Question # 01: Factorize each of the following expressions:
\((v)\) \(\; \) \(x^4-30x^2y^2+9y^4 \)
Solution:
\(\ \ \ \ x^4-30x^2y^2+9y^4 \)
\(=x^{4} -30x^{2} y^{2} +9y^{4} +36x^{2} y^{2} -36x^{2} y^{2}\)
\(=x^{4} +6x^{2} y^{2} +9y^{4} -36x^{2} y^{2}\)
\(=\left( x^{2}\right)^{2} +2\left( x^{2}\right)\left( 3y^{2}\right) +\left( 3y^{2}\right)^{2} -( 6x y)^{2}\)
\(=\left( x^{2} +3y^{2}\right)^{2} -( 6x y)^{2}\)
\(=\left( x^{2} +3y^{2} +6xy\right)\left( x^{2} +3y^{2} +6xy\right)\)