Exercise 7.1
Find the greatest common factors (GCF / HCF) of the following polynomials : (1 – 14)
Question 1.
2x2 and 12x2
Solution:
2x2 and 12x2
HCF of 2 and 12 =2
HCF of x2,x2=x2
∴ HCF = 2x2
Question 2.
(6xy3 and 18x2y3
Solution:
6x3y and 18xy
HCF of 6, 18 = 6
HCF of x3 and x2 = x2
HCF of y and y3 -y
∴ HCF = 6x2y
Question 3.
7x, 21x2 and 14xy2
Solution:
7x, 21x2 and 14xy2
HCF of 7, 21 and 14 = 7
HCF of x, x2, x = x
∴ HCF = 7x
Question 4.
42x2yz and 63x3y2z3
Solution:
42x2yz and 63x3y2z3
HCF of 42 and 63 = 21
HCF of x2, x3 = x2
HCF of y,y2=y
HCF of z,z3 = z
∴ HCF = 21 x2yz
Question 5.
12ax2,6a2x3 and 2a3 x5
Solution:
12ax2, 6a2x3 and 2a3x5
HCF of 12, 6,2 = 2
HCF of a, a2, a3 = a
HCF of x2, x3, x5 = x2
∴ HCF = 2ax2
Question 6.
9x2, 15x2y3, 6xy2 and 21x2y2
Solution:
9x2, 15xV, 6xy2 and 21x2y2
HCF of 9, 15, 6,21 = 3
HCF of x2, x2, x, x2 = x
HCF of 1, y3, y2, y2 =2
∴ HCF = 3x
Question 7.
4a2b3 -12a3b, 18a4b3
Solution:
4a2b3, -12a3b, 18a4b3
HCF of 4, 12, 18 = 2
HCF of a2, a3, a4 = a2
HCF of b3,b, b3 = b
∴ HCF = 2a2b
Question 8.
6x2y2, 9xy3, 3x3y2
Solution:
6x2y2, 9xy3, 3x3y2
HCF of 6, 9, 3 = 3
HCF of x2, x, x3 = x
HCF of y2,y3,y2=y2
∴ HCF = 3xy2
Question 9.
a2b3, a3b2
Solution:
a2b3, a3b2
HCF of a2, a3 = a2
HCF of b3, b2 = b2
∴ HCF = a2b2
Question 10.
36a2b2c4, 54a5c2,90a4b2c2
Solution:
36a2b2c4, 54a5c2,90a4b2c2
HCF of 36, 54, 90 = 18
HCF of a2, a5, a4 = a2
HCF of b2, 1,b2= 1
HCF of c4,c2,c2 = c2
∴ HCF = 18a2 x 1 x c2 = 18a2c2
Question 11.
x3, – yx2
Solution:
x3, – yx2
HCF of x3, x2 = x2
HCF of 1, y= 1
∴ HCF = x2
Question 12.
15a3, -45a2, -150a
Solution:
15a3,-45a2,-150a
HCF of 15,45, 150 = 15
HCF of a3, a2, a = a
∴ HCF = 15a
Question 13.
2x3y2, 10x2y3, 14xy
Solution:
2x3y2, 10x2y3, 14xy
HCF of 2, 10, 14 = 2
HCF of x3, x2, x = x
HCF of y2,y3,y=y
∴ HCF = 2xy
Question 14.
14x3y5, 10x5y3, 2x2y2
Solution:
14x3y5, 10x5y3, 2x2y2
HCF of 14, 10, 2, = 2
HCF of x3, x5, x2 = x2
HCF of y5,y3,y2=y2
∴ HCF = 2xy
Find the greatest common factor of the terms in each of the following expressions:
Question 15.
5a4 + 10a3 – 15a2
Solution:
5a4 + 10a3– 15a2
HCF of 5, 10, 15 = 5
HCF of a4, a3, a2 = a2
∴ HCF = 5a2
Question 16.
2xyz + 3x2y + 4y2
Solution:
2xyz + 3x2y + 4y2
HCF of 2, 3,4 = 1
HCF of x, x2, 1 = 1
HCF of y,y,y2 =y
HCF of z, 1, 1 = 1
∴ HCF = y
Question 17.
3a2b2 + 4b2c2 + 12a2b2c2
Solution:
3a2b2 + 4b2c2 + 12a2b2c2
HCF of 3, 4, 12 = 1
HCF of a2, 1, a2 = 1
HCF of b2, b2, b2 = b2
HCF of 1, c2, c2 = 1
∴ HCF = b2
Exercise 7.2
Factorize the following :
Question 1.
3x-9
Solution:
3x – 9 = 3 (x – 3) (HCF of 3, 9 = 3)
Question 2.
5x – 15x2
Solution:
5x- 15x2 = 5x (1 – 3x)
{HCF of 5, 15 = 5 and of x, x2 = x}
Question 3.
20a12b2 – 15a8b4
Solution:
20a12b2 – 15a8b4
{HCF of 20, 15 = 5, a12, a8 = a8, b2, b4 = b2}
= 5a8 b2(4a4 – 3b2)
Question 4.
72xy – 96x7y6
Solution:
72xy – 96x7y6
HCF of 72, 96 = 24 of x6x7 = x6, y7,y6 = y6
∴ 72x7y6 – 96x7y6 = 24x6y6 (3y – 4x)
Question 5.
20X3 – 40x2 + 80x
Solution:
20x3 – 40x2 + 80x
HCF of 20, 40,80 = 20
HCF of x3, x2, x = x
∴ 20x3 – 40x2 + 80x = 20x (x2 – 2x + 4)
Question 6.
