Question

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:
$\left(i\right)p\left(x\right)=2x^3+x^2-2x-1,g\left(x\right)=x+1$
$\left(ii\right)p\left(x\right)=x^3+3x^2+3x+1,g\left(x\right)=x+2$ [4 MARKS]

Answer 1

Concept: 1 Mark
Application: 1.5 Marks each

$\left(i\right)p\left(x\right)=2x^3+x^2-2x-1,g\left(x\right)=x+1$

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

Therefore, by the Factor Theorem, (x+1) is a factor of p(x).

$\left(ii\right)p\left(x\right)=x^3+3x^2+3x+1,g\left(x\right)=x+2$

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

Therefore, by the Factor Theorem, (x+2) is not a factor of p(x).