Solve 25x^{4}-x^2-25x^2y^2+y^2

$$25 x ^ { 4 } - x ^ { 2 } - 25 x ^ { 2 } y ^ { 2 } + y ^ { 2 }$$

Factor

$$\left(5x-1\right)\left(x-y\right)\left(5x+1\right)\left(x+y\right)$$

Solution Steps

Do the grouping 25x^{4}-x^{2}-25x^{2}y^{2}+y^{2}=\left(25x^{4}-x^{2}\right)+\left(-25x^{2}y^{2}+y^{2}\right), and factor out x^{2} in the first and -y^{2} in the second group.

$$x^{2}\left(25x^{2}-1\right)-y^{2}\left(25x^{2}-1\right)$$

Factor out common term 25x^{2}-1 by using distributive property.

$$\left(25x^{2}-1\right)\left(x^{2}-y^{2}\right)$$

Consider 25x^{2}-1. Rewrite 25x^{2}-1 as \left(5x\right)^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(5x-1\right)\left(5x+1\right)$$

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

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

Rewrite the complete factored expression.

$$\left(5x-1\right)\left(5x+1\right)\left(x-y\right)\left(x+y\right)$$

Evaluate

$$\left(25x^{2}-1\right)\left(x^{2}-y^{2}\right)$$