Question

If $(x+2)$ is a factor of $x^2+ax+2b$ and $a+b=4$ find $a$ and $b$.

Answer 1

Let $p\left(x\right)=x^2+ax+2b$

If $x+2$ is a factor of $p\left(x\right)$

Then by factor theorem

$p\left(-2\right)=0$

$\Rightarrow {\left(-2\right)}^2+a\left(-2\right)+2b=0$

$\Rightarrow 4-2a+2b=0$

$\Rightarrow a-b=2$ ---(i)

Also given that $a+b=4$ ---(ii)

Adiing (i) and (ii) we get

$2a=6$

$\Rightarrow a=3$

putting $a=3$ in (i) we get

$a-b=2$

$\Rightarrow 3-b=2$

$\Rightarrow b=1$

Therefore we get $a=3$ and $b=1$