] = -40s+150

3. 37w2+(13w2-18w+9)+w(-12w+19) = 38w2+w+9

5. 2-[2+3(3)-5]-4 = -44

7. x(3x-15)+4x(x-12)-(12x+27x+10) = 7x2-10x+10

9. 13a+[15a-(20a-5b)]+25 = 8a+5b+25

11. 15x2-13(x2-6)+2(x2-18) = 4x2+42

13. a-{a-[a-(-a)-a]-a}-a = a

15. 4a{a2-5a[2-5(6a+12)+30]-20a2} = 524a3+560a2

CHAPTER 2: SPECIAL PRODUCTS & BINOMIAL EXPANSION

EXERCISE 2.1

1. (xy )2 = x2 y2

3. (−12 x3)2 = 144 x6

5. (4e3 f 6gh2)2 = 16e6 f 12 g2h4

7. (0.1n2 p3)2 = 0.01n4 p6

9. ( 13b2 yn)

2

= 19b4 y2n

11. ( 23m¿¿ x y3 z2)

2

¿ = 49m2x y6 z4

13. (2 xa+2 yb)2 = 4 x2a+4 y2b

15. (4 t2u3 x)2 = 16 t 4u6x

17. [2a(b+c)3]2 = 4 a2(b+c)6

19. (−2 x2m−1 ya+5b)2 = 4 x4m+1 y2a+10b

EXERCISE 2.2

1. (a−x)2 = a2+2 (a ) (−x )+x2

= a2−2ax+ x2

3. (2 x2−3 y z2)2 = 4 x4 2 (2 x2 ) (−3 y z2 )−9 y z4

= 4 x4+12x2 yz2−9 y z4

5. (2m−5n3)2 = 4m2+2 (2m ) (−5n3 )−25n6

= 4m2−20mn3−25n6

7. (1−5q2 r3 z4)2 = 1+2(1)(5q2r3 z4)-25q4 r6 z8

= 15q2r3 z4−25q4 r6 z8

9. (0.01c2+0.2de)2 = 0.0001c4+2(0.01c2)(0.2de)+0.04d2 e2

= 0.0001c4+0.04c2de+0.04d2 e2

11. (2 s2−23t)

2

= 4s4+2(2s2)(- 23

t) - 49

t

= 4s4-223s2t-

49

t

13. (2 x2n+1+3 y3b)2 = 4x4a+2+2(2x2a+1)(3y3b)+9y6b

= 4x4a+2+12x2a+1 y3b+9y6b

15. [2a−(2x−3)]2 = 4a2-4ax-12a+4x2+a

EXERCISE 2.3

1. (a+b−1)2 = a2+b2+1+2ab-2a-2b

3. (4 r2+s−2 t)2 = 16r 4+s2+4t 2+8r2s-16r2t-4st

5. (2m−3n+5)2 = 4m2-9n2+25-12mn+20m-30n

7. (3 p2−4q−t 2)2 = 9p4+16q2+t 4-24p2q-6p2 t2−8qt 2

9. (1−e3+ f 4)2 = 1+e6+f 8-2e3+2f 4-2e3 f 4

11. (0.1+0.2 x−0.3 yz )2 = 0.01+0.04x2+0.09y2 z2+0.04x-0.06yz-0.12xyz

13. ( 12x+1

3y−1

4)

2

= 14x2+ 1

9y2+ 1

16+ 1

3xy− 1

4x−1

6y

17. (2m+3n−4 p+2)2 = 4m2+9n2+16p2+4+12mn-16mp+8m-24np+12n-16p

19. (3a+b−3c−3d+e)2 = 9a2+b2+9c2+9d2+e2+6ab-18ac-18ad+6ae-6bc-6bd+2be+18cd-6ce-6de

