Question

Determine which of the following polynomials has (x+1) as a factor:
(i) $x^3$+$x^2$+x+1 (ii) $x^4$+$x^3$+$x^2$+x+1

Answer 1

(i) Let p(x)=$x^3$+$x^2$+x+1

Zero of (x+1) is −1

p(−1)=${(-1)}^3$+${(-1)}^2$−1+1=−1+1−1+1=0

Therefore, by the Factor Theorem, (x+1) is the factor of $x^3$+$x^2$+x+1.

(ii) Let p(x)=$x^4$+$x^3$+$x^2$+x+1

Zero of (x+1) is −1

p(−1)=${(-1)}^4$+${(-1)}^3$+${(-1)}^2$−1+1 = 1−1+1−1+1 = 1 which is not equal to 0.

Therefore, by the Factor theorem, (x+1) is not the factor of p(x) = $x^4$+$x^3$+$x^2$+x+1.