Question # 01: Factorize each of the following expressions:
\((vi)\) \(\; \) \(x^4+11x^2y^2+y^4\)
Note: The question given in the book is incorrect. The correct form of the question should be: \(x^4-7x^2y^2+y^4\)
Solution:
\(\ \ \ \ x^{4} -7x^{2} y^{2} +y^{4} \)
\(=x^{4} -7x^{2} y^{2} +y^{4} +2x^{2} y^{2} -2x^{2} y^{2}\)
\(=x^{4} +y^{4} +2x^{2} y^{2} -9x^{2} y^{2}\)
\(=\left( x^{2}\right)^{2} +\left( y^{2}\right)^{2} +2x^{2} y^{2} -( 3xy)^{2}\)
\(=\left( x^{2} +y^{2}\right)^{2} -( 3xy)^{2}\)
\(=\left( x^{2} +y^{2} +3xy\right)\left( x^{2} +y^{2} -3xy\right)\)