EXERCISE 2.4

1. (x+15)(x-15) = x2−225

3. (2a+5b)(2a-5b) = 4 a2−25b2

5. (r2s+10t)(r2 s-10t) = r 4 s2-100t 2

7. (13+2 x2 y3

)(13−2x2 y3

) ¿19−4 x4 y6

9. (0.03m2+np2)(0.03m2−np2) = 0.0009m4-n2 p4

11. (x+y-z)(x+y-z) = x2+ y2+z2-2xy-2xz-2yz

13. (a+2b+3c)(a-2b-3c) = a2−4 b2−12bc−9c2

15. [2(a+b)2+3(c+d)][2(a+b)2−3(c+d)] = 4(a+b)2-9(c+d )2

EXERCISE 2.5

1. (a+b)(3a+b) = 3a2+4ab+b2

3. (5m-n)(3m+7n) = 15m2+32mn-7n2

5. (r2 s)(2r2 5 s) = 2r 4+3r2 s-5s2

7. (0.1x+3)(0.2x-5) = 0.02x2+0.1x-15

9. (12x+4)(

13

x-8) = 16x2

-2 23x-32

11. (34p2

-3b)(43p2

+5b) = p4- 14p2

b-15b2

13. (2r+s+t)(2r+s-5t) = 4r2+4rs-8rt+s2-4s2

15. [2(a+b)-3(x+y)][3(a+b)+2(x+y)] = 6a2+12ab-5ax-5ay+6b2-5bx-5by-6x2-12xy-6y2

EXERCISE 2.6

1. (x+3 y)3 = x3+9x2 y+81xy3+9y3

3. (2 x2−7 y)3 = 8x6+84x4y-42x2 y2-343y3

5. (5a2b−3 y2)3 = 125a6b3+45a4b2 y2-45a4b2 y4-27y6

7. (1−2a2b3)3 = 1+6a2b3-14a6b9

11. (3x+y)(9x2-3xy+y2) = 27x3+y3

13. (2a2+5b3)(4a4-10a2b3+25b6) = 8a6+125b8

15. [2x+(a+b)2]3 = 8x+a2b2

CHAPTER 2 TEST

A.

1. (−5 x2 y3)2 = 25x4 y6

3. (−34a2 y3)

2

= 9

16a4 y6

5. (0.3m3+0.5n)2 = 0.09m4+0.3m2n+0.25n2

7. (2a−3b−2c )2 = 4a2+9b2+4c2-12ab-8ac+12bc

9. (3x+4y)(3x-4y) = 9x2-16y2

11. (14

x+43

2)(14x−4

322

) = ( 14)

2

- ( 43 z2 )

2

+ 1

16 x2 - 179z4

13. [2a+(7b+c)][2a-(7b+c)] = 4a2-49b2+14bc+c2

15. (8x+yz2)(7x-9yz2) = 52x−2-2xy2+7x2x-9

17. (2xa+3yb)(5x4-7yb) = 10xa2-14xy6b+15x3y

19. [(a+b)-3][(a+b)+5] = a2b2+2

CHAPTER 3: FACTORING

EXERCISE 3. 1

1. 6ab+8bc = 2b(3a+4c)

3. 15x3+-24x2+6x = 3x(5x2+8x+2)

5. 18x3 y3+27x2 y2+-4xy = xy(18x2 y2+27xy-4)

7. 4x2+-8x3+10x4y+-12x5 y2 = 2x2(2-4x+5x2y-6x3y2)

9. 24x2(a+b)+-32x3(a+b) = 8x2(a+b)(3-4x)

11. 5a+10b+15c+-20d = 5(a+2b-3c-4d)

15. 14xn−1-21xn−1 = 42n+1

17. 0.5x4-1.5x3+2x2+4.5x = 3x2

19. 36a2b(3x+y)-18ab2(3x+y)+-81ab(3x+y) = 9ab(3x+y)(4a-2b-9)

EXERCISE 3.2

1. 4a2+-25b2 = (2a+5b)(2a+-5b)

3. 49m4n2+100p6 = (100p3+7m2n)(-10p3+7m2)

5. 81x4+-36y2 z 4 = 9(3x2+2yz2)(3x2+-2yz2)

7. 4/9 v2-m2n4 = (2v/3-mn2)(mn2+2v/3)

