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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