Precalculus Examples

$x^{4} - 26 x^{2} y^{2} + 25 y^{4}$

Step 1

Rewrite the middle term.

$x^{4} + 2 x^{2} (5 y^{2}) - 36 x^{2} y^{2} + 25 y^{4}$

Step 2

Rearrange terms.

$x^{4} + 2 x^{2} (5 y^{2}) + 25 y^{4} - 36 x^{2} y^{2}$

Step 3

Factor first three terms by perfect square rule.

${(x^{2} + 5 y^{2})}^{2} - 36 x^{2} y^{2}$

Step 4

Rewrite $36 x^{2} y^{2}$ as ${(6 x y)}^{2}$ .

${(x^{2} + 5 y^{2})}^{2} - {(6 x y)}^{2}$

Step 5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2} + 5 y^{2}$ and $b = 6 x y$ .

$(x^{2} + 5 y^{2} + 6 x y) (x^{2} + 5 y^{2} - (6 x y))$

Step 6

Simplify.

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Step 6.1

Factor by grouping.

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Step 6.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 1 \cdot 5 = 5$ and whose sum is $b = 6$ .

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Step 6.1.1.1

Reorder $5 y^{2}$ and $6 x y$ .

$(x^{2} + 6 x y + 5 y^{2}) (x^{2} + 5 y^{2} - (6 x y))$

Step 6.1.1.2

Factor $6$ out of $6 x y$ .

$(x^{2} + 6 (x y) + 5 y^{2}) (x^{2} + 5 y^{2} - (6 x y))$

Step 6.1.1.3

Rewrite $6$ as $1$ plus $5$

$(x^{2} + (1 + 5) (x y) + 5 y^{2}) (x^{2} + 5 y^{2} - (6 x y))$

Step 6.1.1.4

Apply the distributive property.

$(x^{2} + 1 (x y) + 5 (x y) + 5 y^{2}) (x^{2} + 5 y^{2} - (6 x y))$

Step 6.1.1.5

Multiply $x y$ by $1$ .

$(x^{2} + x y + 5 (x y) + 5 y^{2}) (x^{2} + 5 y^{2} - (6 x y))$

Step 6.1.1.6

Remove unnecessary parentheses.

$(x^{2} + x y + 5 x y + 5 y^{2}) (x^{2} + 5 y^{2} - (6 x y))$

Step 6.1.2

Factor out the greatest common factor from each group.

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Step 6.1.2.1

Group the first two terms and the last two terms.

$((x^{2} + x y) + 5 x y + 5 y^{2}) (x^{2} + 5 y^{2} - (6 x y))$

Step 6.1.2.2

Factor out the greatest common factor (GCF) from each group.

$(x (x + y) + 5 y (x + y)) (x^{2} + 5 y^{2} - (6 x y))$

Step 6.1.3

Factor the polynomial by factoring out the greatest common factor, $x + y$ .

$(x + y) (x + 5 y) (x^{2} + 5 y^{2} - (6 x y))$

Step 6.2

Multiply $6$ by $- 1$ .

$(x + y) (x + 5 y) (x^{2} + 5 y^{2} - 6 (x y))$

Step 7

Factor.

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Step 7.1

Factor by grouping.

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Step 7.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 1 \cdot 5 = 5$ and whose sum is $b = - 6$ .

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Step 7.1.1.1

Reorder terms.

$(x + y) (x + 5 y) (x^{2} + 5 y^{2} - 6 x y)$

Step 7.1.1.2

Reorder $5 y^{2}$ and $- 6 x y$ .

$(x + y) (x + 5 y) (x^{2} - 6 x y + 5 y^{2})$

Step 7.1.1.3

Factor $- 6$ out of $- 6 x y$ .

$(x + y) (x + 5 y) (x^{2} - 6 (x y) + 5 y^{2})$

Step 7.1.1.4

Rewrite $- 6$ as $- 1$ plus $- 5$

$(x + y) (x + 5 y) (x^{2} + (- 1 - 5) (x y) + 5 y^{2})$

Step 7.1.1.5

Apply the distributive property.

$(x + y) (x + 5 y) (x^{2} - 1 (x y) - 5 (x y) + 5 y^{2})$

Step 7.1.1.6

Remove unnecessary parentheses.

$(x + y) (x + 5 y) (x^{2} - 1 x y - 5 (x y) + 5 y^{2})$

Step 7.1.1.7

Remove unnecessary parentheses.

$(x + y) (x + 5 y) (x^{2} - 1 x y - 5 x y + 5 y^{2})$

Step 7.1.2

Factor out the greatest common factor from each group.

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Step 7.1.2.1

Group the first two terms and the last two terms.

$(x + y) (x + 5 y) ((x^{2} - 1 x y) - 5 x y + 5 y^{2})$

Step 7.1.2.2

Factor out the greatest common factor (GCF) from each group.

$(x + y) (x + 5 y) (x (x - y) - 5 y (x - y))$

Step 7.1.3

Factor the polynomial by factoring out the greatest common factor, $x - y$ .

$(x + y) (x + 5 y) ((x - y) (x - 5 y))$

Step 7.2

Remove unnecessary parentheses.

$(x + y) (x + 5 y) (x - y) (x - 5 y)$