Question

Express in the product of two polynomial$x^4+y^4+x^2{y}^2$.

Answer 1

Let us add and subtract $x^2{y}^2$ in the given expression $x^4+y^4+x^2{y}^2$ and then solve as shown below:

$(x^4+y^4+x^2{y}^2+x^2{y}^2)-x^2{y}^2=\left[\right(x^2{)}^2+(y^2{)}^2+2x^2{y}^2]-x^2{y}^2=\left[\right(x^2+y^2{)}^2]-x^2{y}^2(\because (a+b{)}^2=a^2+b^2+2ab)=(x^2+y^2{)}^2-(xy{)}^2=(x^2+y^2-xy\left)\right(x^2+y^2+xy\left)\right(\because a^2-b^2=(a-b)(a+b))$

Hence, $x^4+y^4+x^2{y}^2=(x^2+y^2-xy)(x^2+y^2+xy)$.