Simplify each of the following products:$ (2 x^{4}-4 x^{2}+1)(2 x^{4}-4 x^{2}-1) $

Given:

\( (2 x^{4}-4 x^{2}+1)(2 x^{4}-4 x^{2}-1) \)

To do:

We have to simplify the given product.

Solution:

We know that,

$(a+b)^2=a^2+b^2+2ab$

$(a-b)^2=a^2+b^2-2ab$

$(a+b)(a-b)=a^2-b^2$

Therefore,

$(2 x^{4}-4 x^{2}+1)(2 x^{4}-4 x^{2}-1)=[(2 x^{4}-4 x^{2})+1][(2 x^{4}-4 x^{2})-1]$

$=(2 x^{4}-4 x^{2})^{2}-(1)^{2}$

$=(2 x^{4})^{2}-2 \times 2 x^{4} \times 4 x^{2}+(4 x^{2})^{2}-1$

$=4 x^{8}-16 x^{6}+16 x^{4}-1$

Hence, $(2 x^{4}-4 x^{2}+1)(2 x^{4}-4 x^{2}-1)=4 x^{8}-16 x^{6}+16 x^{4}-1$.

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Updated on: 10-Oct-2022

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