2x3y2 – 4x2y3 + 8xy4
Solution:
2x3y2 – 4x2y3 + 8xy4
HCF of 2, 4, 8 = 2
HCF of x3, x2, x = 1
and HCF of y2, y3, y4 = y2
∴ 2x3y2 – 4x2y3 + 8xy4
= 2xy2 (x2 – 2xy + 4y2)
Question 7.
10m3n2 + 15m4n – 20m2n3
Solution:
10m3n2 + 15m4n – 20m2n3
HCF of 10, 15, 20 = 5
HCF of m3, m4, m2 = m2
HCF of n2, n, n3 = n
10m3n2 + 15m4n – 20m2n3
5m2n(2mn + 3m2– 4n2)
Question 8.
2a4b4 – 3a3b5 + 4a2b5
Solution:
2a4b4 – 3a3b5 + 4a2b5
HCF of 2, 3, 4= 1
HCF of a4, a3, a2 = a2
HCF of b4, b5 b5 = b4
∴ 2a4b4 – 3a3b5 + 4a2b5 = a2b4
(2a2 – 3ab + 4b)
Question 9.
28a2 + 14a2b2 – 21a4
Solution:
28a2 + 14a2b2 – 21a4
HCF of 28, 14,21 =7
HCF of a2, a2, a4 = a2
HCF of 1, b2, 1 = 1
∴ 28a2 + 14a2b2-21a4 = 7a2
(4 + 2b2 – 3a2)
Question 10.
a4b – 3a2b2 – 6ab3
Solution:
a4b – 3a2b2 – 6ab3
HCF of 1,3,6 = 1
HCF of a4, a2, a = a
HCF of b, b2, b3 = b
∴ a4b – 3a2b2 – 6ab3 = ab (a3 – 3ab – 6b2)
Question 11.
2l2mn – 3lm2n + 4lmn2
Solution:
2l2mn – 3lm2n + 4lmn2
HCF 2, 3,4 = 1,
HCF of l2,l,l = l
HCF of m, m2, m = m
HCF of n, n, n2 = n
∴ 2l2 mn – 3lm2n + 4lmn2
= lmn (21 -3m + 4n)
Question 12.
x4y2 – x2y4 – x4y4
Solution:
x4y2 – x2y4 – x4y4
HCF of x4, x2, x4 = x2
HCF of y2, y4, y4 =y2
∴ x4y2 – x2y4 – x4y4 = x2y2 (x2 -y2 -x2y2)
Question 13.
9 x2y + 3 axy
Solution:
9 x2y + 3 axy
HCF of 9, 3 = 3
HCF of x2, x = x
HCF of y,y = y
HCF of 1,a = 1
∴ 9x2y + 3axy = 3xy (3x + a)
Question 14.
16m – 4m2
Solution:
16m – 4m2
HCF of 16, 4 = 4
HCF of m, m2 = m
∴ 16m – 4m2 = 4m (4 – m)
Question 15.
-4a2 + 4ab – 4ca
Solution:
-4a2 + 4ab – 4ca
HCF of 4, 4, 4 = 4
HCF of a2, a, a = a
∴ -4a2 + 4ab – 4ca = -4a (a – b + c)
Question 16.
x2yz + xy2z + xyz2
Solution:
x2yz + xy2z + xyz2
HCF of x2, x, x = x
HCF of y,y2,y=y
HCF of z, z,z2 = z
∴ x2yz + xy2z + xyz2 = xyz (x + y + z)
Question 17.
ax2y + bxy2 + cxyz
Solution:
ax2y + bxy2 + cxyz
HCF of x2, x, x = x,
HCF of y,y2,y = y
ax2y + bxy2 + cxyz = xy (ax + by + cz)
Exercise7.3
Factorize each of the following algebraic expressions.
Question 1.
6x (2x – y) + 7y (2x – y)
Solution:
6x (2x – y) + 7y (2x – y)
= (2x – y) (6x + 7y)
[∵ (2x – y) is common]
Question 2.
2r (y – x) + s (x – y)
Solution:
2r (y – x) + s (x – y)
-2r (x – y) +s (x – y)
= (x – y) (-2r + s) [(x – y) is common]
= (x-y) (s-2r)
Question 3.
7a (2x – 3) + 2b (2x – 3)
Solution:
7a (2x – 3) + 3b (2x – 3)
= (2x – 3) (7a + 3b) [(2x – 3) is common]
Question 4.
9a (6a – 5b) – 12a2 (6a – 5b)
Solution:
9a (6a – 5b) – 12a2 (6a – 5b)
HCF of 9 and 12 = 3
∴ 3a (6a – 5b) (3 – 4a)
{(6a – 5b) is common}
Question 5.
5 (x – 2y)2 + 3 (x – 2y)
Solution:
5 (x – 2y)2 + 3 (x – 2y)
= 5 (x – 2y) (x – 2y) + 3 (x – 2y)
= (x – 2y) {5 (x – 2y) + 3}
{(x – 2y) is common}
= (x – 2y) (5x – 10y + 3)
Question 6.
16 (2l – 3m)2 – 12 (3m – 2l)
Solution:
16 (2l – 3m)2 – 12 (3m-2l)
= 16 (2l – 3m) (2l – 3m) + 12 (2l – 3m)
HCF of 16, 12 = 4 4 (2l-3m) {4 (2l- 3m) + 3}
{(2l – 3m) is common}
= 4 (2l -3m) (8l- 12m+ 3)
Question 7.
3a (x – 2y) – b (x – 2y)
Solution:
3a (x – 2y) – b (x – 2y)
= (x – 2y) (3a – b)
{(x – 2y) is common}
Question 8.
a2 (x + y) + b2 (x + y) + c2 (x + y)
Solution:
a2 (x + y) + b2 (x + y) + c2 (x + 3’)
= (x + y) (a2 + b2 + c2)
{(x + y) is common}
Question 9.
