Question

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

Answer 1

Concept: 0.5 Mark each
Application: 0.5 Mark each

(i) Let $p\left(x\right)=x^3+x^2+x+1$
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 $p\left(x\right)=x^4+x^3+x^2+x+1$ Zero of (x+1) is −1

$p(-1)={(-1)}^4+{(-1)}^3+{(-1)}^2-1+1$
$=1-1+1-1+1=1\ne 0$.

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