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Question

If x21 is a factor of ax4+bx3+cx2+dx+e, show that a+c+e=b+d=0.

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Solution

p(x)=ax4+bx3+cx2+dx+e

Then,
(x21) is a factor of p(x)

(x1)(x+1) is a factor of p(x)

(x1) and (x+1) are factors of p(x)

p(1)=0 and p(1)=0

a+b+c+d+e=0 and ab+cd+e=0

Adding and subtracting these two equations, we get,
2(a+c+e)=0 and 2(b+d)=0

a+c+e=0 and b+d=0a+c+e=b+d=0

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