(x-y)2 + (x -y)
Solution:
(x – y)2 + (x- y) = (x – y) (x – y) + (x – y)
= (x – y) (x – y + 1) {(a – y) is common}
Question 10.
6 (a + 2b) – 4 (a + 2b)2
Solution:
6 (a + 2b) – 4 (a + 2b)2
= 6 (a + 2b) – 4 (a + 2b) (a + 2b)
HCF of 6, 4 = 2
= 2 {a + 2b) {3 – 2 {a + 2b)
{2 (a + b) is common}
= 2 (a + 2b) (3-2 a- 4b)
Question 11.
a (x -y) + 2b (y – x) + c (x -y)2
Solution:
a (x -y) + 2b (y – x) + c (x -y)2
= a (x – y) – 2b (x – y) + c (x – y) {x – y)
= (x – y) {x – 2b + c (x – y)}
{(a – y) is common}
= (a – y) (a – 2b + cx – cy)
Question 12.
– 4 (a – 2y)2 + 8 (a – 2y)
Solution:
– 4 (x – 2y)2 + 8 (x – 2y)
= – 4 (x – 2y) (x – 2y) + 8 (x – 2y)
{- 4 (x – 2y) is common}
= – 4 (x – 2y) (x – 2y – 2)
= 4 (x – 2y) (2 – x + 2y)
Question 13.
x3 (a – 2b) + a2 (a – 2b)
Solution:
x3 (a – 2b) + x2 (a – 2b)
HCF of x3, x2 = x2
∴ x2 (a – 2b) (x + 1)
{x2 (x – 2b) is common}
= x2 (x – 2b) (x + 1)
Question 14.
(2x – 3y) (a + b) + (3x – 2y) (a + b)
Solution:
(2x – 3y) (a + b) + (3x – 2y) (a + b)
= (a + b) {2x – 3y + 3x – 2y}
{(x + b) is common}
= (a + b) (5x – 5y)
= 5 (a + b) (x – y)
Question 15.
4 (x + y) (3a – b) + 6 (a + y) (2b – 3a)
Solution:
4 (x + y) (3a – b) + 6 (a + y) (2b – 3a)
= 4 (x + y) (3a – b) – 6 (x + y) (3a – 2b)
HCF of 4, 6 = 2
= 2 (x + y) {2 (3a – b) – 3 (3a – 2b)}
= 2 (x + 3) {6a – 2b – 9a + 6b}
= 2 (x +y) {-3a + 4b}
= 2 (x + y) (4b – 3a)
Exercise 7.4
Factorize each of the following expressions :
Question 1.
qr-pr + qs – ps
Solution:
qr- pr + qs-ps
Arranging in suitable groups = r(q-p) +s (q-p) {(q – p) is common}
= (q-p) (r + s)
Question 2.
p2q -pr2-pq + r2
Solution:
p2q -pr2-pq + r2
= p2q -pq-pr2 + r2 (Arranging in group)
= pq(p- 1)-r2(p-1) {(p – 1) is common}
= (p – 1) (pq – r2)
Question 3.
1 + x + xy + x2y
Solution:
1 + x + xy + x 2y
= 1 (1 + x) +xy (1 +x)
= (1 + x) (1 + xy) {(1 + x) is common}
Question 4.
ax + ay – bx – by
Solution:
ax + ay – bx – by
= a (x + y) – b (x + y) {(x + y) is coinmon}
= (x+y) (a- b)
Question 5.
xa2 + xb2 -ya2 – yb2
Solution:
xa2 + xb2 – ya2 – yb2
= x (a2 + b2) -y (a2 + b2) {(a2 + b2) is common}
= {a2 + b2) (x -y)
Question 6.
x2 + xy + xz + yz
Solution:
x2 + xy + xz + yz
= x (x + y) + z(x + y) {(x + y) is common}
= (x + y) (x + z)
Question 7.
2ax + bx + 2ay + by
Solution:
2ax + bx + 2ay + by
= x {2a + b) + y (2a + b) {(2a + b) is common}
= (2a + b) (x + y)
Question 8.
ab- by- ay +y2
Solution:
ab – by – ay + y2
= b(a-y)-y(a-y) {(a -y) is common}
= (a-y) (b – y)
Question 9.
axy + bcxy -az- bcz
Solution:
axy + bcxy – az – bcz
= xy (a + bc) – z (a + bc) {(a + bc) is common}
= (a + bc) (xy – z)
Question 10.
lm2 – mn2 – lm + n2
Solution:
lm2 – mn2 – lm + n2
= m (lm – n2)- 1 (lm – n2) {(lm – n2) is common}
= (lm – n2) (m – 1)
Question 11.
x3 – y2 + x – x2y2
Solution:
x3 -y2 + x – x2y2
⇒ x3 + x – x2y2 – y2
= x(x2+ 1)-y2(x2+ 1) {(x2 + 1) is common}
= (x2 + 1) (x -y2)
Question 12.
6xy + 6 – 9y – 4x
Solution:
6xy + 6 – 9y – 4x
= 6 xy – 4x – 9y + 6
= 2x (3y – 2) – 3 (3y – 2) {(3y – 2) is common}
= (3y-2) (2x – 3)
Question 13.
x2 – 2ax – 2ab + bx
Solution:
x2 – 2ax – 2ab + bx
⇒ x2 – 2ax + bx – 2ab
= x (x – 2a) + b (x – 2a) {(x – 2a) is common}
= (x – 2a) (x + b)
Question 14.
x3 – 2x2y + 3xy2 – 6y3
Solution:
x3 – 2x2y + 3xy2 – 6y3
= x2 (x – 2y) + 3y2 (x – 2y) {(x – 2y) is common}
= (x – 2y) (x2 + 3y2)
Question 15.
abx2 + (ay – b) x-y
Solution:
abx2 + (ay – b) x-y
= abx2 + ayx – bx -y
= ax (bx + y) – 1 (bx + y) {(bx +y) is common}
= (bx + y) (ax – 1)
Question 16.
