Rs Aggarwal 2019 Solutions for Class 9 Math Chapter 3 Factorisation Of Polynomials are provided here with simple step-by-step explanations. These solutions for Factorisation Of Polynomials are extremely popular among Class 9 students for Math Factorisation Of Polynomials Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rs Aggarwal 2019 Book of Class 9 Math Chapter 3 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rs Aggarwal 2019 Solutions. All Rs Aggarwal 2019 Solutions for class Class 9 Math are prepared by experts and are 100% accurate.

Page No 99:

Question 1:

Factorize:
9x2 + 12xy

Answer:

We have:
9x2+12xy=3x3x+4y

Page No 99:

Question 2:

Factorize:
18x2y − 24xyz

Answer:

We have:
18x2y-24xyz=6xy3y-4z

Page No 99:

Question 3:

Factorize:
27a3b3 − 45a4b2

Answer:

We have:
27a3b3-45a4b2=9a3b23b-5a

Page No 99:

Question 4:

Factorize:
2a(x + y) − 3b(x + y)

Answer:

We have:
2ax+y-3bx+y=x+y2a-3b

Page No 99:

Question 5:

Factorize:
2x(p2 + q2) + 4y(p2 + q2)

Answer:

We have:
2xp2+q2+4yp2+q2=2xp2+q2+2yp2+q2=2p2+q2x+2y
                            

Page No 99:

Question 6:

Factorize:
x(a − 5) + y(5 − a)

Answer:

We have:
xa-5+y5-a=xa-5-ya-5
                    =a-5x-y

Page No 99:

Question 7:

Factorize:
4(a + b) − 6(a + b)2

Answer:

We have:
4a+b-6a+b2=2a+b2-3a+b
                     =2a+b2-3a-3b

Page No 99:

Question 8:

Factorize:
8(3a − 2b)2 − 10(3a − 2b)

Answer:

We have:
83a-2b2-103a-2b=23a-2b43a-2b-5
                           =23a-2b12a-8b-5

Page No 99:

Question 9:

Factorize:
x(x + y)3 − 3x2y(x + y)

Answer:

We have:
xx+y3-3x2yx+y=xx+yx+y2-3xy
                        =xx+yx2+y2+2xy-3xy=xx+yx2+y2-xy

Page No 99:

Question 10:

Factorize:
x3 + 2x2 + 5x + 10

Answer:

We have:
x3+2x2+5x+10=x3+2x2+5x+10
                   =x2x+2+5x+2=x+2x2+5

Page No 99:

Question 11:

Factorize:
x2 + xy − 2xz − 2yz

Answer:

We have:
x2+xy-2xz-2yz=x2+xy-2xz+2yz                           =xx+y-2zx+y                           =x+yx-2z

Page No 99:

Question 12:

Factorize:
a3ba2b + 5ab − 5b

Answer:

We have:
a3b-a2b+5ab-5b=ba3-a2+5a-5                              =ba3-a2+5a-5

                      =ba2a-1+5a-1=ba-1a2+5

Page No 99:

Question 13:

Factorize:
8 − 4a − 2a3 + a4

Answer:

We have:
8-4a-2a3+a4= 8-4a-2a3-a4                         = 42-a- a32-a                         = 2-a 4 - a3
                     

Page No 99:

Question 14:

Factorize:
x3 − 2x2y + 3xy2 − 6y3

Answer:

We have:
x3-2x2y+3xy2-6y3=x3-2x2y+3xy2-6y3
                       =x2x-2y+3y2x-2y=x-2yx2+3y2

Page No 99:

Question 15:

Factorize:
px − 5q + pq − 5x

Answer:

We have:
px-5q+pq-5x=px-5x+pq-5q
                   =xp-5+qp-5=p-5x+q

Page No 99:

Question 16:

Factorize:
x2 + yxyx

Answer:

We have:
x2+y-xy-x=x2-xy-x-y
               =xx-y-1x-y=x-yx-1

Page No 99:

Question 17:

Factorize:
(3a − 1)2 − 6a + 2

Answer:

We have:
3a-12-6a+2=3a-12-23a-1
                   =3a-13a-1-2=3a-13a-1-2=3a-13a-3=33a-1a-1

Page No 99:

Question 18:

Factorize:
(2x − 3)2 − 8x + 12

Answer:

We have:
2x-32-8x+12=2x-32-42x-3
                    =2x-32x-3-4=2x-32x-3-4=2x-32x-7

Page No 99:

Question 19:

Factorize:
a3 + a − 3a2 − 3

Answer:

We have:
a3+a-3a2-3=a3-3a2+a-3
                =a2a-3+1a-3=a-3a2+1

Page No 99:

Question 20:

Factorize:
3ax − 6ay − 8by + 4bx

Answer:

We have:
3ax-6ay-8by+4bx=3ax-6ay+4bx-8by
                       =3ax-2y+4bx-2y=x-2y3a+4b

Page No 99:

Question 21:

Factorize:
abx2 + a2x + b2x + ab

Answer:

We have:
abx2+a2x+b2x+ab=abx2+b2x+a2x+ab
                       =bxax+b+aax+b=ax+bbx+a

Page No 99:

Question 22:

Factorize:
x3x2 + ax + xa − 1

Answer:

We have:
x3-x2+ax+x-a-1=x3-x2+ax-a+x-1
                        =x2x-1+ax-1+1x-1=x-1x2+a+1



Page No 100:

Question 23:

Factorize:
2x + 4y − 8xy − 1

Answer:

We have:
2x+4y-8xy-1=2x-8xy-1-4y
                  =2x1-4y-11-4y=1-4y2x-1

Page No 100:

Question 24:

Factorize:
ab(x2 + y2) − xy(a2 + b2)

Answer:

We have:
abx2+y2-xya2+b2=abx2+aby2-a2xy-b2xy
                          =abx2-a2xy-b2xy-aby2=axbx-ay-bybx-ay=bx-ayax-by

Page No 100:

Question 25:

Factorize:
a2 + ab(b + 1) + b3

Answer:

We have:
a2+abb+1+b3=a2+ab2+ab+b3
                    =a2+ab2+ab+b3=aa+b2+ba+b2=a+b2a+b

Page No 100:

Question 26:

Factorize:
a3 + ab(1 − 2a) − 2b2

Answer:

We have:
a3+ab1-2a-2b2=a3+ab-2a2b-2b2
                      =a3-2a2b+ab-2b2=a2a-2b+ba-2b=a-2ba2+b

Page No 100:

Question 27:

Factorize:
2a2 + bc − 2abac2

Answer:

We have:
2a2+bc-2ab-ac=2a2-2ab-ac-bc
                    =2aa-b-ca-b=a-b2a-c

Page No 100:

Question 28:

Factorize:
(ax + by)2 + (bxay)2

Answer:

We have:
ax+by2+bx-ay2=ax2+2×ax×by+by2+bx2-2×bx×ay+ay2
                        =a2x2+2abxy+b2y2+b2x2-2abxy+a2y2=a2x2+b2y2+b2x2+a2y2=a2x2+b2x2+a2y2+b2y2=x2a2+b2+y2a2+b2=a2+b2x2+y2

Page No 100:

Question 29:

Factorize:
a(a + bc) − bc

Answer:

We have:
aa+b-c-bc=a2+ab-ac-bc
                  =a2-ac+ab-bc=aa-c+ba-c=a-ca+b

Page No 100:

Question 30:

Factorize:
a(a − 2bc) + 2bc

Answer:

We have:
aa-2b-c+2bc=a2-2ab-ac+2bc
                    =a2-2ab-ac-2bc=aa-2b-ca-2b=a-2ba-c

Page No 100:

Question 31:

Factorize:
a2x2 + (ax2 + 1)x + a

Answer:

We have:
a2x2+ax2+1x+a=ax2+1x+a2x2+a
                      =xax2+1+aax2+1=ax2+1x+a

Page No 100:

Question 32:

Factorize:
ab(x2 + 1) + x(a2 + b2)

Answer:

We have:
abx2+1+xa2+b2=abx2+ab+a2x+b2x
                        =abx2+a2x+b2x+ab=axbx+a+bbx+a=bx+aax+b

Page No 100:

Question 33:

Factorize:
x2 − (a + b)x + ab

Answer:

We have:
x2-a+bx+ab=x2-ax-bx+ab
                   =x2-ax-bx-ab=xx-a-bx-a=x-ax-b

Page No 100:

Question 34:

Factorize:
x2+1x2-2-3x+3x

Answer:

We have: x2+1x2-2-3x+3x= x2-2+1x2-3x+3x=x2-2×x×1x+1x2-3x-1x=x-1x2-3x-1x=x-1xx-1x-3



Page No 105:

Question 1:

Factorise:
9x2 – 16y2

Answer:

9x2-16y2=3x2-4y2=3x+4y3x-3y                  a2-b2=a+ba-b

Page No 105:

Question 2:

Factorise:
254x2-19y2

Answer:

254x2-19y2=52x2-13y2=52x+13y52x-13y                  a2-b2=a+ba-b

Page No 105:

Question 3:

Factorise:
81 – 16x2

Answer:

81-16x2=92-4x2=9+4x9-4x                  a2-b2=a+ba-b

Page No 105:

Question 4:

Factorise:
5 – 20x2

Answer:

5-20x2=51-4x2=512-2x2=51+2x1-2x                  a2-b2=a+ba-b

Page No 105:

Question 5:

Factorise:
2x4 – 32

Answer:

2x4-32=2x4-16=2x22-42=2x2+4x2-4                  a2-b2=a+ba-b

=2x2+4x2-22=2x2+4x+2x-2                  a2-b2=a+ba-b

Page No 105:

Question 6:

Factorize:
3a3b − 243ab3

Answer:

3a3b-243ab3=3aba2-81b2                         =3aba2-9b2                         =3aba-9ba+9b                         

Page No 105:

Question 7:

Factorize:
3x3 − 48x

Answer:

3x3-48x=3xx2-16                =3xx2-42                =3xx-4 x+4

Page No 105:

Question 8:

Factorize:
27a2 − 48b2

Answer:

27a2-48b2=39a2-16b2                    =33a2-4b2                    =33a-4b3a+4b

Page No 105:

Question 9:

Factorize:
x − 64x3

Answer:

x-64x3=x1-64x2              =x1-8x2              =x1-8x 1+8x

Page No 105:

Question 10:

Factorize:
8ab2 − 18a3

Answer:

8ab2-18a3=2a4b2-9a2                    =2a2b2-3a2                    =2a2b-3a2b+3a

Page No 105:

Question 11:

Factorize:
150 − 6x2

Answer:

150-6x2=625-x2                 =652-x2                 =65-x5+x

Page No 105:

Question 12:

Factorise:
2 – 50x2

Answer:

2-50x2=21-25x2=212-5x2=21+5x1-5x                  a2-b2=a+ba-b

Page No 105:

Question 13:

Factorise:
20x2 – 45

Answer:

20x2-45=54x2-9=52x2-32=52x+32x-3                  a2-b2=a+ba-b

Page No 105:

Question 14:

Factorise:
(3a + 5b)2 – 4c2

Answer:

3a+5b2-4c2=3a+5b2-2c2=3a+5b+2c3a+5b-2c                  a2-b2=a+ba-b

Page No 105:

Question 15:

Factorise:
a2 b2 a – b

Answer:

a2-b2-a-b=a+ba-b-1a+b                  a2-b2=a+ba-b=a+ba-b-1=a+ba-b-1

Page No 105:

Question 16:

Factorise:
4a2 – 9b2 – 2a – 3b

Answer:

4a2-9b2-2a-3b=2a2-3b2-12a+3b=2a+3b2a-3b-12a+3b                  a2-b2=a+ba-b=2a+3b2a-3b-1=2a+3b2a-3b-1

Page No 105:

Question 17:

Factorise:
a2 b2 + 2bcc2

Answer:

a2-b2+2bc-c2=a2-b2-2bc+c2=a2-b-c2                                     a2-2ab+b2=a-b2=a+b-ca-b-c                  a2-b2=a+ba-b=a+b-ca-b+c

Page No 105:

Question 18:

Factorise:
4a2 – 4b2 + 4a + 1

Answer:

4a2-4b2+4a+1=4a2+4a+1-4b2=2a2+2×2a×1+12-4b2=2a+12-2b2                                  a2+2ab+b2=a+b2

=2a+1+2b2a+1-2b             a2-b2=a+ba-b=2a+1+2b2a+1-2b=2a+2b+12a-2b+1

Page No 105:

Question 19:

Factorize:
a2 + 2ab + b2 − 9c2

Answer:

a2+2ab+b2-9c2=a+b2-3c2                               =a+b-3ca+b+3c

Page No 105:

Question 20:

Factorize:
108a2 − 3(bc)2

Answer:

108a2-3b-c2=336a2-b-c2                            =36a2-b-c2                            =36a-b+c6a+b-c

Page No 105:

Question 21:

Factorize:
(a + b)3ab

Answer:

a+b3-a-b=a+b3-a+b                        =a+ba+b2-1                        =a+ba+b2-12                        =a+ba+b-1a+b+1

Page No 105:

Question 22:

Factorise:
x2 + y2z2 – 2xy

Answer:

x2+y2-z2-2xy=x2+y2-2xy-z2=x-y2-z2                                  a2-2ab+b2=a-b2=x-y+zx-y-z                     a2-b2=a+ba-b

Page No 105:

Question 23:

Factorise:
x2 + 2xy + y2 a2 + 2abb2

Answer:

x2+2xy+y2-a2+2ab-b2=x2+2xy+y2-a2-2ab+b2=x+y2-a-b2             a2+2ab+b2=a+b2 and a2-2ab+b2=a-b2=x+y+a-bx+y-a-b          a2-b2=a+ba-b=x+y+a-bx+y-a+b

Page No 105:

Question 24:

Factorise:
25x2 – 10x + 1 – 36y2

Answer:

25x2-10x+1-36y2=5x2-2×5x×1+12-6y2=5x-12-6y2                                        a2-2ab+b2=a-b2=5x-1+6y5x-1-6y                   a2-b2=a+ba-b=5x+6y-15x-6y-1

Page No 105:

Question 25:

Factorize:
aba2 + b2

Answer:

a-b-a2+b2=a-b-a2-b2                       =a-b-a-ba+b                       =a-b1-a+b                       =a-b1-a-b

Page No 105:

Question 26:

Factorize:
a2b2 − 4ac + 4c2

Answer:

a2-b2-4ac+4c2=a2-4ac+4c2-b2                              =a2-2×2a×c +2c2-b2                              =a-2c2-b2                              =a-2c+ba-2c-b

Page No 105:

Question 27:

Factorize:
9 − a2 + 2abb2

Answer:

9-a2+2ab-b2=9-a2-2ab+b2                            =32-a-b2                            =3-a-b3+a-b                            =3-a+b3+a-b

Page No 105:

Question 28:

Factorize:
x3 − 5x2x + 5

Answer:

x3-5x2-x+5=x2x-5-1x-5                         =x-5x2-1                         =x-5x2-12                         =x-5x-1x+1

Page No 105:

Question 29:

Factorise:
1 + 2ab – (a2 + b2)

Answer:

1+2ab-a2+b2=1+2ab-a2-b2=1-a2+2ab-b2=12-a2-2ab+b2

=12-a-b2                                        a2-2ab+b2=a-b2=1+a-b1-a-b                    a2-b2=a+ba-b=1+a-b1-a+b

Page No 105:

Question 30:

Factorise:
9a2 + 6a + 1 – 36b2

Answer:

9a2+6a+1-36b2=3a2+2×3a×1+12-6b2=3a+12-6b2                              a2+2ab+b2=a+b2=3a+1-6b3a+1+6b              a2-b2=a-ba+b=3a-6b+13a+6b+1

Page No 105:

Question 31:

Factorize:
x2y2 + 6y − 9

Answer:

 x2-y2+6y-9=x2-y2-6y+9                          =x2-y2-2×y×3 +32                          =x2-y-32                                                    =x+y-3x-y-3                          =x+y-3x-y+3

Page No 105:

Question 32:

Factorize:
4x2 − 9y2 − 2x − 3y

Answer:

4x2-9y2-2x-3y=4x2-9y2-2x+3y                               =2x2-3y2-2x+3y                               =2x-3y2x+3y-12x+3y                               =2x+3y2x-3y-1                                

Page No 105:

Question 33:

Factorize:
9a2 + 3a − 8b − 64b2

Answer:

9a2+3a-8b-64b2=9a2-64b2+3a-8b                                   =3a2-8b2+3a-8b                                   =3a-8b3a+8b+13a-8b                                   =3a-8b3a+8b+1

Page No 105:

Question 34:

Factorise:
x2+1x2-3

Answer:

x2+1x2-3=x2+1x2-2-1=x2+1x2-2×x×1x-1=x-1x2-12                                   a2-2ab+b2=a-b2=x-1x+1x-1x-1                   a2-b2=a-ba+b

Page No 105:

Question 35:

Factorise:
x2-2+1x2y2

Answer:

x2-2+1x2-y2=x2-2×x×1x+1x2-y2=x-1x2-y2                                   a2-2ab+b2=a-b2=x-1x+yx-1x-y                   a2-b2=a-ba+b

Disclaimer: The expression of the question should be x2-2+1x2-y2. The same has been done before solving the question.

Page No 105:

Question 36:

Factorise:
x4+4x4

Answer:

x4+4x4=x4+4x4+4-4=x22+2x22+2×x2×2x2-22=x2+2x22-22                                      a2+2ab+b2=a+b2=x2+2x2+2x2+2x2-2                   a2-b2=a+ba-b

Page No 105:

Question 37:

Factorise:
x8 – 1

Answer:

x8-1=x42-12=x4+1x4-1                               a2-b2=a+ba-b=x4+1x22-12

=x4+1x2+1x2-1                   a2-b2=a+ba-b=x4+1x2+1x2-12=x4+1x2+1x+1x-1            a2-b2=a+ba-b

Page No 105:

Question 38:

Factorise:
16x4 – 1

Answer:

16x4-1=4x22-12=4x2+14x2-1                               a2-b2=a+ba-b=4x2+12x2-12=4x2+12x+12x-1                     a2-b2=a+ba-b

Page No 105:

Question 39:

81x4y4

Answer:

81x4-y4=9x22-y22=9x2+y29x2-y2                               a2-b2=a+ba-b=9x2+y23x2-y2=9x2+y23x+y3x-y                      a2-b2=a+ba-b

Page No 105:

Question 40:

x4 – 625

Answer:

x4-625=x22-252=x2+25x2-25                               a2-b2=a+ba-b=x2+25x2-52=x2+25x+5x-5                         a2-b2=a+ba-b



Page No 114:

Question 1:

Factorize:
x2 + 11x + 30

Answer:

