Question

Find the HCF of the following pair if polynomials:
$a^3+125$ and $a^2-25$

Answer 1

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

Factorise $a^3+125$ as follows:

$a^3+125=a^3+5^3=(a+5)[a^2+5^2-(a\times 5\left)\right]=(a+5)(a^2+25-5a)$

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

Now, factorise $a^2-25$ as follows:

$a^2-25=(a+5)(a-5)$ (using identity $a^2-b^2=(a+b)(a-b)$)

Since the common factor between the polynomials $a^3+125$ and $a^2-25$ is $(a+5)$.

Hence, the $HCF$ is $(a+5)$.