Question

Check whether $-2,-5$ are the zeros of the polynomial $p\left(x\right)=x^2+7x+10$

Answer 1

If $a$ is a zero of $p\left(x\right)$, then by factor theorem, $p\left(a\right)=0$

Now, $p\left(x\right)=x^2+7x+10$

For $x=-2$

$\Rightarrow p(-2)=(-2{)}^2+7(-2)+10=4-14+10=14-14=0$

Since $p(-2)=0,-2$ is a factor of $p\left(x\right)$

For $x=-5$

$\Rightarrow p(-2)=(-5{)}^2+7(-5)+10=25-35+10=35-35=0$

Since $p(-5)=0,-5$ is a factor of $p\left(x\right)$

Hence, $-2,-5$ are the zeroes/factors of $p\left(x\right)$.