Solve x^4-8x^2y^2+16y^4

$$x ^ { 4 } - 8 x ^ { 2 } y ^ { 2 } + 16 y ^ { 4 }$$

Factor

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

Solution Steps

Consider x^{4}-8x^{2}y^{2}+16y^{4} as a polynomial over variable x.

$$x^{4}-8y^{2}x^{2}+16y^{4}$$

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 16y^{4}. One such factor is x^{2}-4y^{2}. Factor the polynomial by dividing it by this factor.

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

Consider x^{2}-4y^{2}. Rewrite x^{2}-4y^{2} as x^{2}-\left(2y\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-2y\right)\left(x+2y\right)$$

Rewrite the complete factored expression.

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

Evaluate

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