(ax + by)2 + (bx – ay)2
Solution:
(ax + by)2 + (bx – ay)2
= a2x2 + b2y2 + 2abxy + b2x2 + a2y2 – 2abxy
= a2x2 + b2y2 + b2x2 + a2y2
= a2x2 + b2x2 + a2y2 + by2
= x2 (a2 + b2) + y2 (a2 + b2) {(a2 + b2) is common}
= (a2 + b2) (x2 + y2)
Question 17.
16 (a – b)3 -24 (a- b)2
Solution:
16 (a – b)3 -24 (a- b)2
HCF of 16, 24 = 8
and HCF of (a – b)3, (a – b)2 = (a – b)2
∴16 (a – b)3 – 24 (a – b)2
= 8 (a-b)2 {2 (a-b)- 3}
{8 (a – b)2 is common}
= 8 (a – b)2 (2a – 2b – 3)
Question 18.
ab (x2 + 1) + x (a2 + b2)
Solution:
ab (x2 + 1) + x (a2 + b2)
= abx2 + ab + a2x + b2x
= abx2 + b2x + a2x + ab
= bx (ax + b) + a (ax + b) {(ax + b) is common}
= (ax + b) (bx + a)
Question 19.
a2x2 + (ax2 + 1) x + a
Solution:
a2x2 + (ax2 + 1) x + a
= a2x2 + ax3 + x + a
= ax3 + a2x2 + x + a
= ax2 (x + a) + 1 (x + a) {(x + a) is common}
= (x + a) (ax2 + 1)
Question 20.
a(a- 2b -c) + 2bc
Solution:
a(a- 2b -c) + 2bc
= a2– 2ab -ac +2bc
= a (a – 2b) – c (a – 2b) {(a – 2b) is common}
= (a – 2b) (a – c)
Question 21.
a (a + b – c)- bc
Solution:
a (a + b – c) – bc
= a2 + ab – ac – bc
= a (a + b) – c (a + b) {(a + b) is common}
= (a + b) (a – c)
Question 22.
x2 – 11xy – x +11y
Solution:
x2 – 11xy-x + 11y
= x2 -x – 11 xy + 11 y
= x (x – 1) – 11y (x – 1) {(x – 1) is common}
= (x- 1) (x- 11y)
Question 23.
ab – a – b + 1
Solution:
ab – a-b + 1
= a (b – 1) – 1 (b – 1) {(b – 1) is common}
= (b – 1) (a – 1)
Question 24.
x2 + y – xy – x
Solution:
x2 + y – xy – x
= x2 – x- xy + y
= x (x – 1) – y (x – 1) {(x – 1) is common}
= (x- 1) (x-y)
Exercise 7.5
Factorize each of the following expressions :
Question 1.
16x2-25y2
Solution:
16x2 – 25y2 = (4x)2 – (5y)2 {∵ a2 – b2 = (a + b) (a – b)}
= (4x + 5y) (4x – 5y)
Question 2.
27x2 – 12y2
Solution:
27x2 – 12y2 = 3 (9x2 – 4y2) {∵ a2 -b2 = (a + b) (a – b)}
= 3 [(3x)2 – (2y)2]
= 3 (3x + 2y) (3x – 2y)
Question 3.
144a2 – 289b2
Solution:
144a2 – 289b2 = (12a)2 – (17b)2 { ∵ a2 – b2 = (a + b) (a – b}
= (12a+ 17b) (12a- 17b)
Question 4.
12m2 – 27
Solution:
12m2 – 27 = 3 (4m2 – 9)
= 3 {(2m)2-(3)2} {∵ a2 – b2 = (a + b) (a – b)}
= 3 (2m + 3) (2m – 3)
Question 5.
125x2 – 45y2
Solution:
125x2 – 45y2 = 5 (25x2 – 9y2)
= 5 {(5x-)2 – (3y)2} {∵ a2 – b2 = (a + b) (a – b}
= 5 (5x + 3y) (5x – 3y)
Question 6.
144a2 – 169b2
Solution:
144a2 – 169b2 = (12a)2 – (13b)2 {∵ a2 -b2 = (a + b) (a – b)}
= (12a + 13b) (12a-13b)
Question 7.
(2a – b)2 – 16c2
Solution:
(2a – b)2 – 16c2 = (2a – b)2 – (4c)2 {∵ a2 – b2 = (a + b) (a – b)}
= (2a – b + 4c) (2a – b – 4c)
Question 8.
(x + 2y)2 – 4 (2x -y)2
Solution:
(x + 2y)2 – 4 (2x – y)2
= (x + 2y)2 – {2 (2x –y)}2
= (x + 2y)2 – (4x – 2y)2 {∵ a2– b2 = (a + b) (a – b)}
= (a + 2y + 4x – 2y) (x + 2y – 4x + 2y)
= 5x (-3x + 4y)
Question 9.
