Question

If $(x+2)$ and $(x-1)$ are factors of $x^3+10x^2+px+q$, find $p$ and $q$.

• A. $p=23,q=-19$
• B. $p=12,q=9$
• C. $p=7,q=-18$
• D. $p=11,q=-13$

Answer 1

The correct option is C $p=7,q=-18$

Let, $f\left(x\right)=x^3+10x^2+px+q$

Given : $(x+2)$ and $(x-1)$ are factors of $f\left(x\right)$

$\therefore f(-2)=0,f\left(1\right)=0$

$f(-2)=(-2{)}^3+10(-2{)}^2+p(-2)+q$

$⟹0=-8+40-2p+q$

$⟹2p-q=32$ -----------$\left(i\right)$

$f\left(1\right)=(1{)}^3+10(1{)}^2+p\left(1\right)+q$

$⟹0=1+10+p+q$

$⟹p+q=-11$ ----------$\left(ii\right)$

Adding $\left(i\right)$ and $\left(ii\right)$,

we get,

$2p-q+p+q=32-11$

$⟹3p=21$

$⟹p=7$

Substituting $p=7$ in $\left(ii\right)$, we get,

$7+q=-11$

$⟹q=-11-7=-18$