Mathematics
Factorise the following:
(x + 1)6 - (x - 1)6
Factorisation
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Answer
(x + 1)6 - (x - 1)6 = [(x + 1)3]2 - [(x - 1)3]2
We know that,
a2 - b2 = (a + b)(a - b)
∴ [(x + 1)3]2 - [(x - 1)3]2 = [(x + 1)3 + (x - 1)3][(x + 1)3 - (x - 1)3]
We know that,
a3 + b3 = (a + b)(a2 + b2 - ab)
a3 - b3 = (a - b)(a2 + b2 + ab)
∴ [(x + 1)3 + (x - 1)3][(x + 1)3 - (x - 1)3] = [(x + 1 + x - 1){(x + 1)2 - (x + 1)(x - 1) + (x - 1)2}][(x + 1) - (x - 1)][(x + 1)2 + (x + 1)(x - 1) + (x - 1)2]
= 2x[x2 + 1 + 2x - (x2 - 1) + x2 + 1 - 2x](x - x + 1 + 1)[x2 + 1 + 2x + x2 - 1 + x2 + 1 - 2x]
= 2x[x2 + 1 + 2x - x2 + 1 + x2 + 1 - 2x]2[x2 + 1 + 2x + x2 - 1 + x2 + 1 - 2x]
= 4x(x2 + 3)(3x2 + 1).
Hence, (x + 1)6 - (x - 1)6 = 4x(x2 + 3)(3x2 + 1).
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