3a5 – 48a3
Solution:
3a5 – 48a3 = 3a3 (a2– 16)
= 3a3 {(a)2 – (4)2} {∵ a2 – b2 = (a + b) (a – b)}
= 3a3 (a + 4) (a – 4)
Question 10.
a4 – 16b4
Solution:
a4 – 16b4 = (a2)2 – (4b2)2
= (a2 + 4b2) (a2 – 4b2)
= (a2 + 4b2) {(a)2 – (2b)2 } { ∵ a2 – b2 = (a + b) (a – b)}
= (a2 + 4b2) (a + 2b) (a – 2b)
Question 11.
x8 – 1
Solution:
x8 – 1 = (x4)2 – (1)2
= (x4 + 1) (x4 – 1)
= (x4+ 1) I (x2)2 – (1)2} {∵ a2 – b2 = (a + b) (a – b)}
= (x4 + 1) (x2 + 1) (x2 – 1)
= (x4 + 1) (x2 + 1) {(x)2 – (1)2}
= (x4+ 1)(x2 + 1)(x+ 1)(x- 1)
= (x-1)(x+ 1) (x2 + 1) (x4 + 1)
Question 12.
64 – (a + 1)2
Solution:
64 – (a + 1)2 = (8)2 – (a + 1)2 {∵ a2 – b2 = (a + b) (a – b)}
= (8 + a + 1) (8 – a – 1)
= (9 + a) (7 – a)
Question 13.
36l2 – (m + n)2
Solution:
36l2 – (m + n)2 = (6l)2 – (m + n)2 {∵ a2 – b2 = (a + b) (a – b)}
= (6l + m + n) (6l – m – n)
Question 14.
25x4y4 – 1
Solution:
25x4y4 – 1 = (5x4y4)2 – (1)2 { ∵ a2 – b2 = (a + b) (a – b)}
= (5x4y4 + 1) (5x2y2 – 1)
Question 15.
Solution:
Question 16.
x3 – 144x
Solution:
x3 – 144x = x (x2 – 144)
= x {(x)2 – (12)2} {∵ a2 – b2 = (a + b) (a – b)}
= x (x + 12) (x – 12)
Question 17.
(x – 4y)2 – 625
Solution:
(x – 4y)2 – 625
= (x – 4y)2 – (25)2 {∵ a2 – b2 = (a + b) (a – b)}
= (x – 4y + 25) (x -4y – 25)
Question 18.
9 (a – b)2 – 100 (x -y)2
Solution:
9(a-b)2– 100(x-y)2
= {3(a-b)}2-{10(x-y)}2 {∵ a2 – b2 = (a + b) (a – b)}
= (3a – 3b)2 – (10x – 10y)2
= (3a – 3b + 10x – 10y) (3a – 3b – 10x + 10y)
Question 19.
(3 + 2a)2 – 25a2
Solution:
(3 + 2a)2 – 25a2
= (3 + 2a)2 – (5a)2 (∵ a2 – b2 = (a + b) (a – b)}
= (3 + 2a + 5a) (3 + 2a – 5a)
= (3 + 7a) (3 – 3a)
= (3 + 7a) 3 (1 – a)
= 3(1-a) (3 +7a)
Question 20.
(x + y)2 – (a – b)2
Solution:
Question 21.
Solution:
Question 22.
75a3b2 – 108ab4
Solution:
75a3b2 – 108ab4
= 3ab2 (25a2 – 36b2)
= 3ab2 {(5a)2 – (6b)2} {∵ a2 – b2 = (a + b) (a – b)}
= 3ab2 (5a + 6b) (5a – 6b)
Question 23.
x5– 16x3
Solution:
x5 – 16x3 = x3 (x2 – 16)
= x3 {(x)2 – (4)2} {∵ a2 – b2 = (a + b) (a – b)}
= x3 (x + 4) (x – 4)
Question 24.
Solution:
Question 25.
256x5 – 81x
Solution:
256x5– 81x = x(256x4– 81)
= x {(16x2)2 – (9)2} {∵ a2 – b2 = {a + b) (a – b)}
= x (16x2 + 9) (16x2 – 9)
= x (16x2 + 9) {(4x)2 – (3)2}
= x (16x2 + 9) (4x + 3) (4x-3)
Question 26.
a4 – (2b + c)4
Solution:
a4 – (2b + c)4
= (a2)2 – [(2b + c)2]2 {∵ a2 – b2 = (a + b) (a – b)}
= {a2 + (2b + c)2} {a2 – (2b + c)2}
= {a2 + (2b + c)2} {(a)2 – (2b + c)2}
= {a2 + (2b + c)2} (a + 2b + c) (a -2b- c)
Question 27.
(3x + 4y)4 – x4
Solution:
(3x + 4y)4 – x4 – [(3x + 4y)2]2 – (x2)2
= [(3x + 4y)2 + x2] [(3x + 4y)2 – x2] {∵ a2 – b2 = (a + b) (a – b)
= [(3x + 4y)2 + x2] [(3x + 4y + x) (3x + 4y – x)]
= [(3x + 4y)2 + x2] (4x + 4y) (2x + 4y)
= [(3x + 4y)2 + x2] 4 (x + y) 2 (x + 2y)
= 8 (x + y) (x + 2y) [(3x + 4y)2 + x2]
Question 28.
p2q2 – p4q4
Solution:
p2q2– p4q4 =p2q2 (1 -p2q2)
=p2q2 [(1)2 – (pq)2] {∵ a2 – b2 = (a + b) (a – b)
= p2q2 (1 +pq) (1 -pq)
Question 29.
3x3y – 243xy3
Solution:
3x3y – 243xy3
= 3xy (x2 – 81y2)
= 3xy [(x)2 – (9y)2]
= 3xy (x + 9y) (x – 9y)
Question 30.
a4b4 – 16c4
Solution:
a4b4 – 16c4 = (a2b2)2 – (4c2)2
= (a2b2 + 4c2) (a2b2 – 4c2)
= (a2b2 + 4c2) [(ab)2 – (2c)2] {∵ a2 – b2 = (a + b) (a – b)
= (a2b2 + 4c2) (ab + 2c) (ab – 2c)
Question 31.
x4-625
Solution:
x4 – 625 = (x2)2 – (25)2 {∵ a2 – b2 – (a + b) (a – b)
= (x2 + 25) (x2 – 25)
= (x2 + 25) [(x)2 – (5)2]
= (x2 + 25) (x + 5) (x – 5)
Question 32.
x4-1
Solution:
x4 – 1 = (x2)2 – (1)2 = (x2 + 1) (x2 – 1)
= (x2 + 1) [(x)2 – (1)2]
= (x2 + 1) (x + 1) (x – 1)
Question 33.