9. 25y6+-12y4 = y4 (-12+25y2)

11. 1/16x41/25y2 z 4 = (yz2+ x2)(x2-yz2)

13. 9(a+b)2 = (3a+3b-c2x) (3a+3b+c2x)

15. a2b2-121c6d4m = (ab-11c3d2m) (ab+11c3d2m)

17. 98x2n-18y4 m = 2(7xn-3y2m) (7xn+3y2m)

19. 4x2(a+b)2-y2 z 4m = (2ax+2bx-yz2m) (2ax+2bx+yz2m)

EXERCISE 3.3

1. 27x3+y3 = (3x+y)(9x2-3xy+y2)

3. 343m3-125n3 = (7m-5n)(48m2+35mn+25n2)

5. 3x4y+-24xy4 = 3xy(x-2y)(x2+2xy+4y2)

7. 54x6+-16y3 = 2(3w2-2y)(9x4+6x2y+4y2)

9. 0.125v3-m6n = (0.5v-m2n)(0.25v2+0.5m2nv+m4n)

15. 8/125a3b3−c6 = (2ab/5-c2)(4a2b2/25+2abc2/5+c4)

17. a3(m+n)-512 = a3m+a3n-512

19. 8a3+27(x+ y )3 = (2a+3x+3y)(4a2-6ax-6ay+9x2+18xy+9y2)

EXERCISE 3.4

1. 4a2+16a+16 = (2a+4)(2a+4)

3. m2+10mn+25n2 = (m+5n)(m+5n)

5. a4+10a2+25 = (a2+5)(a2+5)

7. 2a2+8ab+8b2 = 2(a+2b)2

9. 4e2+48ef 2+144f 4 = (2e+12f 2)(2e+12f 2)

11. 1

16x2

- 92x+81 = (

14x−9)(

14x−9)

13. 9a2n+48anbn+64b2n = (3an+8bn) (3an+8bn)

17. 1.44y6+2.4y3 z2+z4 = (1.2y3+z3) (1.2y3+z3)

19. 1+30m4n2+225m8n2 = (1+15m4n) (1+15m4n)

EXERCISE 3.5

1. x2-8x+7 = (x-1)(x-7)

3. x2-4x-21 = (x+3)(x-7)

5. x2+4x-45 = (x+9)(x-5)

7. x2a+xa-42 = (xa+7)(xa−6)

9. 2 x2-24x+54 = (2x-6)(x-9)

11. 2x2-11x+5 = (2x+1)(x+5)

13. 7x2-12x-4 = (7x+2)(x-2)

15. 12x2+20x+7 = (6x+7)(2x+1)

17. 13x2-14xyb+1 = (13xyb-1)(xyb-1)

19. 3x2+10xyb-25y2b = (3x-5y2)(x+5yb)

EXERCISE 3.6

1. a5 + b5 = (x-7)(x-1)

3. x7-128 = (x-2)(x6+2x5+4x4+8x3+16x2+32x+64)

5. 128x7-1 = (2x-1)(64x6+32x5+16x4+8x3+4x2+2x+1)

7. x5 y5-1 = (xy-1)(x4 y4+x3 y3+x2 y2+xy+1)

9. 1

32+x5 = (

12

+x)(1

16+

18

x+14x2

+12x3

+x4)

11. x9-y9 = (x-y)(x8+x7 y+x6 y2+ x5 y3+x4 y4+x3 y5+x2 y6+ xy7+ xy8)

13. 32(a+b)5-1 = (2a+2b-1)(16a4+64a3b+8a3+96a2b2+24a2b+4a2+64ab3+24ab2+8ab+2a+16b4+8b3+4b2+2b+1)

15. (a+b)5-c5 = a5b5−c5

EXERCISE 3.7

1. c2+3c-cd-3d = (c-d)(c+3)

3. 6a2+9ab-8a-12b = (3a-4)(2a+3b)

5. x2y-yz+2ax2-2az = (y+2a)(x2-z)

