Question

Suppose the polynomials $p\left(y\right)=(y-1)(2y^2+ay+2)$ and $q\left(y\right)=(y+2)(3y^2-by+1)$ have their HCF $h\left(y\right)=y^2+y-2$. Find the values of $a$ and $b$.

Answer 1

Observe $h\left(y\right)=y^2+y-2=(y+2)(y-1)$. Thus both $y+2$ and $y-1$ are factors of both $p\left(y\right)$ and $q\left(y\right)$. Hence $(y-1)(y+2)$ is a factor of $p\left(y\right)=(y-1)(2y^2+ay+2)$. This shows that $(y+2)$ is a factor of $2y^2+ay+2$. We may write
$2v+ay+2=(y+2)f\left(y\right)$
for some polynomial $f\left(y\right)$. Putting $y=-2$ in this, we obtain
$2\times {(-2)}^2+2=0$.
We obtain $10-2a=0$. Hence $a=5$. Similarly $(y-1)(y+2)$ is a factor of $q\left(y\right)=(y+2)(3y^2-by+1)$. This shows that $y-1$ is a factor of $3y^2-by+1$ and we may write
$3y^2-by+1=(y-1)g\left(y\right)$
for some polynomial $g\left(y\right)$. Taking $y=1$, we obtain
$3\times {\left(1\right)}^2-b\times \left(1\right)+1=0$.
This gives $4-b=0$. Hence $b=4$.
Thus $a=5$ and $b=4$.