Mathematics
Factorise the following:
(x2 - 2x)2 - 23(x2 - 2x) + 120
Factorisation
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Answer
Let (x2 - 2x) = t.
(x2 - 2x)2 - 23(x2 - 2x) + 120 = t2 - 23t + 120.
t2 - 23t + 120 = t2 - 15t - 8t + 120.
= t(t - 15) - 8(t - 15)
= (t - 15)(t - 8)
= (x2 - 2x - 15)(x2 - 2x - 8)
Factorising x2 - 2x - 15 we get,
x2 - 2x - 15 = x2 - 5x + 3x - 15 = x(x - 5) + 3(x - 5) = (x - 5)(x + 3).
Factorising x2 - 2x - 8 we get,
x2 - 2x - 8 = x2 - 4x + 2x - 8 = x(x - 4) + 2(x - 4) = (x - 4)(x + 2).
∴ (x2 - 2x - 15)(x2 - 2x - 8) = (x - 5)(x + 3)(x - 4)(x + 2).
Hence, (x2 - 2x)2 - 23(x2 - 2x) + 120 = (x - 5)(x + 3)(x - 4)(x + 2).
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