Solve x^4-x^2y^2-72y^4

$$x ^ { 4 } - x ^ { 2 } y ^ { 2 } - 72 y ^ { 4 }$$

Factor

$$\left(x-3y\right)\left(x+3y\right)\left(x^{2}+8y^{2}\right)$$

Solution Steps

Consider x^{4}-x^{2}y^{2}-72y^{4} as a polynomial over variable x.

$$x^{4}-y^{2}x^{2}-72y^{4}$$

Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{4} and m divides the constant factor -72y^{4}. One such factor is x^{2}-9y^{2}. Factor the polynomial by dividing it by this factor.

$$\left(x^{2}-9y^{2}\right)\left(x^{2}+8y^{2}\right)$$

Consider x^{2}-9y^{2}. Rewrite x^{2}-9y^{2} as x^{2}-\left(3y\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-3y\right)\left(x+3y\right)$$

Rewrite the complete factored expression.

$$\left(x-3y\right)\left(x+3y\right)\left(x^{2}+8y^{2}\right)$$

Evaluate

$$x^{4}-72y^{4}-\left(xy\right)^{2}$$