Page 80 - Maths Skills - 8

78 Maths

Example 2: Find the product of (x – y) and (x + xy + y ) and verify the result for x = y = 1.
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2
Solution: We have, (x – y)(x + xy + y ) = x(x + xy + y ) – y(x + xy + y )
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2
2
2
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2
= (x × x + x × xy + x × y ) – (y × x + y × xy + y × y )
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= (x + x y + xy ) – (yx + xy + y )
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= x + x y + xy – yx – xy – y = x – y [On cancellation]
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Verification: LHS = (1 – 1) × (1 + 1 × 1 + 1 ) = 0 × 3 = 0
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RHS = 1 – 1 = 0
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3
Thus, LHS = RHS
Example 3: Multiply : (3x – 2y) by (5x + 2x – 3).
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Solution: We have, (3x – 2y) × (5x + 2x – 3) = 3x(5x + 2x – 3) – 2y(5x + 2x – 3)
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= (3x × 5x + 3x × 2x – 3x × 3) – (2y × 5x + 2y × 2x – 2y × 3)
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= (15x + 6x – 9x) – (10x y + 4xy – 6y)
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= 15x + 6x – 9x – 10x y – 4xy + 6y
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Example 4: Simplify : (y + 1)(2y – 1) – (3y – 1) (y + 2).
Solution: We have, (y + 1)(2y – 1) – (3y – 1)(y + 2) = {y (2y – 1) + 1 (2y – 1)} – {3y(y + 2) – 1 (y + 2)}
= (2y – y + 2y – 1) – (3y + 6y – y – 2)
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= 2y – y + 2y – 1 – 3y – 6y + y + 2
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= 2y – 3y – y + 2y – 6y + y – 1 + 2 = – y – 4y + 1
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Exercise 5.3
1. Find the products of the following.
(i) (2x – y)(3x + y ) (ii) (x – 3y)(x + 3xy) (iii) x 1 x 1
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2
3

3
x 2
(iv) (x – a )(x – a) (v) 2 x 3 x y ) (vi) (a b + ab )(b c + c b)
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2

y (
7 5
2. Find the following products and verify the results for x = – 2 and y = – 5.
(i) (2x + 3y + 3)(2y – 3x ) (ii) (x – y)(x + xy + y )
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2
2
2
(iii) (2x + 3y)(4x – 6xy + 9y ) (iv) (x + y)(x – x y + y )
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2
2
4
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(v) (3x ) y 4 x 5 xy 2 y 2 (vi) (x – y )(x – xy + y )
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4
4
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2

3 6 5
3. Find the following products.
(i) 3x (3x + 2)(4x – 1) (ii) (2x + 7)(2 – x)(5 + x)
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2
(iii) (3 – 5x)(8 – 3x)(2x + 5) (iv) (x + 3)(x – 3)(x – 1)