Page 81 - Maths Skills - 8

Algebraic Expressions and Identities 79

4. Find the products of the following.

(i) (3x – 2)(5x + 6x + 2) (ii) (x + y + z )(xy + yz) (iii) (x + y)(x – xy + y )
2
2
2
2
2
2
3 1
(iv) (5x + y)(3x + 2y) (v) (x + y )(x – xy + y ) (vi) x 3y 2 x y
3
2
2
2
3
2
5 3
5. Simplify.
(i) (3y + 2)(y – 2) – (7y + 3)(y – 4) (ii) (2x – 3y)(x + y) – (5x + 2y)(x – y)
(iii) x + (3x – y)(3x + y + y ) (iv) (a – 3a + 5)(2a – 3) – (5a + 3a – 3)(a – 1)
2
2
2
2
STANDARD IDENTITIES
An identity is an equality, which is true for all values of the variables.
The basic important identities are;
Identity 1: (a + b) = a + 2ab + b 2 Identity 2: (a – b) = a – 2ab + b 2
2
2
2
2
Proof: We have, (a + b) = (a + b) × (a + b) Proof: We have, (a – b) = (a – b) × (a – b)
2
2
= a (a + b) + b (a + b) = a(a – b) – b(a – b)
= a + ab + ba + b 2 = a – ab – ba + b 2
2
2
= a + 2ab + b 2 = a – 2ab + b 2
2
2
2
2
2
\ (a + b) = a + 2ab + b 2 ∴ (a – b) = a – 2ab + b 2
2
Identity 3: (a + b)(a – b) = a – b 2 Identity 4: (x + a) (x + b) = x + (a + b)x + ab
2
2
Proof: We have, (a + b)(a – b) = a(a – b) + b(a – b) Proof: (x + a) (x + b) = x (x + b) + a (x + b)
2
= a – ab + ba – b 2 = x + bx + ax + ab
2
2
= a – b 2 = x + (a + b)x + ab
2
2
∴ (a + b)(a – b) = a – b 2 ∴ (x + a) (x + b) = x + (a + b)x + ab
2
Let’s Attempt
Example 1: Find the following products.

(i) 4 x 7 4 x 7 (ii) (2x + 3y)(2x + 3y)

5 5

Solution: (i) We have, 4 x 7 4 x 7 4 x 7 2

5 5 5
4 2 4
= x + 2 × x × 7 + 7 2 [ (a + b) = a + 2ab + b ]
2
2
2
5 5
= 16 x + 2 56 x + 49
25 5
(ii) We have,
(2x + 3y)(2x + 3y) = (2x + 3y) = (2x) + 2 × 2x × 3y + (3y) [ (a + b) = a + 2ab + b ]
2
2
2
2
2
2
= 4x + 12xy + 9y 2
2