Mathematics
Factorise the following:
x6 - 7x3 - 8
Factorisation
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Answer
Factorising x6 - 7x3 - 8 we get,
x6 - 7x3 - 8 = x6 - 8x3 + x3 - 8
= x3(x3 - 8) + 1(x3 - 8)
= (x3 - 8)(x3 + 1)
= [(x)3 - (2)3][(x)3 + 13]
We know that,
a3 + b3 = (a + b)(a2 - ab + b2)
a3 - b3 = (a - b)(a2 + ab + b2)
∴ x3 + 13 = (x + 1)(x2 - x + 1)
and,
(x)3 - (2)3 = (x - 2)(x2 + 2x + 4)
∴ [(x)3 - (2)3][(x)3 + 13] = (x - 2)(x2 + 2x + 4)(x + 1)(x2 - x + 1).
Hence, x6 - 7x3 - 8 = (x - 2)(x2 + 2x + 4)(x + 1)(x2 - x + 1).
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