myhelper: RD Sharma solution class 8 chapter 7 Factorization Exercise 7.2

Exercise 7.2

Page-7.5

Question 1:

Factorize the following:
3x − 9

Answer 1:

The greatest common factor of the terms 3x and -9 of the expression 3x - 9 is 3.
Now.
3x = 3x
and
-9 = 3.-3
Hence, the expression 3x - 9 can be factorised as 3(x - 3).

Question 2:

Factorize the following:
5x − 15x²

Answer 2:

The greatest common factor of the terms 5x and 15x² of the expression 5x - 15x² is 5x.
Now,
5x = 5x $\times$ 1
and
-15x² = 5x $\times$ -3x
Hence, the expression 5x - 15x²can be factorised as 5x(1 - 3x).

Question 3:

Factorize the following:
20a¹²b² − 15a⁸b⁴

Answer 3:

The greatest common factor of the terms 20a¹²b² and -15a⁸b⁴ of the expression 20a¹²b² - 15a⁸b⁴ is 5a⁸b².
20a¹²b² = 5×4×a⁸×a⁴×b² = 5a⁸×b² $\times$ 4a⁴ and -15a⁸b⁴= 5×-3×a⁸×b²×b² = 5a⁸b² $\times$ -3b²

Hence, the expression 20a¹²b² - 15a⁸b⁴can be factorised as 5a⁸b²(4a⁴-3b²)

Question 4:

Factorize the following:
72x⁶y⁷ − 96x⁷y⁶

Answer 4:

The greatest common factor of the terms 72x⁶y⁷ and -96x⁷y⁶ of the expression 72x⁶y⁷ - 96x⁷y⁶⁴ is 24x⁶y⁶.

Now,
72x⁶y⁷ = 24x⁶y⁶ $\times$ 3y
and
-96x⁷y⁶ = 24x⁶y⁶ $\times$ -4x

Hence, the expression 72x⁶y⁷ - 96x⁷y⁶ can be factorised as 24x⁶y⁶(3y - 4x).

Question 5:

Factorize the following:
20x³ − 40x² + 80x

Answer 5:

The greatest common factor of the terms 20x³, -40x² and 80x of the expression 20x³ - 40x² + 80x is 20x.
Now,
20x³ = 20x $\times$ x²
-40x² = 20x $\times$ -2x
and
80x = 20x $\times$ 4
Hence, the expression 20x³ - 40x² + 80x can be factorised as 20x(x² - 2x + 4).

Question 6:

Factorize the following:
2x³y² − 4x²y³ + 8xy⁴

Answer 6:

The greatest common factor of the terms 2x³y², -4x²y³ and 8xy⁴of the expression 2x³y² - 4x²y³+ 8xy⁴y⁶⁴ is 2xy².

Now,
2x³y² = 2xy² $\times$ x²
-4x²y³= 2xy² $\times$ -2xy
8xy⁴ = 2xy² $\times$ 4y²

Hence, the expression 2x³y² - 4x²y³ + 8xy⁴ can be factorised as 2xy²(x² - 2xy + 4y²).

Question 7:

Factorize the following:
10m³n² + 15m⁴n − 20m²n³

Answer 7:

The greatest common factor of the terms 10m³n², 15m⁴n and -20m²n³of the expression 10m³n² + 15m⁴n - 20m²n³ is 5m²n.

Now,
10m³n²= 5m²n $\times$ 2mn
15m⁴n = 5m²n $\times$ 3m²
-20m²n³= 5m²n $\times$ -4n²

Hence, 10m³n² + 15m²n - 20m²n³ can be factorised as 5m²n(2mn + 3m² - 4n²).

Question 8:

Factorize the following:
2a⁴b⁴ − 3a³b⁵ + 4a²b⁵

Answer 8:

The greatest common factor of the terms 2a⁴b⁴, -3a³b⁵ and 4a²b⁵ of the expression 2a⁴b⁴ - 3a³b⁵ + 4a²b⁵ is a²b⁴.

Now,
2a⁴b⁴= a²b⁴ $\times$ 2a²
-3a³b⁵= a²b⁴ $\times$ -3ab
4a²b⁵= a²b⁴ $\times$ 4b

Hence, (2a⁴b⁴ - 3a³b⁵ + 4a²b⁵) can be factorised as [a²b⁴(2a² - 3ab + 4b)].

Question 9:

Factorize the following:
28a² + 14a²b² − 21a⁴

Answer 9:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s 28 a^{2}, 14 a^{2} b^{2} a n d 21 a^{4} o f t h e e x p r e s s i o n 28 a^{2} + 14 a^{2} b^{2} - 21 a^{4} i s 7 a^{2} . A l s o, w e c a n w r i t e 28 a^{2} = 7 a^{2} \times 4, 14 a^{2} b^{2} = 7 a^{2} \times 2 b^{2} a n d 21 a^{4} = 7 a^{2} \times 3 a^{2} . ∴ 28 a^{2} + 14 a^{2} b^{2} - 21 a^{4} = 7 a^{2} \times 4 + 7 a^{2} \times 2 b^{2} - 7 a^{2} \times 3 a^{2} = 7 a^{2} (4 + 2 b^{2} - 3 a^{2})$

