Rs Aggarwal 2019 2020 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 2020 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 2020 Solutions. All Rs Aggarwal 2019 2020 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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