We have:
x2+11x+30
We have to split 11 into two numbers such that their sum of is 11 and their product is 30.
Clearly, 5+6=11 and 5×6=30.

 x2+11x+30 = x2+5x+6x+30                             = x(x+5)+6(x+5)                             =(x+5)(x+6)

Page No 114:

Question 2:

Factorize:
x2 + 18x + 32

Answer:

We have:
x2+18x+32
We have to split 18 into two numbers such that their sum is 18 and their product is 32.
Clearly, 16+2=18 and 16×2=32.

x2+18x+32=x2+16x+2x+32                           =x(x+16)+2(x+16)                           =(x+16)(x+2)

Page No 114:

Question 3:

Factorise:
x
2 + 20x – 69

Answer:


x2+20x-69=x2+23x-3x-69=xx+23-3x+23=x+23x-3

Page No 114:

Question 4:

x2 + 19x – 150

Answer:


x2+19x-150=x2+25x-6x-150=xx+25-6x+25=x+25x-6

Page No 114:

Question 5:

Factorise:
x
2 + 7x – 98

Answer:


x2+7x-98=x2+14x-7x-98=xx+14-7x+14=x+14x-7

Page No 114:

Question 6:

Factorise:
x2+23x24

Answer:


x2+23x24=x2+43x-23x-24=xx+43-23x+43=x+43x-23

Page No 114:

Question 7:

Factorise:
x
2 21x + 90

Answer:


x2-21x+90=x2-15x-6x+90=xx-15-6x-15=x-6x-15

Page No 114:

Question 8:

Factorise:
x
2 – 22x + 120

Answer:


x2-22x+120=x2-12x-10x+120=xx-12-10x-12=x-10x-12

Page No 114:

Question 9:

Factorise:
x
2 4x + 3

Answer:


x2-4x+3=x2-3x-x+3=xx-3-1x-3=x-1x-3

Page No 114:

Question 10:

Factorise:
x2+76x+60

Answer:


x2+76x+60=x2+56x+26x+60=xx+56+26x+56=x+56x+26

Page No 114:

Question 11:

Factorise:
x2+33x+6

Answer:


x2+33x+6=x2+23x+3x+6=xx+23+3x+23=x+23x+3

Page No 114:

Question 12:

Factorise:
x2+66x+48

Answer:


x2+66x+48=x2+46x+26x+48=xx+46+26x+46=x+46x+26

Page No 114:

Question 13:

Factorise:
x2+55x+30

Answer:


x2+55x+30=x2+35x+25x+30=xx+35+25x+35=x+35x+25

Page No 114:

Question 14:

Factorise:
x2-24x-180

Answer:


x2-24x-180=x2-30x+6x-180=xx-30+6x-30=x-30x+6

Page No 114:

Question 15:

Factorise:
x
2 – 32x – 105

Answer:


x2-32x-105=x2-35x+3x-105=xx-35+3x-35=x-35x+3

Page No 114:

Question 16:

Factorise:
x
2 – 11x – 80

Answer:


x2-11x-80=x2-16x+5x-80=xx-16+5x-16=x-16x+5

Page No 114:

Question 17:

Factorise:
6 – x – x2

Answer:


-x2-x+6=-x2-3x+2x+6=-xx+3+2x+3=x+3-x+2=x+32-x

Page No 114:

Question 18:

Factorise:
x2-3x-6

Answer:


x2-3x-6=x2-23x+3x-6=xx-23+3x-23=x-23x+3

Page No 114:

Question 19:

Factorise:
403x – x2

Answer:


-x2+3x+40=-x2+8x-5x+40=-xx-8-5x-8=x-8-x-5=8-xx+5

Page No 114:

Question 20:

Factorise:
x226x + 133

Answer:


x2-26x+133=x2-19x-7x+133=xx-19-7x-19=x-19x-7

Page No 114:

Question 21:

Factorise:
x2-23x-24

Answer:


x2-23x-24=x2-43x+23x-24=xx-43+23x-43=x-43x+23

Page No 114:

Question 22:

Factorise:
x2-35x-20

Answer:


x2-35x-20=x2-45x+5x-20=xx-45+5x-45=x-45x+5

Page No 114:

Question 23:

Factorise:
x2+2x-24

Answer:


x2+2x-24=x2+42x-32x-24=xx+42-32x+42=x+42x-32

Page No 114:

Question 24:

Factorise:
x2-22x-30

Answer:


x2-22x-30=x2-52x+32x-30=xx-52+32x-52=x-52x+32

Page No 114:

Question 25:

Factorize:
x2x − 156

Answer:

We have:
x2-x-156
We have to split (-1) into two numbers such that their sum is (-1) and their product is (-156).
Clearly, -13+12=-1 and -13×12=-156.

x2-x-156=x2-13x+12x-156                        =x(x-13)+12(x-13)                        =(x-13)(x+12)

Page No 114:

Question 26:

Factorise:
x2 – 32x – 105

Answer:


x2-32x-105=x2-35x+3x-105=xx-35+3x-35=x-35x+3

Page No 114:

Question 27:

Factorise:
9x2 + 18x + 8

Answer:


9x2+18x+8=9x2+12x+6x+8=3x3x+4+23x+4=3x+43x+2

Page No 114:

Question 28:

Factorise:
6x2 + 17x + 12

Answer:


6x2+17x+12=6x2+9x+8x+12=3x2x+3+42x+3=2x+33x+4

Page No 114:

Question 29:

Factorize:
18x2 + 3x − 10

Answer:

We have:
18x2+3x-10
We have to split 3 into two numbers such that their sum is 3 and their product is (-180), i.e., 18×-10.
Clearly, 15+-12=3 and 15×-12=-180.

​18x2+3x-10=18x2+15x-12x-10                             =3x6x+5-26x+5                             =6x+53x-2

Page No 114:

Question 30:

Factorize:
2x2 + 11x − 21

Answer:

We have:
2x2+11x-21
We have to split 11 into two numbers such that their sum is 11 and their product is (-42), i.e., 2×-21.
Clearly, 14+-3=11 and 14×-3=-42.

2x2+11x-21=2x2+14x-3x-21                             =2xx+7-3x+7                             =x+72x-3

Page No 114:

Question 31:

Factorize:
15x2 + 2x − 8

Answer:

We have:
15x2+2x-8
We have to split 2 into two numbers such that their sum is 2 and their product is (-120), i.e., 15×-8.
Clearly, 12+-10=2 and 12×-10=-120.

15x2+2x-8=15x2+12x-10x-8                          =3x5x+4-25x+4                          =5x+43x-2

Page No 114:

Question 32:

Factorise:
21x2 + 5x – 6

Answer:


21x2+5x-6=21x2+14x-9x-6=7x3x+2-33x+2=3x+27x-3

Page No 114:

Question 33:

Factorize:
24x2 − 41x + 12

Answer:

We have:
24x2-41x+12
We have to split (-41) into two numbers such that their sum is (-41) and their product is 288, i.e., 24×12.
Clearly, -32+-9=-41 and -32×-9=288.

24x2-41x+12=24x2-32x-9x+12                               =8x3x-4-33x-4                               =3x-48x-3

Page No 114:

Question 34:

Factorise:
3x2 – 14x + 8

Answer:

3x2-14x+8=3x2-12x-2x+8                      =3xx-4-2x-4                      =x-43x-2

Hence, factorisation of 3x2 – 14x + 8 is x-43x-2.

Page No 114:

Question 35:

Factorize:
2x2 + 3x − 90

Answer:

We have:
2x2+3x-90
We have to split 3 into two numbers such that their sum is 3 and their product is (-180), i.e., 2×-90.
Clearly, -12 + 15 = 3 and -12×15 = -180.

2x2+3x-90=2x2-12x+15x-90                          =2xx-6+15x-6                          =x-62x+15

Page No 114:

Question 36:

Factorize:
5x2+2x-35

Answer:

We have:
5x2+2x-35
We have to split 2 into two numbers such that their sum is 2 and product is (-15), i.e.,5×-35.
Clearly, 5+-3=2 and 5×-3=-15.

5x2+2x-35=5x2+5x-3x-35                                  =5xx+5-3x+5                                  =x+55x-3

Page No 114:

Question 37:

Factorize:
23x2+x-53

Answer:

We have:
23x2+x-53
We have to split 1 into two numbers such that their sum is 1 and product is 30, i.e.,23×-53.
Clearly, 6+-5=1 and 6×-5=-30.

23x2+x-53=23x2+6x-5x-53                                  =23xx+3-5x+3                                  =x+323x-5

Page No 114:

Question 38:

Factorize:
7x2+214x+2

Answer:

We have:
7x2+214x+2
We have to split 214 into two numbers such that their sum is 214 and product is 14.
Clearly, 14+14=214 and 14×14=14.
7x2+214x+2=7x2+14x+14x+2                                 =7x7x+2+27x+2                                 =7x+27x+2                                 =7x+22

Page No 114:

Question 39:

Factorize:
63x2-47x+53

Answer:

We have:
63x2-47x+53
Now, we have to split (-47) into two numbers such that their sum is (-47) and their product is 90.
Clearly, -45+-2=-47 and -45×-2=90.

