Question

Find the HCF of the following pair of polynomials:
$x^3+27$ and $x^2-9$

Answer 1

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

Factorise $x^3+27$ as follows:

$x^3+27=x^3+3^3=(x+3)[x^2+3^2-(x\times 3\left)\right]=(x+3)(x^2+9-3x)$

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

Now, factorise $x^2-9$ as follows:

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

Since the common factor between the polynomials $x^3+27$ and $x^2-9$ is $(x+3)$.

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