Page 159 - ICSE Math 8

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Solution: (a) l x + l (x – y) – l(y + z) – z = l x + l x – l y – ly – lz – z (Simplifying the expression)
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= (l x + l x) – (l y + ly) – (lz + z) (Regrouping)
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= l x(l + 1) – ly(l + 1) – z(l + 1)
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= (l x – ly – z)(l + 1) {Taking (l + 1) common}
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(b) (x – 2x) – 3(x –2x) – y(x – 2x) + 3y
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= {(x – 2x) – 3(x – 2x)} – {y(x – 2x) – 3y} (Regrouping)
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= (x – 2x)(x – 2x – 3) – y(x – 2x – 3)
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= (x – 2x – y)(x – 2x – 3) {Taking (x – 2x – 3) common}
EXERCISE 14.2
Factorize the following:
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(a) ab – cb + ad – cd (b) 6xy – y + 12xz – 2yz (c) b – ab(1 – a) – a 3
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(d) a – a(x + 2y) + 2xy (e) x – 2x y + 3xy – 6y 3 (f) xy(a + 1) – a(x + y )
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(g) 8(p – q) – 12(p – q) 2 (h) (lx – my) + (mx + ly) (i) 1 + xy – x – y
Factorization of Difference of Squares
The factorization of binomial expressions expressible as the difference of two squares involves the following
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identity: p – q = (p + q)(p – q). The expression of the form p – q is known as difference of two squares.
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Example 7: Factorize: (a) 16x – 25y 2 (b) 81x – 625
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Solution: (a) 16x – 25y = (4x) – (5y) = (4x + 5y)(4x – 5y)
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(b) 81x – 625 = (9x ) – (25) = (9x + 25)(9x – 25)
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= (9x + 25){(3x) – 5 } = (9x + 25)(3x + 5)(3x – 5)
Point to remember
Make sure that both the terms are perfect squares and there is a minus sign between them.
Example 8: Factorize the following.
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(a) 36l m – m (b) 81(a + b) – 64(x + y) 2
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m Ê 1 ˆ Ï Ê 1 ˆ Ô
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Solution: (a) 36lm - = m Á 36l - 2 ˜ = m Ì () 2 - Á ˜ ˝
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49l 2 Ë 49l ¯ Ó Ô Ë 7l ¯ ˛ Ô
 1  1 
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= m 6 ++  l 6 −−  {Using a – b = (a + b)(a – b)}
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 l 7  l 7 
Point to remember
In case there are common factors in the binomial, take the common factors out and then factorize.
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(b) 81(a + b) – 64(x + y) Try This
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= {9(a + b)} – {8(x + y)} 2
Fractorize:
= {9(a + b) + 8 (x + y)}{9(a + b) – 8(x + y)} (a) l – (m + n) (b) x y – x y
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{Using a – b = (a + b)(a – b)} (c) a(a + c) – b(b + c)
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