Question 10:

Factorize the following:
a⁴b − 3a²b² − 6ab³

Answer 10:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s a^{4} b, 3 a^{2} b^{2} a n d 6 a b^{3} o f t h e e x p r e s s i o n a^{4} b - 3 a^{2} b^{2} - 6 a b^{3} i s a b . A l s o, w e c a n w r i t e a^{4} b = a b \times a^{3}, 3 a^{2} b^{2} = a b \times 3 a b a n d 6 a b^{3} = a b \times 6 b^{2} . ∴ a^{4} b - 3 a^{2} b^{2} - 6 a b^{3} = a b \times a^{3} - a b \times 3 a b - a b \times 6 b^{2} = a b (a^{3} - 3 a b - 6 b^{2})$

Question 11:

Factorize the following:
2l²mn - 3lm²n + 4lmn²

Answer 11:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s 2 l^{2} m n, 3 l m^{2} n a n d 4 l m n^{2} o f t h e e x p r e s s i o n 2 l^{2} m n - 3 l m^{2} n + 4 l m n^{2} i s l m n . A l s o, w e c a n w r i t e 2 l^{2} m n = l m n \times 2 l, 3 l m^{2} n = l m n \times 3 m a n d 4 l m n^{2} = l m n \times 4 n . ∴ 2 l^{2} n m - 3 l m^{2} n + 4 l m n^{2} = l m n \times 2 l - l m n \times 3 m + l m n \times 4 n = l m n (2 l - 3 m + 4 n)$

Question 12:

Factorize the following:
x⁴y² − x²y⁴ − x⁴y⁴

Answer 12:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s x^{4} y^{2}, x^{2} y^{4} a n d x^{4} y^{4} o f t h e e x p r e s s i o n x^{4} y^{2} - x^{2} y^{4} - x^{4} y^{4} i s x^{2} y^{2} . A l s o, w e c a n w r i t e x^{4} y^{2} = x^{2} y^{2} \times x^{2}, x^{2} y^{4} = x^{2} y^{2} \times y^{2} a n d x^{4} y^{4} = x^{2} y^{2} \times x^{2} y^{2} . ∴ x^{4} y^{2} - x^{2} y^{4} - x^{4} y^{4} = x^{2} y^{2} \times x^{2} - x^{2} y^{2} \times y^{2} - x^{2} y^{2} \times x^{2} y^{2} = x^{2} y^{2} (x^{2} - y^{2} - x^{2} y^{2})$

Question 13:

Factorize the following:
9x²y + 3axy

Answer 13:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s 9 x^{2} y a n d 3 a x y o f t h e e x p r e s s i o n 9 x^{2} y + 3 a x y i s 3 x y . A l s o, w e c a n w r i t e 9 x^{2} y = 3 x y \times 3 x a n d 3 a x y = 3 x y \times a . ∴ 9 x^{2} y + 3 a x y = 3 x y \times 3 x + 3 x y \times a = 3 x y (3 x + a)$

Question 14:

Factorize the following:
16m − 4m²

Answer 14:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s 16 m a n d 4 m^{2} o f t h e e x p r e s s i o n 16 m - 4 m^{2} i s 4 m . A l s o, w e c a n w r i t e 16 m = 4 m \times 4 a n d 4 m^{2} = 4 m \times m . ∴ 16 m - 4 m^{2} = 4 m \times 4 - 4 m \times m = 4 m (4 - m)$

Question 15:

Factorize the following:
−4a² + 4ab − 4ca

Answer 15:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s - 4 a^{2}, 4 a b a n d - 4 c a o f t h e e x p r e s s i o n - 4 a^{2} + 4 a b - 4 c a i s - 4 a . A l s o, w e c a n w r i t e - 4 a^{2} = - 4 a \times a, 4 a b = - 4 a \times (- b) a n d 4 c a = - 4 a \times c . ∴ - 4 a^{2} + 4 a b - 4 c a = - 4 a \times a + (- 4 a) \times (- b) - 4 a \times c = - 4 a (a - b + c)$

Question 16:

Factorize the following:
x²yz + xy²z + xyz²

Answer 16:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s x^{2} y z, x y^{2} z a n d x y z^{2} o f t h e e x p r e s s i o n x^{2} y z + x y^{2} z + x y z^{2} i s x y z . A l s o, w e c a n w r i t e x^{2} y z = x y z \times x, x y^{2} z = x y z \times y a n d x y z^{2} = x y z \times z . ∴ x^{2} y z + x y^{2} z + x y z^{2} = x y z \times x + x y z \times y + x y z \times z = x y z (x + y + z)$

Question 17:

Factorize the following:
ax²y + bxy² + cxyz

Answer 17:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s a x^{2} y, b x y^{2} a n d c x y z o f t h e e x p r e s s i o n a x^{2} y + b x y^{2} + c x y z i s x y . A l s o, w e c a n w r i t e a x^{2} y = x y \times a x, b x y^{2} = x y \times b y a n d c x y z = x y \times c z . ∴ a x^{2} y + b x y^{2} + c x y z = x y \times a x + x y \times b y + x y \times c z = x y (a x + b y + c z)$

RD Sharma solution class 8 chapter 7 Factorization Exercise 7.2