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Question
Factorise the following polynomials using synthetic division:
x3 + x2 – 14x – 24
Solution
p(x) = x3 + x2 – 14x – 24
p(1) = (1)3 + (1)2 – 14(1) – 24
= 1 + 1 – 14 – 24
= – 36
≠ 0
x + 1 is not a factor.
p(–1) = (–1)3 + (–1)2 – 14(–1) – 24
= –1 + 1 + 14 – 24
= 15 – 25
≠ 0
x – 1 is not a factor.
p(2) = (–2)3 + (–2)2 – 14(–2) – 24
= – 8 + 4 + 28 – 24
= 32 – 32
= 0
∴ x + 2 is a factor
x2 – x – 12 = x2 – 4x + 3x – 12
= x(x – 4) + 3(x – 4)
= (x – 4)(x + 3)
This (x + 2)(x + 3)(x – 4) are the factors.
x3 + x2 – 14x – 24 = (x + 2)(x + 3)(x – 4)
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