Question

Find the HCF of$p\left(x\right)=x^4-2x^3-15x^2$ and $q\left(x\right)=x^3-9x$

Answer 1

We know that $HCF$ is the highest common factor.

Factorise $x^4-2x^3-15x^2$ as follows:

$x^4-2x^3-15x^2=x^2(x^2-2x-15)=x^2(x^2-5x+3x-15)=x^2\left[x\right(x-5)+3(x-5\left)\right]=x^2(x+3)(x-5)$

Now, factorise $x^3-9x$ as follows:

$x^3-9x=x(x^2-9)=x(x+3)(x-3)$ (using identity $a^2-b^2=(a+b)(a-b)$)

Since the common factor between the polynomials $x^4-2x^3-15x^2$ and $x^3-9x$ is $x(x+3)$.

Hence, the $HCF$ is $x(x+3)$.