7. 2x2-x+4x-2 = (x+2)(2x-1)

9. ax+bx+ay+by = (x+y)(a+b)

11. 4a2-9b4+2ax-3b2x = 5ab2+ax2

13. 10x2+4xy+45x+18y = (2x+9)(5x+2y)

15. a2+3a-2a-6 = (a+3)(a-2)

CHAPTER 4

EXERCISE 4.1

1. 2x−x

= -2

3. −1y

= −1y

5. −r

−s−t+u =

rs+t−u

7. x− yy−z

= xz

9. −3

(5)2(−2)2 = −329

EXERCISE 4.2

1. 24 x2 x4

9x2 y2 = 8

3 y2

3. 3( y+2)3( y+4)

= y+1y+2

5. 4 x2−12x+1

= 2x-1

7. a2+3a−10a2+a−6

= a+5a+3

9. m3−273−m

= 9+3m+m2

11. a2−6 a+8a2+a−6

= a−4a+3

13. 125a3−x3

x−5a = 25a2−ax+x2

15. 14a2−31a−107a2−33a−10

= (2a+5)(a+5)

EXERCISE 4.3

1. 18 y2

25 z∙

5 z2

3 y2 = 6 z2

5

3. 3 y+18y2 ∙

4 y4

6 y+36= 4 y2

2

5. w+3 ∙w

2w2+5w−3 =

w2w−1

7. 3a2−13a+49a2−6a+1

∙28+7 aa2−16

= 7 (4+a)

(3a−1 )(a−4)

9. 4 a+126a−18

∙5a−153a+9

= 109

EXERCISE 4.4

1. x2,y6,z9

= 1xyz

3.a

52,r

169,s4

= a1r1

s4 (52)(169)

5.xx+ y

,yx− y

,xy

x2− y2 = x1y1

xy

( x− y)(x+ y )(x2− y2)

7.2

x2+4 x+4,

3x+2

,3

3m−3n = 21

31

4

(x2+4 x+4)(x+2)(x2+2 x)

9.1

x2+4 x+4,

2

x2+ x−20 =

11

2

(x2+ x−20)(x2−x−12)

11.1

(c−d)2,ba−b

,c

a2−2ab+b2

= 11

21

3

3 [ (c−d )3 ][ (c−d )2](c−d)

13.1

x2+xy,

2

xy2+ y3 = 11

1

(xy2+ y3)(x2+xy )

15.a

(a−b)2,ba−b

,c

a2−2ab+b2 = a1b1

c

[ (a−b )2](a−b)(a2−2ab+b2)

CHAPTER 5: EXPONENTS

EXERCISE 5.1

1. x5 ∙ x3 = x8

3. (2 x2 z5)4 = 4x8 z20

5. 18x3 y5 z3

6 x y2 z0 = 3x2 y3 z3

7. (a2b3)2

a3b2 = ab4

9. (−2 x2a−1 y1−2a)3 = 2 x2 y3

11. (m−1)3(1)3

m2−m = m2−2m+1m+1

13. [(−2)3]2∙ [(−2)2]2 = 65.536

15. 44 ∙36∙(x5)2

28 93(x2)3 = x4

EXERCISE 5.2

1. x2 y−3

x3 y−2 = 1xy

3. a3 ∙3−2 = 3a

5. (m−2n2

mn−2 ) = n4

m3

7. x−2+xx−2 = x3

9. (50 x−2)2(23 x3)0 = 1

x4

11. a−mma

a2mm−2a = am

a2m2

13. 5−1b−2 c0

(2bc )−2 ∙( 1a−2b

)−1

= 4bc2

5a2

15. ( a−2m+2bn

a−3m+3b−2n )−2

¿ = am−1b3n

EXERCISE 5.3

1. 2512 = 5

3. 3265 = 64

5. 3265 = 64

7. 813 = 2

9. 6413 = 4

11. [8(a+ y)]23 = 8

13. (x2 y−2)12 = xy2

15.(4m2)(9n4)

= 2m

9n2