Factor 25x^4-10x^2y^2+y^4-100

$25 x^{4} - 10 x^{2} y^{2} + y^{4} - 100$

Step 1

Factor using the perfect square rule.

Rewrite $25 x^{4}$ as ${(5 x^{2})}^{2}$ .

${(5 x^{2})}^{2} - 10 x^{2} y^{2} + y^{4} - 100$

Rewrite $y^{4}$ as ${(y^{2})}^{2}$ .

${(5 x^{2})}^{2} - 10 x^{2} y^{2} + {(y^{2})}^{2} - 100$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x^{2} y^{2} = 2 \cdot (5 x^{2}) \cdot y^{2}$

Rewrite the polynomial.

${(5 x^{2})}^{2} - 2 \cdot (5 x^{2}) \cdot y^{2} + {(y^{2})}^{2} - 100$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 5 x^{2}$ and $b = y^{2}$ .

${(5 x^{2} - y^{2})}^{2} - 100$

Step 2

Rewrite $100$ as $10^{2}$ .

${(5 x^{2} - y^{2})}^{2} - 10^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 x^{2} - y^{2}$ and $b = 10$ .

$(5 x^{2} - y^{2} + 10) (5 x^{2} - y^{2} - 10)$

Factor 25x^4-10x^2y^2+y^4-100

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