49 (a – b)2 -25 (a + b)2
Solution:
49 (a – by -25 (a + b)2
= [7 (a – b)]2 – [5 (a + b)]2
= (7a – 7b)2 – (5a + 5b)2 {∵ a2 – b2 = (a + b) (a – b)
= (7a -7b + 5a + 5b) (7a – 7b -5a- 5b)
=(12a – 2b)(2a – 12b)
= 2 (6a – b) 2 (a – 6b)
= 4 (6 a- b) (a – 6b)
Question 34.
x – y – x2 + y2
Solution:
x-y-x2 + y2 = (x-y)-(x2-y2) {∵ a2 – b2 = (a + b) (a – b)
= {x-y)-(x + y)(x-y)
= (x-y)(1 – x – y)
Question 35.
16 (2x – 1)2 – 25y2
Solution:
16 (2x – 1)2 – 25y2
= [4 (2x – 1)]2 – (5y)2
= (8x – 4)2 – (5y)2
= (8x – 4 + 5y) (8x -4-5y)
= (8x + 5y – 4) (8x – 5y – 4)
Question 36.
4 (xy + 1)2 – 9 (x – 1)2
Solution:
4 (xy + 1)2 – 9 (x – 1)2
= [2 (xy + 1)]2 – [3 (x – 1)]2
= (2xy + 2)2 – (3x – 3)2 {∵ a2 – b2 = (a + b) (a – b)
= (2xy + 2 + 3x – 3) (2xy + 2 – 3x + 3)
= (2xy + 3x – 1) (2xy – 3x + 5)
Question 37.
(2x + 1)2 – 9x4
Solution:
(2x + 1)2 – 9x4 = (2x + 1)2 – (3x2)2 {∵ a2 – b2 = (a + b) (a – b)
= (2x + 1 + 3x2) (2x + 1 – 3x2)
= (3x2 + 2x + 1) (-3x + 2x + 1)
Question 38.
x4 – (2y- 3z)2
Solution:
x4 – (2y – 3z)2 = (x2)2 – (2y – 3z)2
= (x2 + 2y- 3z) (x2 – 2y + 3z)
Question 39.
a2-b2 +a-b
Solution:
a2 – b2 + a – b
= (a + b) {a – b) + 1 (a – b)
= (a – b) (a + b + 1)
Question 40.
16a4 – b4
Solution:
16a4 – b4
= (4a2)2 – (b2)2 { ∵ a2 – b2 = (a + b) (a – b)
= (4a2 + b2) (4a2 – b2)
= (4a2 + b2) {(2a)2 – (b)2}
= (4a2 + b2) (2a + b) (2a – b)
Question 41.
a4 – 16 (b – c)4
Solution:
a4 – 16 (b- c)4 = (a2)2 – [4 (b – c)2]2 { ∵ a2 – b2 = (a + b) (a – b)
= [a2 + 4 (b – c)2] [a2 – 4 (b – c)2]
= [a2 + 4 (b – c)2] [(a)2 – [2 (b – c)]2]
= [a2 + 4 (b – c)2] [(a)2 – (2b – 2c)2]
= [a2 + 4 (b – c)2] (a + 2b – 2c) (a – 2b + 2c)
Question 42.
2a5 – 32a
Solution:
2a5 – 32a = 2a (a4 – 16)
= 2a [(a2)2 – (4)2] {∵ a2 – b2 = (a + b) (a – b)
= 2a (a2 + 4) (a2 – 4)]
= 2a (a2 + 4) [(a)2 – (2)2]
= 2a (a2 + 4) (a + 2) (a – 2)
Question 43.
a4b4 – 81c4
Solution:
a4b4 – 81c4 = (a2b2)2 – (9c2)2
= (a2b2 + 9c2) (a2b2 – 9c2) {∵ a2 – b2 = (a + b) (a – b)
= (a2b2 + 9c2) {(ab)2 – (3c)2}
= (a2b2 + 9c2) (ab + 3c) (ab – 3c)
Question 44.
xy9-yx9
Solution:
xy9 – yx9 = xy (y8 – x8)
= xy [(y4)2 – (x4)2] {∵ a2 – b2 = (a + b) (a – b)}
= xy(y4 + x4)(y4-x4)
= xy (y4 + x4) {(y2)2 – (x2)2}
= xy (y4 + x4) (y2 + x2) (y2 – x2)
= xy (y4 + x4) (y2 + x2) (y + x) (y – x)
Question 45.
x3 -x
Solution:
x3-x = x(x2– 1)
= x [(x)2 – (1)2] = x (x + 1) (x – 1)
Question 46.
18a2x2 – 32
Solution:
18a2x2 – 32
= 2 [9a2x2 – 16]
= 2 [(3ax)2 – (4)2] {∵ a2 – b2 = (a + b) (a – b)
= 2 (3ax + 4) (3ax – 4)
Exercise 7.6
Factorize each of the following algebraic expressions :
Question 1.
4x2 + 12xy + 9y2
Solution:
4x2 + 12xy + 9y2 = (2x)2 + 2 x 2x x 3y + (3y)2 {∵ a2 + 2ab + b2 = (a +b)2}
= (2x + 3y)2
Question 2.
