Find the HCF of the following pair of polynomials:
$4 x^{2} - 1$ and $4 x^{2} + 4 x + 1$

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Solution

We know that $H C F$ is the highest common factor.

Factorise $4 x^{2} - 1$ as follows:

$4 x^{2} - 1 = (2 x + 1) (2 x - 1)$ (using identity $a^{2} - b^{2} = (a + b) (a - b)$ )

Now, factorise $4 x^{2} + 4 x + 1$ as follows:

$4 x^{2} + 4 x + 1 = (2 x)^{2} + (1)^{2} + (2 \times 2 x \times 1) = (2 x + 1)^{2} = (2 x + 1) (2 x + 1)$
(using identity $(a + b)^{2} = a^{2} + b^{2} + 2 a b$ )

Since the common factor between the polynomials $4 x^{2} - 1$ and $4 x^{2} + 4 x + 1$ is $(2 x + 1)$ .

Hence, the $H C F$ is $(2 x + 1)$ .

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