63x2-47x+53 =63x2-2x-45x+53                                        =2x33x-1-5333x-1                                        =33x-12x-53

Page No 114:

Question 40:

Factorize:
55x2+20x+35

Answer:

We have:
55x2+20x+35
We have to split 20 into two numbers such that their sum is 20 and their product is 75.
Clearly, 
15+5=20 and 15×5=75

55x2+20x+35=55x2+15x+5x+35                                       =5x5x+3+5(5x+3)                                       =5x+35x+5

Page No 114:

Question 41:

Factorise:
3x2+10x+83

Answer:

3x2+10x+83=3x2+6x+4x+83                                 =3xx+23+4x+23                                 =x+233x+4

Hence, factorisation of 3x2+10x+83 is x+233x+4.

Page No 114:

Question 42:

Factorize:
2x2+3x+2

Answer:

We have:
2x2+3x+2
We have to split 3 into two numbers such that their sum is 3 and their product is 2, i.e., 2×2.
Clearly, 2+1=3 and 2×1=2.

2x2+3x+2=2x2+2x+x+2                                =2xx+2+1x+2                                =x+22x+1

Page No 114:

Question 43:

Factorize:
2x2+33x+3

Answer:

We have:
2x2+33x+3
We have to split 33 into two numbers such that their sum is 33 and their product is 6, i.e.,2×3.
Clearly, 23+3=33 and 23×3=6.

2x2+33x+3=2x2+23x+3x+3                              =2xx+3+3x+3                              =x+32x+3

Page No 114:

Question 44:

Factorize:
15x2x − 128

Answer:

We have:
15x2-x-28
We have to split (-1) into two numbers such that their sum is (-1) and their product is (-420), i.e., 15×-28.
Clearly, -21+20=-1 and -21×20=-420.

15x2-x-28=15x2-21x+20x-28                          =3x(5x-7)+4(5x-7)                          =(5x-7)(3x+4)

Page No 114:

Question 45:

Factorize:
6x2 − 5x − 21

Answer:

We have:
6x2-5x-21
We have to split (-5) into two numbers such that their sum is (-5) and their product is (-126), i.e., 6×-21.
Clearly, 9+-14=-5 and 9×-14=-126.

6x2-5x-21=6x2+9x-14x-21                          =3x2x+3-72x+3                          =2x+33x-7

Page No 114:

Question 46:

Factorize:
2x2 − 7x − 15

Answer:

We have:
2x2-7x-15
We have to split (-7) into two numbers such that their sum is (-7) and their product is (-30), i.e., 2×-15.
Clearly, -10+3=-7 and -10×3=-30.

2x2-7x-15=2x2-10x+3x-15                          =2xx-5+3x-5                          =x-52x+3

Page No 114:

Question 47:

Factorize:
5x2 − 16x − 21

Answer:

We have:
5x2-16x-21
We have to split (-16) into two numbers such that their sum is (-16) and their product is (-105), i.e., 5×-21.
Clearly, -21+5=-16 and -21×5=-105.

5x2-16x-21=5x2+5x-21x-21                             =5xx+1-21x+1                             =x+15x-21

Page No 114:

Question 48:

Factorise:
6x2 – 11x – 35

Answer:

6x2-11x-35=6x2-21x+10x-35                         =3x2x-7+52x-7                         =2x-73x+5

Hence, factorisation of 6x2 – 11x – 35 is 2x-73x+5.
 

Page No 114:

Question 49:

Factorise:
9x2 – 3x – 20

Answer:

9x2-3x-20=9x2-15x+12x-20                       =3x3x-5+43x-5                       =3x-53x+4

Hence, factorisation of 9x2 – 3x – 20 is 3x-53x+4.

Page No 114:

Question 50:

Factorize:
10x2 − 9x − 7

Answer:

We have:
10x2-9x-7

We have to split (-9) into two numbers such that their sum is (-9) and their product is (-70), i.e., 10×-7.
Clearly, -14+5=-9 and -14×5=-70.

10x2-9x-7=10x2+5x-14x-7                          =5x2x+1-72x+1                          =2x+15x-7

Page No 114:

Question 51:

Factorize:
x2-2x+716

Answer:

We have:x2-2x+716=16x2-32x+716=11616x2-32x+7

Now, we have to split (-32) into two numbers such that their sum is (-32) and their product is 112, i.e., 16×7.
Clearly, -4+-28=-32 and -4×-28=112.

x2 - 2x + 716 =116(16x2-32x+7)                                =116(16x2-4x-28x+7)                                =1164x(4x-1)-7(4x-1)                                =116(4x-1)(4x-7)

Page No 114:

Question 52:

Factorise:
13x2-2x-9

Answer:

13x2-2x-9=x2-6x-273                       =x2-9x+3x-273                       =xx-9+3x-93                       =x-9x+33                       =x-93×x+31                       =13x-3x+3

Hence, factorisation of 13x2-2x-9 is 13x-3x+3.

Page No 114:

Question 53:

Factorise:
x2+1235x+135

Answer:

x2+1235x+135=35x2+12x+135                           =35x2+7x+5x+135                           =7x5x+1+15x+135                           =5x+17x+135                           =5x+17x+15×7                           =5x+15×7x+17                           =x+15x+17

Hence, factorisation of x2+1235x+135 is x+15x+17.

Page No 114:

Question 54:

Factorise:
21x2-2x+121

Answer:

21x2-2x+121=21x2-x-x+121                           =21xx-121-1x-121                           =x-12121x-1

Hence, factorisation of 21x2-2x+121 is x-12121x-1.

Page No 114:

Question 55:

Factorise:
32x2+16x+10

Answer:

32x2+16x+10=32x2+15x+x+10                           =3x12x+5+1x+10                           =32xx+10+1x+10                           =x+1032x+1

Hence, factorisation of 32x2+16x+10 is x+1032x+1.

Page No 114:

Question 56:

Factorise:
23x2-173x-28

Answer:

23x2-173x-28=23x2-8x+73x-28                             =2x13x-4+713x-4                             =13x-42x+7

Hence, factorisation of 23x2-173x-28 is 13x-42x+7.

Page No 114:

Question 57:

Factorise:
35x2-195x+4

Answer:

35x2-195x+4=35x2-3x-45x+4                           =3x15x-1-415x-1                           =15x-13x-4

Hence, factorisation of 35x2-195x+4 is 15x-13x-4.

Page No 114:

Question 58:

Factorise:
2x2-x+18

Answer:

2x2-x+18=2x2-12x-12x+18                    =2xx-14-12x-14                    =x-142x-12

Hence, factorisation of 2x2-x+18 is x-142x-12.

Page No 114:

Question 59:

Factorize:
2(x + y)2 − 9(x + y) − 5

Answer:

We have:
2x+y2-9x+y-5Let:(x+y)=u
Thus, the given expression becomes
2u2-9u-5
Now, we have to split (-9) into two numbers such that their sum is (-9) and their product is (-10).
Clearly, -10+1=-9 and -10×1=-10.

2u2-9u-5=2u2-10u+u-5                         =2u(u-5)+1(u-5)                         =(u-5)(2u+1)
Putting u=(x+y), we get:
2x+y2 - 9x+y - 5 = x+y-52x+y+1                                          = x+y-52x+2y+1

Page No 114:

Question 60:

Factorize:
9(2ab)2 − 4(2ab) − 13

Answer:

We have:
9(2a-b)2-4(2a-b)-13Let:(2a-b)=p
Thus, the given expression becomes
9p2-4p-13
Now, we must split (-4) into two numbers such that their sum is (-4) and their product is (-117).
Clearly, -13+9=-4 and -13×9=-117.
9p2-4p-13=9p2+9p-13p-13                           =9p(p+1)-13(p+1)                           =(p+1)(9p-13)
Putting p=(2a-b), we get:
92a-b2-42a-b-13=2a-b+192a-b-13                                           =2a-b+118a-9b-13



Page No 115:

Question 61:

Factorise:
7x-2y2-25x-2y+12

Answer:

7x-2y2-25x-2y+12=7x-2y2-21x-2y-4x-2y+12                                              =7x-2yx-2y-3-4x-2y-3                                              =7x-2y-4x-2y-3                                              =7x-14y-4x-2y-3

Hence, factorisation of 7x-2y2-25x-2y+12 is 7x-14y-4x-2y-3.

Page No 115:

Question 62:

Factorise:
103x+1x2-3x+1x-3

Answer:

103x+1x2-3x+1x-3=103x+1x2-63x+1x+53x+1x-3                                                =23x+1x53x+1x-3+153x+1x-3                                                =53x+1x-323x+1x+1                                                =15x+5x-36x+2x+1

Hence, factorisation of 103x+1x2-3x+1x-3 is 15x+5x-36x+2x+1.

Page No 115:

Question 63:

Factorise:
62x-3x2+72x-3x-20

Answer:

62x-3x2+72x-3x-20=62x-3x2+152x-3x-82x-3x-20                                                  =32x-3x22x-3x+5-422x-3x+5                                                  =22x-3x+532x-3x-4                                                  =4x-6x+56x-9x-4

Hence, factorisation of 62x-3x2+72x-3x-20 is 4x-6x+56x-9x-4.

Page No 115:

Question 64:

Factorise:
a+2b2+101a+2b+100

Answer:

a+2b2+101a+2b+100=a+2b2+100a+2b+1a+2b+100                                                 =a+2ba+2b+100+1a+2b+100                                                 =a+2b+1a+2b+100                                                 =a+2b+1a+2b+100

Hence, factorisation of a+2b2+101a+2b+100 is a+2b+1a+2b+100.

Page No 115:

Question 65:

Factorise:
4x4 + 7x2 – 2

Answer:

4x4+7x2-2=4x4+8x2-x2-2                      =4x2x2+2-1x2+2                      =4x2-1x2+2

Hence, factorisation of 4x4 + 7x2 – 2 is 4x2-1x2+2.