9a2 – 24ab + 16b2
Solution:
9a2 – 24ab + 16b2
= (3a)2 – 2 x 3a x 4b + (4b)2 {∵ a2 – 2ab + b2 = (a – b)2}
= (3a – 4b)2
Question 3.
36a2 – 6pqr + 9r2
Solution:
p2q2 – 6pqr + 9r2
= (pq)2 – 2 x pq x3r + (3r)2 {∵ a2 – 2ab + b2 = (a -b)2}
= (pq-3r)2
Question 4.
36a2 + 36a + 9
Solution:
36a2 + 36a + 9
= (6a)2 + 2 x 6a x 3 + (3)2 {∵ a2 + 2ab + b2 = (a + b)2
= (6a + 3)2
Question 5.
a2 + 2ab + b2 – 16
Solution:
a2 + 2ab + b2 – 16
= (a + b)2 – (4)2 {∵ a2 + 2ab + b2 = (a + b)2 and a2 – b2 = (a + b) (a – b)}
= (a + b + 4) (a + b – 4)
Question 6.
9z2 – x2 + 4xy – 4y2
Solution:
9z2 – x2 + 4xy – 4y2 {∵ a2 – b2 = (a + b) (a – b) and a2 – 2ab + b2 (a – b)2}
= 9z2 – (x2 – 4xy + 4y2)
= (3z)2 – [(x)2 – 2 x x x 2y + (2y)2]
= (3z)2-(x-2y)2
= (3z + x – 2y) (3z – x + 2y)
Question 7.
9a4 – 24a2b2 + 16b4 – 256
Solution:
Question 8.
16 – a6 + 4a3b3 – 4b6
Solution:
Question 9.
a2 – 2ab + b2 – c2
Solution:
Question 10.
x2 + 2x + 1 – 9y2
Solution:
Question 11.
a2 + 4ab + 3b2
Solution:
a2 + 4ab + 3b2
= a2 + 4ab+ 4b2 – b2
= (a)2 + 2 x a x 2b + (2b)2 – b2 (∵ 3b2 = 4b2 – b2)
= (a + 2b)2 – (b)2 {∵ a2 – b2 = (a +b) (a – b)}
= (a + 2b + b) (a + 2b- b)
Question 12.
96 – 4x-x2
Solution:
96 – 4x – x2 = 96 – (4x + x2)
= 96 – [(x)2 + 2 x x x 2 + (2)2] + (2)2 (on completing the square)
= 96 + 4 – (x + 2)2 = 100 – (x + 2)2
= (10)2 – (x + 2)2
= (10 + x + 2) (10 – x- 2)
= (x + 12) (-x + 8)
Question 13.
a4 + 3a2 + 4
Solution:
a4 + 3a2 + 4
= (a2)2 + (2)2 + 2 x a2 x 2 – a2 (on completing the square)
= (a2 + 2)2 – (a)2
= (a2 + 2 + a) (a2 + 2 – a)
= (a1 + a + 2) (a2 – a + 2)
Question 14.
4a4 + 1
Solution:
4x4 + 1 = (2a2)2 + (1)2 + 2 x 2x2 x 1 – 2 x 2x2 x 1 (completing the square)
= (2x2 + 1)2 – 4a2
= (2x2 + 1)2 – (2a)2 {a2 – b2 = (a + b) (a – b)}
= (2x2 + 1 + 2a) (2a2 + 1 – 2a)
= (2a2 + 2a + 1) (2a2 – 2a + 1)
Question 15.
4x4+y4
Solution:
4a4 + y4 = (2x2)2 + (y2)2 + 2 x 2x2y2 – 2 x 2x2y2
= (2x2 + y2)2 – 4x2y2
= (2x2 + y2)2 – (2xy)2
= (2x2 + y2 + 2xy) (2x2 + y2 – 2xy)
= (2x2 + 2xy + y2) (2x2 – 2xy + y2)
Question 16.
(x+ 2)2 – 6 (a + 2) + 9
Solution:
(x + 2)2 – 6 (x + 2) + 9
= (x + 2)2 – 2 x (x + 2) x 3 + (3)2
= (x + 2 – 3)2
= (x-1)2 = (x-1)(x-1)
Question 17.
25 – p2 – q2 – 2pq
Solution:
Question 18.
a2 + 9y2 – 6xy – 25a2
Solution:
a2 + 9y2 – 6xy – 25a2
Question 19.
49 – a2 + 8ab – 16b2
Solution:
Question 20.
a2 – 8ab + 16b2 – 25c2
Solution:
Question 21.
x2 -y2+ 6y- 9
Solution:
Question 22.
25x2 – 10x + 1 – 36y2
Solution:
Question 23.
a2-b2 + 2bc – c2
Solution:
Question 24.
a2 + 2ab + b2 -c2
Solution:
Question 25.
49 -x2 – y2 + 2xy
Solution:
Question 26.
a2 + 4b2 – 4ab – 4c2
Solution:
Question 27.
x2 -y2 – 4xz + 4z2
Solution:
Exercise 7.7
Factorize each of the following algebraic expressions :
Question 1.
x2 + 12x – 45
Solution:
Question 2.
40 + 3x – x2
Solution:
Question 3.
a2 + 3a-88
Solution:
Question 4.
a2 – 14a – 51
Solution:
Question 5.
x2 + 14x + 45
Solution:
Question 6.
x2 – 22x + 120
Solution:
Question 7.
x2– 11x – 42
Solution:
Question 8.
a2 + 2a – 3
Solution:
Question 9.
a2 + 14a + 48
Solution:
Question 10.
x2 – 4x – 21
Solution:
Question 11.
y2 – 5y-36
Solution:
Question 12.
(a2-5a)2-36
Solution:
Question 13.
