Page 158 - ICSE Math 8

= 4(5x – 9y) {2(5x – 9y) – 3} {Taking 4(5x – 9y) common}
= 4(5x – 9y)(10x – 18y – 3)

2
2
(b) (2x – 3y) – 6x + 9y = (2x – 3y) – 3 × 2x + 3 × 3y
2
= (2x – 3y) – 3(2x – 3y) {Taking (–3) common from –6x + 9y}
= (2x – 3y)(2x – 3y – 3) {Taking (2x – 3y) common}
Example 4: Factorize: (2x + y)(5x – 3y) – (x + y)(5x – 3y)
Solution: (2x + y)(5x – 3y) – (x + y)(5x – 3y)

= {(2x + y) – (x + y)}(5x – 3y) {Taking (5x – 3y) common}
= (2x + y – x – y)(5x – 3y)
= x(5x – 3y)

EXERCISE 14.1

1. Find the HCF of the following monomials.
7 4 2
2
3
6 7
5
4 3
2
(a) 3y and 15y (b) 4xy and 16x y (c) 21x y , 35x y z and 42x y z
6 6 7 5
5 7 6 4
2 3
3 5 3
7 8 5 4
8 6 4 3
4 2
(d) –2x y z, –8xy z and –14x y z (e) –9x y z u , 18x y z u , –27x y z u , 45x y z u
2. Factorize the following.
2
3 3
3 2
2
2
(a) 6x y + 2xy (b) 2x – 16x + 24 (c) –4x y + 8x y + 2x y
2 3 4
2
3 4 2
(d) xy + 3xz + x (e) 32x y z – 48x y z (f) ax y + bxy + cxyz
2
2
3
5 2
4 2 2
2 2 4
2 3
(g) 36x y z – 54x z + 90x y z (h) 48x y z – 36x y (i) 2x (x + 4) – 3(x + 4)
2
2
(j) 9x(6x – 5y) – 12x (6x – 5y) (k) 2(x – 2y) – 3(x – 2y) (l) l(x – y) + 3m(y – x) + n(x – y) 2
(m) x (3x – y) + x (3x – y) (n) (x + y)(4x – 5y) – (x + y)(2x + 3y)
3
4
Factorization by Regrouping
Sometimes the factorization of an algebraic expression with more than three terms is done by grouping terms
such that each group has a common factor.
Step 1: Arrange the terms of the algebraic expression in groups such that each group has a common factor.
Step 2: Factorize each group.
Step 3: Take out the factor which is common to each group and re-arrange the expression.
Example 5: Factorize by grouping the terms.
2
(a) x + xy + 8x + 8y (b) 15pq + 15 + 9q + 25p
2
Solution: (a) x + xy + 8x + 8y
Group two terms in such a way that they have a common factor.
2
Thus, (x + xy) + (8x + 8y) = x(x + y) + 8(x + y) = (x + 8)(x + y)
(b) 15pq + 15 + 9q + 25p
Pairing the 1st and 3rd terms and the 2nd and 4th terms, we get
(15pq + 9q) + (15 + 25p) = 3q(5p + 3) + 5(5p + 3) = (3q + 5)(5p + 3)
Example 6: Factorize the following.
2
2
2
3
2
2
(a) l x + l (x – y) – l(y + z) – z (b) (x – 2x) – 3(x – 2x) –y(x – 2x) + 3y
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