Page 126 - Start Up Mathematics_8 (Non CCE)

2
2
(d) 9(x – y) + 6(y –x) = 9(x – y) – 6(x – y) {Taking (–1) common from y – x}
2
= 3 × 3 × (x – y) – 3 × 2(x – y)
= 3(x – y) {3(x – y) – 2} {Taking 3(x – y) common}
= 3(x – y)(3x – 3y – 2)

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 7.1

1. Find the HCF of the following monomials:
3
5
4 3
7 4 2
6 7
2
2
(a) 3y and 15y (b) 4xy and 16x y (c) 21x y , 35x y z and 42x y z
4 2
5 7 6 4
(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
6 6 7 5
2 3
7 8 5 4
8 6 4 3
3 5 3
2. Factorize the following:
3 3
2
3 2
2
2
(a) 6x y + 2xy (b) 2x – 16x + 24 (c) –4x y + 8x y + 2x y (d) xy + 3xz + x 2
2
3 4 2
2
5 2
2 2 4
(e) 32x y z – 48x y z (f) ax y + bxy + cxyz (g) 36x y z – 54x z + 90x y z
2 3 4
4 2 2
2
2 3
3
(h) 48x y z – 36x y (i) 2x (x + 4) – 3(x + 4) (j) 9x(6x – 5y) – 12x (6x – 5y)
2
(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)
4
3
Factorization of Algebraic Expressions by Grouping
Sometimes the factorization of an algebraic expression 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 (NCERT)
2
Solution: (a) x + xy + 8x + 8y
Step 1: There is no factor common to all the terms. There are four terms in the given
expression. Pair two terms in such a way that they have a common factor.
2
Step 2: Consider the first two terms: x + xy = x(x + y)
Step 3: Now take the last two terms: 8x + 8y = 8(x + y)
Note that (x + y) is a common factor in step 2 and step 3.
Step 4: Combine step 2 and step 3 together.
2
(x + xy) + (8x + 8y) = x(x + y) + 8(x + y) = (x + 8)(x + y)
(b) 15pq + 15 + 9q + 25p
Step 1: There is no factor common to all the terms. There are four terms in the given
expression. On combining first and second terms or third and fourth terms,
there is no common factor. So terms need to be regrouped.
Step 2: Pair the first and third term: 15pq + 9q = 5 × 3pq + 3 × 3q = 3q(5p + 3)
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