# Prove that if x2 – 1 is a factor of ax3 + bx3 + cx + d, then a = –c and b = –d.

Given, ax3 + bx2 + cx + d

Need to show a = – c and b = – d if x2 – 1 is a factor

consider, x2 – 1 is a factor of ax3 + bx2 + cx + d

then x = + 1, x = – 1

substitute x value in the equation ax3 + bx2 + cx + d

We get as follows

for x = 1 we get a(1)3 + b(1)2 + c(1) + d

= a + b + c + d ……..eq(1)

for x = – 1 we get a(– 1)3 + b(– 1)2 + c(– 1) + d

= – a + b – c + d ………eq(2)

Solving the two equations we get

(a + b + c + d) + (– a + b – c + d) = 0

b + d = 0

b = – d

And if (a + b + c + d) – (– a + b – c + d) = 0

a + c = 0

a = – c

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