(a + 7) (a – 10) + 16
Solution:
Exercise 7.8
Resolve each of the following quadratic trinomials into factors :
Question 1.
2x2 + 5x + 3
Solution:
Question 2.
2x2– 3x – 2
Solution:
Question 3.
3x2 + 10x + 3
Solution:
Question 4.
7x – 6 – 2x2
Solution:
Question 5.
7x2 – 19x – 6
Solution:
Question 6.
28-31x -5x2
Solution:
Question 7.
3 + 23y – 8y2
Solution:
Question 8.
11x2 – 54x + 63
Solution:
Question 9.
7x-6x2 + 20
Solution:
Question 10.
3x2 + 22x + 35
Solution:
Question 11.
12x2 – 17xy + 6y2
Solution:
Question 12.
6x2 – 5xy – 6y2
Solution:
Question 13.
6x2 + 13xy + 2y2
Solution:
Question 14.
14x2 + 11xy – 15y2
Solution:
Question 15.
6a2 + 17ab – 3b2
Solution:
Question 16.
36a2 + 12abc – 15b2c2
Solution:
Question 17.
15x2 – 16xyz – 15y2z2
Solution:
Question 18.
(x – 2y)2 -5 (x- 2y) + 6
Solution:
Question 19.
(2a – b)2 + 2 (2a – b) – 8
Solution:
Exercise 7.9
Factorize each of the following quadratic polynomials by using the method of completing the square.
Question 1.
p2 + 6p + 8
Solution:
p2 + 6p + 8
= p2 + 2 x p x 3 + 32 – 32 + 8 (completing the square)
= (p2 + 6p + 32) – 1
= (p + 3)2 – 12
= (P + 3)2 – (1)2 { ∵ a2 + b2 = (a+b) (a-b)}
= (p +3+1) (p + 3 -1)
= (p+4) (p+ 2)
Question 2.
q2 – 10q + 21
Solution:
q2 – 10q + 21
= (q)2 – 2 x q x 5 + (5)2 – (5)2 + 21 (completing the square)
= (q)2 – 2 x q x 5 + (5)2 -25+21
= (q)2-2 x q x 5 + (5)2 – 25 +21
= (q)2-2 x q x 5 + (5)2 – 4
= (q – 5)2 – (2) {∵ a2 – b2 = (a + b) (a – b)}
= (q- 5 + 2) (q-5-2)
=(q- 3) (q-7)
Question 3.
4y2 + 12y + 5
Solution:
4y2 +12y + 5
= (2y)2 + 2 x 2y x 3 + (3)2 – (3)2 + 5 (completing the square)
= (2y + 3)2 – 9 + 5
= (2y + 3)2 – 4
= (2y + 3)2-(2)2 {∵ a2 – b2 = (a + b) (a – b)}
= (2y + 3 + 2) (2y + 3 – 2)
= (2y + 5) (2y+ 1)
Question 4.
p2 + 6p- 16
Solution:
p2 + 6p – 16
= (p)2 + 2 x p x 3 + (3)2 – (3)2 – 16 (completing the square)
= (p)2 + 2 x p x 3 + (3)2 – 9 – 16
= (p + 3)2 – 25
= (p + 3)2 – (5)2 {∵ a2 -b2 = {a + b) (a – b)}
= (p + 3 + 5)(p + 3-5)
= (p + 8) (p – 2)
Question 5.
x2 + 12x + 20
Solution:
x2 + 12x + 20
= (x)2 + 2 x x x 6 + (6)2 – (6)2 + 20 (completing the square)
= (x)2 + 2 x x x6 + (6)2 -36 + 20
= (x + 6)2 -16
= (x + 6)2 – (4)2 {∵ a2 – b2 = (a + b) (a – b)}
= (x + 6 + 4) (x + 6 – 4)
= (x + 10) (x + 2)
Question 6.
a2 – 14a – 51
Solution:
a2 – 14a-51
= (a)2 – 2 x x 7 + (7)2 – (7)2 – 51 (completing the square)
= (a)2 – 2 x a x 7 + (7)2 – 49 – 51
= (a – 7)2 – 100
= (a – 7)2 – (10)2 {∵ a2 – b2 = (a + b) (a – b)}
= (a – 7 + 10) (a – 7 – 10)
= (a + 3) (a – 17)
Question 7.
a2 + 2a – 3
Solution:
a2 + 2a – 3
= (a)2 + 2 x a x 1 + (1)2 – (1)2 – 3 (completing the square)
= (a)2 + 2 x a x 1 + (1)2 – 1 – 3
= (a + 1)2 – 4
= (a + 1)2 – (2)2 {∵ a2 – b2 = (a + b) (a – b)}
= (a + 1 + 2) (a + 1 – 2)
= (a + 3) (a – 1)
Question 8.
4x2 – 12x + 5
Solution:
4x2 – 12x + 5
= (2x)2 – 2 x 2x x 3 + (3)2 – (3)2 + 5 (completing the square)
= (2x)2 – 2 x 2x x 3 + (3)2 -9 + 5
= (2x – 3)2 – 4
= (2x – 3)2 – (2)2 {∵ a2 – b2 = (a + b) (a – b)}
= (2x – 3 + 2) (2x – 3 – 2)
= (2x – 1) (2x – 5)
Question 9.
y2 – 7y + 12
Solution:
Question 10.
z2-4z-12
Solution:
z2 – 4z – 12
= (z)2 – 2 x z x 2 + (2)2 – (2)2 – 12 (completing the square)
= (z)2 – 2 x z x 2 + (2)2 – 4 – 12
= (z-2)2-16
= (z-2)2-(4)2 {∵ a2 – b2 = (a + b) (a – b)}
= (z – 2 + 4) (z – 2 – 4)
= (z + 2)(z-6)
