NCERT Solutions (Ex: 14.1 - 14.2) - Factorisation - Class 8 PDF Download

Exercise 14.1 

Question 1: 

Find the common factors of the given terms. 

(i)    12 x , 36 
 (ii)    2y,22 x y
 (iii)    14pq,28p²q² 
 (iv)    2 x ,3 x ²,4
 (v)    6abc, 24ab² , 12a²b 
 (vi)    16 x ³ ,-4 x ², 32 x
 (vii)  10 pq, 20qr , 30rp
 (viii)    3 x ²y³ ,10 x ³y² , 6 x ²y²z
 

Answer 1:

(i)

12 x = 2 x 2 x 3 x x
36 = 2 x 2 x 3 x 3
Hence, the common factors are 2,2 and 3 = 2 x 2 x 3 = 12

(ii) 2y =2 x y
22 x y = 2 x 11 x x x y
Hence, the common factors are 2 and y  = 2 x y =2y

(iii) 14pq = 2 x 7 x p x q
28p²q² = 2 x 2 x 7 x p x p x q x q
Hence, the common factors are 2 x 7 x p x q = 14pq

(iv) 2 x = 2 x x x 1

3 x ² = 3 x x x x x 1 

 4 = 2 x 2 x 1
Hence, the common factor is 1.

(v)    6abc = 2 x 3 x a x b x c
24ab²   = 2 x 2 x 2 x 3 x a x b x b
12a²b = 2 x 2 x 3 x a x a x b
Hence, the common factors are 2 x 3 x a x b = 6ab

(vi)    16 x ³ = 2 x 2 x 2 x 2 x x x x x x
-4 x ² = (-l) x 2 x 2 x x x x
32 x = 2 x 2 x 2 x 2 x 2 x x
Hence, the common factors are 2 x 2 x x = 4 x

(vii) 10pq =2 x5 x p x q
20qr =2x 2 x5 x qx r
30rp =2x3x5x r x p
Hence.the common factors are 2x 5 = 10

(viii) 3x²y³ =3x x x x x y x y x y
10x³y² =2x 5x x x x x x x y x y
6x²y²z = 2x3 x x x x x y x y x z
Hence, the common factors are x x x x y x y =x²y²

Question 2: 

Factorize the following expressions. 

(i)    7x-42    
 (ii)   6p - 12q
 (iii)   7a² +14a    
 (iv)   -16z 20z³
 (v)   20/²m+ 30alm    
 (vi)   5x²y - 15xy²
 (vii)  10a² -15b² + 20c²    
 (viii)  -4a² + 4ab - 4ca
 (ix)  x²yz+xy²z + xyz²    
 (x) ax²y+ bxy² +cxyz 

Answer 2: 

(i)    7x-42 = 7*x- 2*3* 7
Taking common factors from each term,
= 7 (x -2*3)
= 7(x - 6)

(ii) 6p - 12q = 2 *3* p -2*2* 3*q
Taking common factors from each term,
= 2*3(p-2q)
= 6( p -2q)

(iii) 7a² + 14a = 7 * a*a + 2 *7*a
Taking common factors from each term,
= 7* a (a + 2)
= 7a(a +2)

(iv)    -16z + 20z³ = (-1) *2*2 * 2*2 * z +2*2 * 5 * z *z*z
Taking common factors from each term,
= 2*2 *z(-2 *2 +5 * z*z)
= 4=(-4+5z² )

(v)    20l²m+30alm = 2 *2* 5*l * l*m +2 * 3*5 * a*l*m
Taking common factors from each term,
= 2*5 * l * m (2 * / +3* a)
= 10lm( 2l+3a)

(vi)    5*2y-15xy² =5 *x  *x  * y +3* 5* x * y * y
Taking common factors from each term,
= 5 * x * y(x -3y)
= 5xy( x -3y )

(vii)

(viii)

(ix)

(x)

Question 3: 

Factorize: 

(i) x² +xy+8x+8y
 (ii)  ax+bx -ay-by
 (iii) 15xy -6x+5y -2
 (iv) 15pq+ 15+ 9q+ 25p
 (v)  z-7+7xy-xyz

Answer 3:

(i)    x²+ xy +8x+ 8y = x( x +y ) + 8(x+ y )
= (x + y)(x + 8)

(ii)    15xy-6x +5y-2 =3x( 5y - 2 ) +1(5y -2)
= (5y -2)( 3x + 1)

(iii)    ax +bx -ay -by =(ax +bx)-( ay+by)
= x( a+ b)- y( a +b)
= ( a +b)( x -y )

(iv)    15pq+ 15+ 9q + 25p = 15pq+ 25p +9q+ 15
= 5p( 3q+5) +3( 3q +5)
= ( 3q+5)( 5p +3)

(v) z-7+7xy-xyz =7xy -7-xyz + z
= 7(xy - 1)- =( xy -1 )
= (xy- 1)(7-z)=(-1)(1-xy)(- 1)(z-7)
= (1-xy)(z -7)

Exercise 14.2 

Question 1: 

Factorize the following expressions: 
(i) a²+8a +16
 (ii) p² -10p+25
 (iii) 25m²+30m +9
 (iv) 49y² +84yz +36z2
 (v) 4x² -8x +4
 (vi) 121b² -88bc+ 16c²
 (vii) (l+m)^'2-4lm  (Hint: Expand (l+ m)² first]
 (viii) a⁴ + 2a²b²+b⁴

Answer 1: 

(i) 

(ii)
 

(iii) 

(ivI)

(v)

(vi)

(vii)

(viii)

Question 2: 

Factorize: 
(i) 4p² -9q² 
 (ii) 63a²- 112b²
 (iii) 49x² -36
 (vi) 16x⁵ -144x⁵
 (v)  (l+m)² -(l-m)² 
 (iv) 9x²y²-16
 (vii) ( x² - 2xy+ y² ) - z²
 (viii) 25a² -4b² + 28bc- 49c²

Answer 2: 

Question 3: 

Factorize the expressions: 

(i) ax² +bx    
 (ii)    7p + 2lq²
 (iii)    2x³+ 2xy² + 2xz²
 (iv)    am²'+ bm² + bn² + an²
 (v)    ( lm+l)+m+ I    
 (vi)    y( y+z)+ 9( y+ z)
 (vii)    5y² -20y - 8z+2yz    
 (viii)    10ab+4a +5b+2
 (ix)    6xy - 4y + 6 -9x

Answer 3:

Question 4: 

Factorize: 

(i)    a⁴-b⁴
 (ii)    p⁴-81
 (iii)    x⁴-(y+z)⁴
 (iv)    x⁴-(x -z) ⁴
 (v)    a⁴-2a²b²+b²

Answer 4: 

(i) 

(ii) 

(iii)

(iv)

(v)

Question 5: 

Factorize the following expressions:

Answer 5: 

(i)    p ² +6p+8 = p ² + (4+ 2) p +4 x 2
= p ² +4p +2p+4 x 2
= p(p + 4) +2(p + 4)
= ( p +4)( p + 2)

(ii)    q²  -10q +21 = q² -(7 +3)q +7 x 3
= q² -7q -3q +7 x 3
= q (q -7)-3(q -7)
= (q -7}(q -3)

(iii)    p² +6p -16 = p² + (8-2)p-8x2
=  p ² + 8p-2p-8x2
= p (p+8) -2(p+ 8)
=  (p+ 8)(p-2)