Question # 01: Factorize each of the following expressions:
\((i)\) \(\;\) \( 4x^4+81y^4\)
Solution:
\(\ \ \ \ 4x^4+81y^4\)
\(=\left( 2x^{2}\right)^{2} +\left( 9y^{2}\right)^{2}\)
\(=\left( 2x^{2}\right)^{2} +\left( 9y^{2}\right)^{2} +2\left( 2x^{2}\right)\left( 9y^{2}\right)\)\(\ -\ \)\(2\left( 2x^{2}\right)\left( 9y^{2}\right)\)
\(=\left( 2x^{2} +9y^{2}\right)^{2} -36x^{2} y^{2}\)
\(=\left( 2x^{2} +9y^{2}\right)^{2} -( 6x y)^{2}\)
\(=\left( 2x^{2} +9y^{2} +6x y\right)\left( 2x^{2} +9y^{2} -6x y\right)\)