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)
1. What is factorisation? |
2. What are the methods of factorisation? |
3. How do I factorise quadratic expressions? |
4. How important is factorisation in mathematics? |
5. What are the applications of factorisation in daily life? |
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