Question

If $(x-3)$ is the HCF of $p\left(x\right)=x^3+a{x}^2+bx-6$ and $q\left(x\right)=x^3-x(b-4)+a$, find the value of $a$ and $b$.

Answer 1

That $(x-3)$ is the HCF, means $(x-3)$ is a factor of both the polynomials. Hence $p\left(x\right)=(x-3)f\left(x\right)$ and $q\left(x\right)=(x-3)g\left(x\right)$ for some polynomials $f\left(x\right)$ and $g\left(x\right)$.
by taking $x=3$. we see that $p\left(3\right)=0$ and $q\left(3\right)=0$. thus
$0=p\left(3\right)=3^3+a\times 3^2+b\times 3-6=21+9a+3b$
$0=q\left(3\right)=3^3-3(b-4)+a=39-3b+a$.
We get two simultaneous linear equations for $a$ and $b$;
$9a+3b=-21$
$a-3b=-39$
Adding these tow we get $10a=-60$ or $a=-6$. From the second equation $3b=a+39=-6+39=33$. hence $b=11$.