Page 128 - Start Up Mathematics_8 (Non CCE)

2
2
= {l + (m + n) } (l + m + n) {l – (m + n)}
2
2
{Using p – q = (p + q)(p – q)}
2
2
= {l + (m + n) } (l + m + n)(l – m – n)
Example 8: Factorize the following:
2
2
(a) 36l m – m (b) 81(a + b) – 64(x + y) 2
49 l 2
(c) x y – x y (d) a(a + c) – b(b + c)
4 12
12 4
¸
2
m Ê 1 ˆ Ï Ê 1 ˆ Ô
Ô
Solution: (a) 36lm - = m Á 36l - 2 ˜ = m Ì () 2 - Á ˜ ˝
2
2
6l
49l 2 Ë 49l ¯ Ó Ô Ë 7l ¯ ˛ Ô
Ê 1 ˆ Ê 1 ˆ 2 2
l
= m 6 + l 7 ¯ Ë l 6 - l 7 ¯ ˜ {Using p – q = (p + q)(p – q)}
Á
˜ Á
Ë
2
2
2
(b) 81(a + b) – 64(x + y) = {9(a + b)} – {8(x + y)} 2
= {9(a + b) + 8 (x + y)}{9(a + b) – 8(x + y)}
2
2
{Using p – q = (p + q)(p – q)}
= (9a + 9b + 8x + 8y)(9a + 9b – 8x – 8y)
4 4
4 12
8
8
12 4
(c) x y – x y = x y (x – y )
4 2
4 4
4 2
= x y {(x ) – (y ) }
4 4
4
4
4
4
2
2
= x y (x + y )(x – y ) {Using p – q = (p + q)(p – q)}
2 2
4 4
4
4
2 2
= x y (x + y ){(x ) – (y ) }
2
4
4
2
2
2
4 4
= x y (x + y )(x + y )(x – y )
2
2
2
4
2
4 4
4
= x y (x + y )(x + y )(x + y)(x – y) {Using p – q = (p + q)(p – q)}
2
2
(d) a(a + c) – b(b + c) = a + ac – b – bc (Simplifying the expression)
2
2
= (a – b ) + (ac – bc) (Regrouping)
2
2
= (a + b)(a – b) + c(a – b) {Using p – q = (p + q)(p – q)}
= (a + b + c)(a – b)
2
2
2
2
Example 9: Evaluate (402) – (398) using the identity p – q = (p + q)(p – q).
Solution: Here p = 402, q = 398
2
2
\ (402) – (398) = (402 + 398)(402 – 398) = (800)(4) = 3,200
EXERCISE 7.3
1. Factorize the following:
2
2
2
2
(a) 48x – 243y (b) 36(x + y) – 49(x – 2y) 2 (c) 1 x - 1
16 169
6
2
4
4
4
8
(d) 24l – (e) a – 81b (f) 256a – b 8 (g) (x + y) – (x – y) 4
25m 2
2 2
2
22
22
(h) 63x y – 7 (i) 4 – (l – m) 2 (j) 1 ab - 9 bc (k) 16(2p – 1) – 25q 2
36 49
2
2
2. Evaluate using p – q = (p + q)(p – q), where
(a) p = 16, q = 4 (b) p = 8.8, q = 1.2 (c) p = 506, q = 494 (d) p = 52, q = 48
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