If (x−1) is a factor of ax3−bx2+cx, what is the value of a3−

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Question

If

(x - 1)

is a factor of

a x^{3} - b x^{2} + c x

, what is the value of

a^{3} - b^{3} + c^{3}

?

Options:

A . 2abc

B . -2abc

C . 3abc

D . -3abc

Answer: Option D
:
D

p (x) = a x^{3} - b x^{2} + c x

If (x - 1) is a factor then x = 1 is the zero of polynomial
p(1) = 0

p (1) = a - b + c = 0

It can be written as

a + (- b) + c = 0

We know that if

a + b + c = 0, t h e n a^{3} + b^{3} + c^{3} = 3 a b c

So,

a^{3} + {(- b)}^{3} + c^{3} = 3 a (- b) c = - 3 a b c

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