Q1 of 86 Page 48

Determine which of the following polynomials has (x + 1) as a factor.

x4 + 2x3 + 2x2 + x + 1

Let f(x) = x4 + 2x3 + 2x2 + x + 1


By Factor Theorem, we know that,


If p(x) is a polynomial and a is any real number, then (x – a) is a factor of p(x), if p(a) = 0


For checking (x+1) to be a factor, we will find f(–1)


⇒ f(–1) = (–1)4 + 2(–1)3 + 2(–1)2 + (–1) + 1


⇒ f(–1) = 1 – 2 + 2 – 1 + 1


⇒ f(–1) = 1


As, f(–1) is not equal to zero, therefore (x+1) is not a factor x4 + 2x3 + 2x2 + x + 1


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