Page No 115:

Question 66:

Evaluate {(999)2 – 1}.

Answer:

9992-1=9992-12                   =999-1999+1                   =9981000                   =998000

Hence, {(999)2 – 1} = 998000.



Page No 119:

Question 1:

Expand:
(i) (a + 2b + 5c)2
(ii) (2bb + c)2
(iii) (a − 2b − 3c)2

Answer:

i a+2b+5c2=a2 + 2b2 +5c2+2a2b+22b5c+25ca                           =a2+4b2+25c2+4ab+20bc+10ac

ii 2a-b+c2=2a+-b+c2                          =2a2+-b2+c2+22a-b+2-bc+4ac                          =4a2+b2+c2-4ab-2bc+4ac

iii a-2b-3c2=a+-2b+-3c2                             =a2+-2b2+-3c2+2a-2b+2-2b-3c+2a-3c                             =a2+4b2+9c2-4ab+12bc-6ac

Page No 119:

Question 2:

Expand:
(i) (2a − 5b − 7c)2
(ii) (−3a + 4b − 5c)2
(iii) 12a-14a+22

Answer:

i 2a-5b-7c2=2a+-5b+-7c2                             =2a2+-5b2+-7c2+22a-5b+2-5b-7c+22a-7c                             =4a2+25b2+49c2-20ab+70bc-28ac

ii -3a+4b-5c2=-3a+4b+-5c2                                 =-3a2+4b2+-5c2+2-3a4b+24b-5c+2-3a-5c                                 =9a2+16b2+25c2-24ab-40bc+30ac

iii 12a-14b+22=a2+-b4+22                                  =a22+-b42+22+2a2-b4+2-b42+2a22                                  =a24+b216+4-ab4-b+2a

Page No 119:

Question 3:

Factorize: 4x2 + 9y2 + 16z2 + 12xy − 24yz − 16xz.

Answer:


We have: 4x2+9y2+16z2+12xy-24yz-16xz=2x2+3y2+-4z2+2(2x)(3y)+2(3y)(-4z)+2(-4z)(2x)=2x+3y+-4z2=2x+3y-4z2

Page No 119:

Question 4:

Factorize: 9x2 + 16y2 + 4z2 − 24xy + 16yz − 12xz

Answer:

We have:9x2+16y2+4z2-24xy+16yz-12xz=-3x2+4y2+2z2+2(-3x)(4y)+2(4y)(2z)+2(2z)(-3x)=-3x+4y+2z2=-3x+4y+2z2

Page No 119:

Question 5:

Factorize: 25x2 + 4y2 + 9z2 − 20xy − 12yz + 30xz.

Answer:

We have:25x2+4y2+9z2-20xy-12yz+30xz=5x2+-2y2+3z2+2(5x)(-2y)+2(-2y)(3z)+2(3z)(5x)=5x+-2y+3z2=5x-2y+3z2

Page No 119:

Question 6:

16x2 + 4y2 + 9z2 – 16xy – 12yz + 24xz

Answer:

16x2+4y2+9z2-16xy-12yz+24xz=4x2+-2y2+3z2+24x-2y+2-2y3z+23z4x=4x-2y+3z2               using a2+b2+c2+2ab+2bc+2ca=a+b+c2

Hence, 16x2 + 4y2 + 9z2 – 16xy – 12yz + 24xz = 4x-2y+3z2.

Page No 119:

Question 7:

Evaluate
(i) (99)2
(ii) (995)2
(iii) (107)2

Answer:

i 992=100-12             =100+-12             = 1002+2×100×-1+-12             =10000-200+1             =9801         

ii 9952=1000-52                =1000+-52                =10002+2×1000×-5+-52                =1000000-10000+25                =990025

iii 1072=100+72                 =1002+2×100×7+72                 =10000+1400+49                 =11449



Page No 123:

Question 1:

Expand
(i) (3x + 2)3
(ii) 3a+14b3
(iii) 1+23a3

Answer:

i 3x+23=3x3+3×3x2x2+3×3x×22+23                   =27x3+54x2+36x+8
ii 3a+14b3=3a3+14b3+33a214b+33a14b2                          =27a3+164b3+27a24b+9a16b2
iii 1+23a3=23a3+3×23a2×1+3a23a×12+13                         =827a3+43a2+2a+1

Page No 123:

Question 2:

Expand
(i) (5a – 3b)3
(ii) 3x-5x3
(iii) 45a-23

Answer:

i 5a-3b3=5a3-3b3-35a23b+35a3b2                        =125a3-27b3-225a2b+135ab2

ii 3x-5x3=3x3-5x3-33x25x+33x5x2                        =27x3-125x3-135x+225x


iii 45a-23=45a3-23-345a22+345a22                        =64125a3-8-9625a2+485a

Page No 123:

Question 3:

Factorise
8a3+27b3+36a2b+54ab2

Answer:

8a3+27b3+36a2b+54ab2=2a3+3b3+32a23b+32a3b2                                                 =2a+3b3

Hence, factorisation of 8a3+27b3+36a2b+54ab2 is 2a+3b3.

Page No 123:

Question 4:

Factorise
64a3-27b3-144a2b+108ab2

Answer:

64a3-27b3-144a2b+108ab2=4a3-3b3-34a23b+34a3b2                                                       =4a-3b3

Hence, factorisation of 64a3-27b3-144a2b+108ab2 is 4a-3b3.

Page No 123:

Question 5:

Factorise
1+27125a3+9a5+27a224

Answer:

1+27125a3+9a5+27a225=13+35a3+31235a+3135a2                                            =1+35a3

Hence, factorisation of 1+27125a3+9a5+27a225 is 1+35a3.

Page No 123:

Question 6:

Factorise
125x3-27y3-225x2y+135xy2

Answer:

125x3-27y3-225x2y+135xy2=5x3-3y3-35x23y+35x3y2                                                        =5x-3y3

Hence, factorisation of 125x3-27y3-225x2y+135xy2 is 5x-3y3.

Page No 123:

Question 7:

Factorise
a3x3-3a2bx2+3ab2x-b3

Answer:

a3x3-3a2bx2+3ab2x-b3=ax3-b3-3ax2b+3axb2                                               =ax-b3

Hence, factorisation of a3x3-3a2bx2+3ab2x-b3 is ax-b3.

Page No 123:

Question 8:

Factorise
64125a3-9625a2+485a-8

Answer:

64125a3-9625a2+485a-8=45a3-23-345a22+345a22                                               =45a-23

Hence, factorisation of 64125a3-9625a2+485a-8 is 45a-23.

Page No 123:

Question 9:

Factorise
a3 – 12a(a – 4) – 64

Answer:

a3-12aa-4-64=a3-12a2+48a-64                                  =a3-43-3a24+3a42                                  =a-43

Hence, factorisation of a3 – 12a(a – 4) – 64 is a-43.

Page No 123:

Question 10:

Evaluate
(i) (103)3
(ii) (99)3

Answer:

i 1033=100+33                =1003+33+310023+310032                =1000000+27+90000+2700                =1092727ii 993=100-13              =1003-13-310021+310012              =1000000-1-30000+300              =1000300-30001              =970299



Page No 129:

Question 1:

Factorize:
x3 + 27

Answer:

x3+27=x3+33            =x+3x2-3x+32            =x+3x2-3x+9

Page No 129:

Question 2:

Factorise
27a3 + 64b3

Answer:

We know that 
x3+y3=x+yx2+y2-xy
Given: 27a3 + 64b3
x = 3a, y = 4b
27a3 + 64b3=3a+4b9a2+16b2-12ab
 

Page No 129:

Question 3:

Factorize:
125a3+18

Answer:

125a3+18=5a3+123                   =5a+125a2-5a×12+122                   =5a+1225a2-5a2+14

Page No 129:

Question 4:

Factorize:
216x3+1125

Answer:

216x3+1125=6x3+153                        =6x+156x2-6x×15+152                        =6x+1536x2-6x5+125

Page No 129:

Question 5:

Factorize:
16x4 + 54x

Answer:


16x4+54x=2x8x3+27                    =2x2x3+33                    =2x2x+32x2-2x×3+32                    =2x2x+34x2-6x+9

Page No 129:

Question 6:

Factorize:
7a3 + 56b3

Answer:

7a3+56b3=7a3+8b3                   =7a3+2b3                   =7a+2ba2-a×2b+2b2                   =7a+2ba2-2ab+4b2

Page No 129:

Question 7:

Factorize:
x5 + x2

Answer:

x5+x2=x2x3+1            =x2x3+13            =x2x+1x2-x×1+12            =x2x+1x2-x+1

Page No 129:

Question 8:

Factorize:
a3 + 0.008

Answer:

a3+0.008=a3+0.23                  =a+0.2a2-a×0.2+0.22                  =a+0.2a2-0.2a+0.04

Page No 129:

Question 9:

Factorise
1 – 27a3

Answer:

1-27a3=13-3a3              =1-3a12+1×3x+3a2              =1-3a1+3a+9a2

Page No 129:

Question 10:

Factorize:
64a3 − 343

Answer:

64a3-343=4a3-73                   =4a-716a2+4a×7+72                   =4a-716a2+28a+49

Page No 129:

Question 11:

Factorize:
x3 − 512

Answer:

x3-512 =x3-83               =x-8x2+8x+82               =x-8x2+8x+64

Page No 129:

Question 12:

Factorize:
a3 − 0.064

Answer:

a3-0.064=a3-0.43                  =a-0.4a2+a×0.4+0.42                  =a-0.4a2+0.4a+0.16

Page No 129:

Question 13:

Factorize:
8x3-127y3

Answer:

8x3-127y3=2x3-13y3                    =2x-13y2x2+2x×13y+13y2                    =2x-13y4x2+2x3y+19y2

Page No 129:

Question 14:

Factorise
x3216-8y3

Answer:

We know
a3-b3=a-ba2+b2+ab
We have,
 x3216-8y3=x63-2y3
So, a=x6,b=2y
x3216-8y3=x6-2yx62+x6×2y+2y2=x6-2yx236+xy3+4y2

Page No 129:

Question 15:

Factorize:
x − 8xy3

Answer:

x-8xy3=x1-8y3              =x13-2y3              =x1-2y12+1×2y+2y2              =x1-2y1+2y+4y2

Page No 129:

Question 16:

Factorise
32x4 – 500x

Answer:

 32x4  500x=4x8x3-125=4x2x3-53we knowa3-b3=a-ba2+b2+aba=2x,b=532x4  500x=4x2x3-53=4x2x-54x2+25+10x

Page No 129:

Question 17:

Factorize:
3a7b − 81a4b4

Answer:

3a7b-81a4b4=3a4ba3-27b3                         =3a4ba3-3b3                         =3a4ba-3ba2+a×3b+3b2                         =3a4ba-3ba2+3ab+9b2

Page No 129:

Question 18:

Factorise
x4 y4xy

Answer:

Using the identity 
a3-b3=a-ba2+b2+ab
x4 y4xy=xyx3y3-1=xyxy-1x2y2+1+xy

Page No 129:

Question 19:

Factorise
8x2 y3x5

Answer:

8x2y3x5=x28y3-x3=x22y-x4y2+x2+2xy

Page No 129:

Question 20:

Factorise
1029 – 3x3

Answer:

10293x3=3343-x3=373-x3=37-x49+x2+7x

Page No 129:

Question 21:

Factorize:
x6 − 729

Answer:

x6-729=x23-93             =x2-9x22+x2×9+92             =x2-32x4+9x2+81             =x+3x-3x4+18x2+81-9x2             =x+3x-3x22+2×x2×9+92-9x2             =x+3x-3x2+92-3x2             =x+3x-3x2+9+3xx2+9-3x             =x+3x-3x2+3x+9x2-3x+9

Page No 129:

Question 22:

Factorise
x9 – y9

Answer:

x9y9=x33-y33we knowa3-b3=a-ba2+b2+aba=x3,b=y3So,x9y9=x33-y33=x3-y3x6+y6+x3y3=x-yx2+y2+xyx6+y6+x3y3

Page No 129:

Question 23:

Factorize:
(a + b)3 − (ab)3

Answer:

a + b3-a-b3=a+b-a-ba+b2+a+ba-b+a-b2                            =a+b-a+ba2+2ab+b2+a2-b2+a2-2ab+b2                            =2b3a2+b2

Page No 129:

Question 24:

Factorize:
8a3b3 − 4ax + 2bx

Answer:

8a3-b3-4ax+2bx=2a3-b3-2x2a-b                                    =2a-b2a2+2ab+b2-2x2a-b                                    =2a-b4a2+2ab+b2-2x2a-b                                    =2a-b4a2+2ab+b2-2x

Page No 129:

Question 25:

Factorize:
a3 + 3a2b + 3ab2 + b3 − 8

Answer:

a3+3a2b+3ab2+b3-8=a3+b3+3a2b+3ab2-8                                            =a3+b3+3aba+b-8                                            =a+b3-23                                            =a+b-2a+b2+2a+b+22                                            =a+b-2a+b2+2a+b+4                                            

Page No 129:

Question 26:

Factorize:
a3-1a3-2a+2a

Answer:

a3-1a3-2a+2a=a3-1a3-2a-1a                                 =a3-1a3-2a-1a                                 =a-1aa2+a×1a+1a2-2a-1a                                 =a-1aa2+1+1a2-2a-1a                                 =a-1aa2+1+1a2-2                                 =a-1aa2-1+1a2

Page No 129:

Question 27:

Factorize:
2a3 + 16b3 − 5a − 10b

Answer:

2a3+16b3-5a-10b=2a3+8b3-5a+2b                                        =2a3+2b3-5a+2b                                        =2a+2ba2-a×2b+2b2-5a+2b                                        =2a+2ba2-2ab+4b2-5a+2b                                        =a+2b2a2-2ab+4b2-5

Page No 129:

Question 28:

Factorise
a6 + b6

Answer:

a6+b6=a23+b23            =a2+b2a22-a2b2+b22            =a2+b2a4-a2b2+b4

Page No 129:

Question 29:

Factorise
a12 – b12

Answer:

a12 – b12
=a6+b6a6-b6=a23+b23a32-b32=a2+b2a4+b4-a2b2a3-b3a3+b3=a2+b2a4+b4-a2b2a-ba2+b2+aba+ba2+b2-ab=a-ba2+b2+aba+ba2+b2-aba2+b2a4+b4-a2b2

 

Page No 129:

Question 30:

Factorise
x6 – 7x3 – 8

Answer:

Let x3=y
So, the equation becomes 
y2-7y-8=y2-8y+y-8=yy-8+y-8=y-8y+1=x3-8x3+1=x-2x2+4+2xx+1x2+1-x

Page No 129:

Question 31:

Factorise
x3 – 3x2 + 3x + 7

Answer:

x3 – 3x+ 3x + 7
=x33x2+3x+7=x33x2+3x+8-1=x33x2+3x-1+8=x33x2+3x-1+8=x-13+23=x-1+2x-12+4-2x-1=x+1x2+1-2x+4-2x+2=x+1x2-4x+7

Page No 129:

Question 32:

Factorise
(x +1)3 + (x – 1)3

Answer:

(x +1)3 + (x – 1)3
=x+1+x-1x+12+x-12-x-1x+1=2xx+12+x-12-x2-1=2xx2+1+2x+x2+1-2x-x2+1=2xx2+3

 

Page No 129:

Question 33:

Factorise
(2a +1)3 + (a – 1)3

Answer:

(2a +1)3 + (a – 1)3  
=2a+1+a-12a+12+a-12-2a+1a-1=3a4a2+1+4a+a2+1-2a-2a2+2a-a+1=3a3a2+3a+3=9aa2+a+1

Page No 129:

Question 34:

Factorise
8(x +y)3 – 27(x y)3

Answer:

8(x +y)3 – 27(x – y)3
=2x+y3-3x-y3=2x+2y-3x+3y4x+y2+9x-y2+6x2-y2=-x+5y4x2+y2+2xy+9x2+y2-2xy+6x2-y2=-x+5y4x2+4y2+8xy+9x2+9y2-18xy+6x2-6y2=-x+5y19x2+7y2-10xy

Page No 129:

Question 35:

Factorise
(x +2)3 + (x – 2)3

Answer:

(x +2)3 + (x – 2)3
=x+2+x-2x+22+x-22-x2-4=2xx2+4+4x+x2+4-4x-x2+4=2xx2+12

Page No 129:

Question 36:

Factorise
(x + 2)3 – (x – 2)3

Answer:

(x + 2)3 – (x – 2)3
=x+2-x+2x+22+x-22+x2-4=4x2+4+4x+x2+4-4x+x2-4=43x2+4

Page No 129:

Question 37:

Prove that 0.85×0.85×0.85+0.15×0.15×0.150.85×0.85-0.85×0.15+0.15×0.15=1.

Answer:

LHS:0.85×0.85×0.85+0.15×0.15×0.150.85×0.85-0.85×0.15+0.15×0.15=0.853+0.1530.852-0.85×0.15+0.152We knowa3+b3=a+ba2+b2-abHere a=0.85,b=0.15
0.853+0.1530.852-0.85×0.15+0.152=0.85+0.150.852-0.85×0.15+0.1520.852-0.85×0.15+0.152=0.85+0.15=1:RHS
Thus, LHS = RHS
 

Page No 129:

Question 38:

Prove that 59×59×59-9×9×959×59+59×9+9×9=50.

