Mathematics
Factorise the following:
27(x + y)3 + 8(2x - y)3
Factorisation
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Answer
27(x + y)3 + 8(2x - y)3 = [3(x + y)]3 + [2(2x - y)]3
We know that,
a3 + b3 = (a + b)(a2 - ab + b2).
∴ [3(x + y)]3 + [2(2x - y)]3 = [3(x + y) + 2(2x - y)][{3(x + y)}2 - 3(x + y) × 2(2x - y) + {2(2x - y)}2]
= [3x + 3y + 4x - 2y][9(x2 + y2 + 2xy) - 6(x + y)(2x - y) + 4(4x2 + y2 - 4xy)]
= (7x + y)[9x2 + 9y2 + 18xy - 6(2x2 - xy + 2xy - y2) + 16x2 + 4y2 - 16xy]
= (7x + y)[9x2 + 9y2 + 18xy - 12x2 + 6xy - 12xy + 6y2 + 16x2 + 4y2 - 16xy]
= (7x + y)(13x2 + 19y2 - 4xy).
Hence, 27(x + y)3 + 8(2x - y)3 = (7x + y)(13x2 + 19y2 - 4xy).
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