Question

Use the Factor Theorem to determine whether $g\left(x\right)$ is a factor of $p\left(x\right)$ in each of the following cases:
(i) $p\left(x\right)=2x^3+x^2-2x-1,g\left(x\right)=x+1$
(ii) $p\left(x\right)=x^3+3x^2+3x+1,g\left(x\right)=x+2$
(iii) $p\left(x\right)=x^3-4x^2+x+6,g\left(x\right)=x-3$

Answer 1

(i) Apply factor theorem

$x+1=0$

So $x=-1$

$2x^3+x^2-2x-1$

Replace $x$ by $-1$, we get

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

Reminder is $0$ so that $x+1$ is a factor of $2x^3+x^2-2x-1$

(ii) Apply factor theorem

$x+2=0$

So $x=-2$

$x^3+3x^2+3x+1$

Replace $x$ by $-2$, we get

$(-2{)}^3+3(-2{)}^2+3(-2)+1=-8+12-6+1=1$

Reminder is $1$ so that $x+2$ is not a factor of $x^3+3x^2+3x+1$

(iii) Apply factor theorem

$x-3=0$

So $x=3$

$x^3-4x^2+x+6$

Replace $x$ by $3$, we get

$(3{)}^3-4(3{)}^2+\left(3\right)+6=27-36+3+6=0$

Reminder is $0$ so that $x-3$ is a factor of $x^3-4x^2+x+6$