Answer:

59×59×59-9×9×959×59+59×9+9×9=593-93592+59×9+92
We knowa3+b3=a+ba2+b2-abHere a=59,b=9So,59-9592+92+59×9592+92+59×9=59-9=50:RHS
Thus, LHS=RHS



Page No 136:

Question 1:

Find the product:
(x + yz) (x2 + y2 + z2xy + yz + zx)

Answer:

   x+y-zx2+y2+z2-xy+yz+zx=x+y+-zx2+y2+-z2-xy-y×-z--z×x=x3+y3+-z3-3x×y×-z=x3+y3-z3+3xyz

Page No 136:

Question 2:

Find the product:
(xyz) (x2 + y2 + z2 + xyyz + xz)

Answer:

(x – y − z) (x2 + y2 + z2 + xy – yz + xz)
=(x+-y+z) (x2+y2+z2+xyyz+xz)We knowa+b+ca2+b2+c2-ab-bc-ca=a3+b3+c3-3abcHere, a=x,b=-y,c=-z(x+-y+z) (x2+y2+z2+xyyz+xz)=x3-y3-z3-3xyz

Page No 136:

Question 3:

Find the product:
(x − 2y + 3) (x2 + 4y2 + 2xy − 3x + 6y + 9)

Answer:

   x − 2y + 3x2+4y2+2xy-3x+6y+9=x − 2y + 3x2+4y2+9+2xy+6y-3x=x+-2y+3x2+-2y2+32-x×-2y--2y×3-3×x=x3+-2y3+33-3x-2y3=x3-8y3+27+18xy

Page No 136:

Question 4:

Find the product:
(3x – 5y + 4) (9x2 + 25y2 + 15xy − 20y + 12x + 16)

Answer:

3x-5y+49x2+25y2+15xy-20y+12x+16=3x+-5y+49x2+25y2+16+15xy-20y+12x
a+b+ca2+b2+c2-ab-bc-ca=a3+b3+c3-3abcHere, a=3x,b=-5y,c=4
3x+-5y+49x2+25y2+16+15xy-20y+12x=3x3+-5y3+43-3×3x-5y4=27x3-125y3+64+180xy

Page No 136:

Question 5:

Factorize:
125a3 + b3 + 64c3 − 60abc

Answer:

125a3+b3+64c3-60abc=5a3+b3+4c3-3×5a×b×4c                                              =5a+b+4c5a2+b2+4c2-5a×b-b×4c-5a×4c                                              =5a+b+4c25a2+b2+16c2-5ab-4bc-20ac                          

Page No 136:

Question 6:

Factorize:
a3 + 8b3 + 64c3 − 24abc

Answer:

a3+8b3+64c3-24abc=a3+2b3+4c3-3×a×2b×4c                                          =a+2b+4ca2+2b2+4c2-a×2b-2b×4c-4c×a                                          =a+2b+4ca2+4b2+16c2-2ab-8bc-4ca

Page No 136:

Question 7:

Factorize:
1 + b3 + 8c3 − 6bc

Answer:

1+b3+8c3-6bc=13+b3+2c3-3×1×b×2c                               =1+b+2c12+b2+2c2-1×b-b×2c-1×2c                               =1+b+2c1+b2+4c2-b-2bc-2c

Page No 136:

Question 8:

Factorize:
216 + 27b3 + 8c3 − 108abc

Answer:

216+27b3+8c3-108abc=63+3b3+2c3-3×6×3b×2c                                               =6+3b+2c62+3b2+2c2-6×3b-3b×2c-2c×6                                               =6+3b+2c36+9b2+4c2-18b-6bc-12c

Page No 136:

Question 9:

Factorize:
27a3b3 + 8c3 + 18abc

Answer:

27a3-b3+8c3+18abc=3a3+-b3+2c3-3×3a×-b×2c                                         =3a+-b+2c3a2+-b2+2c2-3a-b--b2c-3a×2c                                         =3a-b+2c9a2+b2+4c2+3ab+2bc-6ac

Page No 136:

Question 10:

Factorize:
8a3 + 125b3 − 64c3 + 120abc

Answer:

8a3+125b3-64c3+120abc=2a3+5b3+-4c3-3×2a×5b×-4c                                                   =2a+5b-4c2a2+5b2+-4c2-2a5b-5b-4c-2a×-4c                                                   =2a+5b-4c4a2+25b2+16c2-10ab+20bc+8ac

Page No 136:

Question 11:

Factorize:
8 − 27b3 − 343c3 − 126bc

Answer:

8-27b3-343c3-126bc=23+-3b3+-7c3-3×2×-3b×-7c                                             =2+-3b+-7c22+-3b2+-7c2-2-3b--3b-7c-2-7c                                             =2-3b-7c4+9b2+49c2+6b-21bc+14c

Page No 136:

Question 12:

Factorize:
125 − 8x3 − 27y3 − 90xy

Answer:

125-8x3-27y3-90xy=53+-2x3+-3y3-3×5×-2x×-3y                                          =5+-2x +-3y52+-2x2+-3y2-5×-2x--2x-3y-5×-3y                                          =5-2x-3y25+4x2+9y2+10x-6xy+15y



Page No 137:

Question 13:

Factorize:
22a3 + 162b3+c3-12abc

Answer:

22a3+162b3+c3-12abc=2a3+22b3+c3-3×2a×22b×c                                                    =2a+22b+c2a2+22b2+c2-2a×22b-22b×c-2a×c                                                    =2a+22b+c2a2+8b2+c2-4ab-22bc-2ac

Page No 137:

Question 14:

Factorise:
27x3 y3z3 – 9xyz

Answer:

27x3-y3z39xyz=3x3-y3-z3-3×3x×-y×-zWe know, a3+b3+c3-3abc=a+b+ca2+b2+c2-ab-bc-caa=3x,b=-y,c=-z3x3-y3-z3-3×3x×-y×-z=3x-y-z9x2+y2+z2+3xy-yz+3xz

Page No 137:

Question 15:

Factorise:
22a3+33b3+c3-36abc

Answer:

22a3+33b3+c3-36abc=2a3+3b3+c3-32a3bcWe knowx3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zxx=2a,y=3b,z=c2a3+3b3+c3-32a3bc=2a+3b+c2a2+3b2+c2-6ab-3bc-2ac

Page No 137:

Question 16:

Factorise:
33a3-b3-55c3-315abc

Answer:

33a3-b3-55c3-315abc=3a3+-b3+-5c3-33a-b-5cWe knowx3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zxHere, x=3a,y=(-b),z=-5c33a3-b3-55c3-315abc=3a3+-b3+-5c3-33a-b-5c=3a-b-5c3a2+b2+5c2+3ab-5bc+15c

Page No 137:

Question 17:

Factorize:
(ab)3 + (bc)3 + (ca)3

Answer:

a-b3+b-c3+c-a3
Putting a-b=x, b-c=y and c-a=z, we get:a-b3+b-c3+c-a3=x3+y3+z3        [Where x+y+z=a-b+b-c+c-a=0]=3xyz                 x+y+z=0 x3+y3+z3=3xyz=3a-bb-cc-a

Page No 137:

Question 18:

Factorise:
a-3b3+3b-c3+c-a3

Answer:

We knowx3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zxx3+y3+z3=x+y+zx2+y2+z2-xy-yz-zx+3xyzHere, x=a-3b,y=(3b-c),z=c-a
a-3b3+3b-c3+c-a3=a-3b+3b-c+c-aa-3b2+3b-c2+c-a2-a-3b3b-c-3b-cc-a-c-aa-3b+3a-3b3b-cc-a=0+3a-3b3b-cc-a=3a-3b3b-cc-a

Page No 137:

Question 19:

Factorize:
(3a − 2b)3 + (2b − 5c)3 + (5c − 3a)3

Answer:

Put 3a-2b=x, 2b-5c=y and 5c-3a=z.We have:x+y+z = 3a-2b+2b-5c+5c-3a=0Now,3a-2b3+2b-5c3+5c-3a3=x3+y3+z3                                                   =3xyz    Here, x+y+z=0. So, x3 + y3 +z3 = 3xyz                                                   =33a-2b2b-5c5c-3a

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Question 20:

Factorize:
(5a − 7b)3 + (9c − 5a)3 + (7b − 9c)3

Answer:

Put 5a-7b=x, 9c-5a=z and 7b-9c=y.Here, x+y+z = 5a - 7b + 9c-5a+7b-9c=0Thus, we have:5a-7b3+9c-5a3+7b-9c3=x3+z3+y3                                                   =3xzy   When x+y+z=0, x3+y3+z3 = 3xyz.                                                   =3 5a-7b9c-5a7b-9c

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Question 21:

Factorize:
a3(bc)3 + b3(ca)3 + c3(ab)3

Answer:

We have:a3b-c3+b3c-a3+c3a-b3 = ab-c3+bc-a3+ca-b3Put ab-c = x      bc-a = y      ca-b = z Here, x+y+z = ab-c+bc-a+ca-b              =ab - ac + bc - ab + ac - bc              =0Thus, we have:a3b-c3+b3c-a3+c3a-b3 =x3 + y3+ z3                                                   =3xyz      When x+y+z =0, x3 + y3+ z3 =3xyz.                                                   =3 ab-cbc-aca-b                                                   =3abca-bb-cc-a

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Question 22:

Evaluate
(i) (–12)3 + 73 + 53
(ii) (28)3 + (–15)3 + (–13)3

Answer:

(i) (–12)+ 7+ 53
We knowx3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zxx3+y3+z3=x+y+zx2+y2+z2-xy-yz-zx+3xyzHere, x=-12,y=7,z=5
-123+73+53=-12+7+5-122+72+52-7-12-35+60+3-12×35=0-1260=-1260

(ii) (28)3 + (–15)3 + (–13)3

We knowx3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zxx3+y3+z3=x+y+zx2+y2+z2-xy-yz-zx+3xyzHere, x=-28,y=-15,z=-13
283+-153+-133=28-15-13282+-152+-132-28-15--15-13-28-13+3×28-15-13=0+16380=16380
 

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Question 23:

Prove that a+b+c3-a3-b3-c3=3a+b b+c c+a

Answer:

a+b+c3=a+b+c3=a+b3+c3+3a+bca+b+ca+b+c3=a3+b3+3aba+b+c3+3a+bca+b+ca+b+c3-a3+b3-c3=3aba+b+3a+bca+b+ca+b+c3-a3+b3-c3=3a+bab+ca+cb+c2a+b+c3-a3+b3-c3=3a+bab+c+cb+ca+b+c3-a3+b3-c3=3a+bb+ca+c
 

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Question 24:

If a, b, c are all nonzero and a + b + c = 0, prove that a2bc+b2ca+c2ab=3.

