Question

Factorise the following expression: $a^4-2a^2{b}^2+b^4$

Answer 1

Factorize by using identities

Given, $a^4-2a^2{b}^2+b^4$

We know that ${(a-b)}^2=a^2+b^2-2ab$

Now, compare $a^4-2a^2{b}^2+b^4$ with this identity, we get

$$\begin{gathered} a^4-2a^2{b}^2+b^4={\left(a^2\right)}^2+{\left(b^2\right)}^2-(2\times a^2\times b^2) \\ {=(a^2-b^2)}^2 \end{gathered}$$ $\mathbf{[}\mathbf{\because }{\mathbf{(}\mathbf{a}\mathbf{-}\mathbf{b}\mathbf{)}}^{\mathbf{2}}\mathbf{=}{\mathit{a}}^{\mathbf{2}}\mathbf{+}{\mathit{b}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathit{a}\mathit{b}\mathbf{]}$

Therefore, $a^4-2a^2{b}^2+b^4$ can be factorized as ${(a^2-b^2)}^2$.