Find the expansions of the following :

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Find the expansions of the following :

(i) (2x + 3y + 5) (2x + 3y – 5)
(ii) (6 – 4a -7b)2
(iii) (7 – 3xy)3
(iv) (x + y + 2)3

Answer :

(i) (2x + 3y + 5) (2x + 3y – 5)

Let us simplify the expression, we get

(2x + 3y + 5) (2x + 3y – 5) = [(2x + 3y) + 5] [(2x – 3y) – 5]

By using the formula, (a)2 – (b)2 = [(a + b) (a – b)]
= (2x + 3y)2 – (5)2

= (2x)2 + (3y) 2 + 2 × 2x × 3y – 5 × 5

= 4x2 + 9y2 + 12xy – 25

(ii) (6 – 4a – 7 b)2

Let us simplify the expression, we get

(6 – 4a – 7 b)2 = [ 6 + (- 4a) + (-7b)]2

= (6)2 + (- 4a)2 + (- 7b)2 + 2 (6) (- 4a) + 2 (- 4a) (-7b) + 2 (-7b) (6)

= 36 + 16a2 + 49b2 – 48a + 56ab – 84b

(iii) (7 – 3xy)3

Let us simplify the expression

By using the formula, we get

(7 – 3xy)3 = (7)3 – (3xy)3 – 3 (7) (3xy) (7 – 3xy)

= 343 – 27x3y3 – 63xy (7 – 3xy)

= 343 – 27x3y3 – 441xy + 189x2y2

(iv) (x + y + 2)3

Let us simplify the expression

By using the formula, we get

(x + y + 2 )3 = [(x + y) + 2]3

= (x + y)3 + (2)3 + 3 (x + y) (2) (x + y + 2)

= x3 + y3 + 3x2y + 3xy2 + 8 + 6 (x + y) [(x + y) + 2]
= x3 + y3 + 3x2y + 3xy2 + 8 + 6 (x + y)2 +12(x + y)

= x3 + y3 + 3x2y + 3xy2 + 8 + 6 (x2 + y2 + 2xy) + 12x + 12y = x3 + y3 + 3x2y + 3xy2 + 8 + 6x2 + 6y2 + 12xy + 12x + 12y

= x3 + y3 + 3x2y + 3xy2 + 8 + 6x2 + 6y2 + 12x + 12y + 12xy

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