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Question
Show that (x – 1) is a factor of x3 – 7x2 + 14x – 8. Hence, completely factorise the given expression.
Solution
Let f(x) = x3 – 7x2 + 14x – 8
f(1) = (1)3 – 7(1)2 + 14(1) – 8
= 1 – 7 + 14 – 8
= 0
Hence, (x – 1) is a factor of f(x).
x2 – 6x + 8
`x - 1")"overline(x^3 - 7x^2 + 14 - 8)`
x3 – x2
– 6x2 + 14x
– 6x2 + 6x
8x – 8
8x – 8
0
∴ x3 – 7x2 + 14x – 8 = (x – 1)(x2 – 6x + 8)
= (x – 1)(x2 – 2x – 4x + 8)
= (x – 1)[x(x – 2) – 4(x – 2)]
= (x – 1)(x – 2)(x – 4)
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