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Question
Show that (x - 1) is a factor of x3 - 7x2 + 14x - 8. Hence, completely factorise the above expression.
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Solution
If (x - 1) is a factor of x3 - 7x2 + 14x - 8 then on putting x - 1 = 0
x = 1
f(1) = 0
= 13 - 7(1)2 + 14(1) - 8
= 1 - 7 + 14 - 8 = 0
Hence, x - 1 is one factor.
To find other factors
= x3 - 7x2 + 14x - 8
= x2(x - 1) - 6x(x - 1) + 8(x - 1)
= (x - 1) (x2 - 6x + 8)
= (x - 1) (x2 - 4x - 2x + 8)
= (x - 1) {x(x - 4) - 2(x - 4)}
= (x - 1) (x - 2) (x - 4).
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