Answer:

a+b+c=0a3+b3+c3=3abc

Thus, we have:
a2bc+b2ca+c2ab=a3+b3+c3abc
                     =3abcabc=3

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Question 25:

If a + b + c = 9 and a2 + b2 + c2 = 35, find the value of (a3 + b3 + c3 – 3abc).

Answer:

a + b + c = 9
a+b+c2=92=81a2+b2+c2+2ab+bc+ca=8135+2ab+bc+ca=81ab+bc+ca=23
We know,
(a+ b3 + c3 – 3abc) = a+b+ca2+b2+c2-ab-bc-ca
=935-23=108
 



Page No 138:

Question 1:

If (x + 1) is factor of the polynomial (2x2 + kx) then the value of k is
(a) –2
(b) –3
(c) 2
(d) 3

Answer:

(c) 2

x+1 is a factor of 2x2+kx.So, -1 is a zero of 2x2+kx.Thus, we have:2×-12+k×-1=02-k=0k=2

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Question 2:

The value of (249)2 – (248)2 is
(a) 12
(b) 477
(c) 487
(d) 497

Answer:

(249)2 – (248)2
We know
a2-b2=a+ba-bSo, 2492-2482=249-248249+248=497
Hence, the correct answer is option (d).

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Question 3:

If xy+yx=-1, where x ≠ 0 and y ≠ 0, then the value of (x3y3) is
(a) 1
(b) −1
(c) 0
(d) 12

Answer:

 (c) 0

   xy+yx=-1x2+y2xy=-1
x2 + y2 = -xy
x2 + y2 + xy = 0

Thus, we have:
x3-y3=x-yx2+y2+xy
         =x-y×0=0

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Question 4:

If a + b + c = 0, then a3 + b3 + c3 = ?
(a) 0
(b) abc
(c) 2abc
(d) 3abc

Answer:

(d) 3abc

  a+b+c=0a+b=-c

a+b3=-c3a3+b3+3aba+b=-c3a3+b3+3ab-c=-c3a3+b3+c3=3abc



Page No 139:

Question 5:

If 3x+12 3x-12=9x2-p then the value of p is
(a) 0
(b) -14
(c) 14
(d) 12

Answer:

3x+12 3x-12=9x2-p
9x2-14=9x2-p                    a2-b2=a+ba-bp=14
Hence, the correct answer is option (c).

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Question 6:

The coefficient of x in the expansion of (x + 3)3 is
(a) 1
(b) 9
(c) 18
(d) 27

Answer:

(x + 3)3
=x3+33+9xx+3=x3+27+9x2+27x
So, the coefficient of x in (x + 3)is 27.
Hence, the correct answer is option (d). 

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Question 7:

Which of the following is a factor of (x + y)3 – (x3 + y3)?
(a) x2 + y2 + 2xy
(b) x2 + y2xy
(c) xy2
(d) 3xy

Answer:

(x + y)3 – (xy3)
=x3+y3+3xyx+y-x3+y3=3xyx+y
Thus, the factors of (x + y)3 – (xy3) are 3xy and (x + y).
Hence, the correct answer is option (d). 
 

Page No 139:

Question 8:

One of the factors of 25x2-1+1+5x2 is
(a) 5 + x
(b) 5 – x
(c) 5x – 1
(d) 10x
 

Answer:

25x2-1+1+5x2=5x-15x+1+1+5x2=5x+15x-1+1+5x=5x+110x
So, the factors of 25x2-1+1+5x2 are (5x + 1) and 10x
Hence, the correct answer is option (d). 

Page No 139:

Question 9:

If (x + 5) is a factor of p(x) = x3 − 20x + 5k, then k = ?
(a) −5
(b) 5
(c) 3
(d) −3

Answer:

(b) 5

x+5 is a factor of px=x3-20x+5k. p-5=0-53-20×-5+5k=0-125+100+5k=05k=25k=5

Page No 139:

Question 10:

If (x + 2) and (x − 1) are factors of (x3 + 10x2 + mx + n), then
(a) m = 5, n = −3
(b) m = 7, n = −18
(c) m = 17, n = −8
(d) m = 23, n = −19

Answer:

(b) m = 7, n = −18

Let:
px=x3+10x2+mx+n
Now,
x+2=0x=-2
(x + 2) is a factor of p(x).
So, we have p(-2)=0
-23+10×-22+m×-2+n=0-8+40-2m+n=032-2m+n=02m-n=32                            .....i
Now,
x-1=0x=1
Also, 
(x - 1) is a factor of p(x).
We have:
p(1) = 0
13+10×12+m×1+n=01+10+m+n=011+m+n=0m+n=-11                              .....iiFrom i and ii, we get:3m=21m=7
By substituting the value of m in (i), we get n = −18.
∴ m = 7 and n = −18

Page No 139:

Question 11:

104 × 96 = ?
(a) 9864
(b) 9984
(c) 9684
(d) 9884

Answer:

(b) 9984

104×96=100+4100-4
         =1002-42=10000-16=9984

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Question 12:

305 × 308 = ?
(a) 94940
(b) 93840
(c) 93940
(d) 94840

Answer:

(c) 93940

305×308=300+5×300+8             =3002+300×5+8+5×8             =90000+3900+40             =93940

Page No 139:

Question 13:

207 × 193 = ?
(a) 39851
(b) 39951
(c) 39961
(d) 38951

Answer:

(b) 39951

  207×193=200+7200-7=2002-72=40000-49=39951

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Question 14:

4a2 + b2 + 4ab + 8a + 4b + 4 = ?
(a) (2a + b + 2)2
(b) (2ab + 2)2
(c) (a + 2b + 2)2
(d) none of these

Answer:

(a) (2a + b + 2)2

    4a2+b2+4ab+8a+4b+4=4a2+b2+4+4ab+4b+8a=2a2+b2+22+2×2a×b+2×b×2+2×2a×2=2a+b+22

Page No 139:

Question 15:

(x2 − 4x − 21) = ?
(a) (x − 7)(x − 3)
(b) (x + 7)(x − 3)
(c) (x − 7)(x + 3)
(d) none of these

Answer:

(c) (x − 7)(x + 3)

x2-4x-21=x2-7x+3x-21
             =xx-7+3x-7=x-7x+3

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Question 16:

(4x2 + 4x − 3) = ?
(a) (2x − 1) (2x − 3)
(b) (2x + 1) (2x − 3)
(c) (2x + 3) (2x − 1)
(d) none of these

Answer:

(c) (2x + 3) (2x − 1)

4x2+4x-3=4x2+6x-2x-3
             =2x2x+3-12x+3=2x+32x-1

Page No 139:

Question 17:

6x2 + 17x + 5 = ?
(a) (2x + 1)(3x + 5)
(b) (2x + 5)(3x + 1)
(c) (6x + 5)(x + 1)
(d) none of these

Answer:

(b) (2x + 5)(3x + 1)

6x2+17x+5=6x2+15x+2x+5
              =3x2x+5+12x+5=2x+53x+1

Page No 139:

Question 18:

(x + 1) is a factor of the polynomial
(a) x3 − 2x2 + x + 2
(b) x3 + 2x2 + x − 2
(c) x3 − 2x2 − x − 2
(d) x3 − 2x2 − x + 2

Answer:

(c) x3 − 2x2 − x − 2

Let:
fx=x3-2x2+x+2
By the factor theorem, (x + 1) will be a factor of f (x) if f (−1) = 0.
We have:
f-1=-13-2×-12+-1+2        =-1-2-1+2        =-20
Hence, (x + 1) is not a factor of fx=x3-2x2+x+2.

Now,
Let:
fx=x3+2x2+x-2

By the factor theorem, (x + 1) will be a factor of f (x) if f (-1) = 0.
We have:
f-1=-13+2×-12+-1-2        =-1+2-1-2        =-20
Hence, (x + 1) is not a factor of fx=x3+2x2+x-2.

Now,
Let:
fx=x3+2x2-x-2

By the factor theorem, (x + 1) will be a factor of f (x) if f (-1) = 0.
We have:
f-1=-13+2×-12--1-2        =-1+2+1-2        =0
Hence, (x + 1) is a factor of fx=x3+2x2-x-2.



Page No 140:

Question 19:

3x3 + 2x2 + 3x + 2 = ?
(a) (3x − 2)(x2 − 1)
(b) (3x − 2)(x2 + 1)
(c) (3x + 2)(x2 − 1)
(d) (3x + 2)(x2 + 1)

Answer:

(d) (3x + 2)(x2 + 1)

3x3+2x2+3x+2=x23x+2+13x+2
                   =3x+2x2+1

Page No 140:

Question 20:

If a + b + c = 0, then a2bc+b2ca+c2ab=?

Answer:

(d) 3

a+b+c=0a3+b3+c3=3abc

Thus, we have:
a2bc+b2ca+c2ab=a3+b3+c3abc
                     =3abcabc=3

Page No 140:

Question 21:

If x + y + z = 9 and xy + yz + zx = 23, then the value of (x3 + y3 + z3 − 3xyz) = ?
(a) 108
(b) 207
(c) 669
(d) 729

Answer:

(a) 108

x3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zx
                   =x+y+zx+y+z2-3xy+yz+zx=9×81-3×23=9×12=108

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Question 22:

If ab+ba=-1 then (a3b3) = ?
(a) −3
(b) −2
(c) −1
(d) 0

Answer:

    ab+ba=-1a2+b2ab=-1
a2 + b2 = -ab
a2 + b2 + ab = 0

Thus, we have:
a3-b3=a-ba2+b2+ab
         =a-b×0=0



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