Precalculus Examples

$2 x^{4} - x^{3} - 51 x^{2} + 25 x + 25$

Step 1

Regroup terms.

$- x^{3} + 25 x + 2 x^{4} - 51 x^{2} + 25$

Step 2

Factor $- x$ out of $- x^{3} + 25 x$ .

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Step 2.1

Factor $- x$ out of $- x^{3}$ .

$- x (x^{2}) + 25 x + 2 x^{4} - 51 x^{2} + 25$

Step 2.2

Factor $- x$ out of $25 x$ .

$- x (x^{2}) - x (- 25) + 2 x^{4} - 51 x^{2} + 25$

Step 2.3

Factor $- x$ out of $- x (x^{2}) - x (- 25)$ .

$- x (x^{2} - 25) + 2 x^{4} - 51 x^{2} + 25$

Step 3

Rewrite $25$ as $5^{2}$ .

$- x (x^{2} - 5^{2}) + 2 x^{4} - 51 x^{2} + 25$

Step 4

Factor.

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Step 4.1

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

$- x ((x + 5) (x - 5)) + 2 x^{4} - 51 x^{2} + 25$

Step 4.2

Remove unnecessary parentheses.

$- x (x + 5) (x - 5) + 2 x^{4} - 51 x^{2} + 25$

Step 5

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

$- x (x + 5) (x - 5) + 2 {(x^{2})}^{2} - 51 x^{2} + 25$

Step 6

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- x (x + 5) (x - 5) + 2 u^{2} - 51 u + 25$

Step 7

Factor by grouping.

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Step 7.1

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 $a \cdot c = 2 \cdot 25 = 50$ and whose sum is $b = - 51$ .

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Step 7.1.1

Factor $- 51$ out of $- 51 u$ .

$- x (x + 5) (x - 5) + 2 u^{2} - 51 (u) + 25$

Step 7.1.2

Rewrite $- 51$ as $- 1$ plus $- 50$

$- x (x + 5) (x - 5) + 2 u^{2} + (- 1 - 50) u + 25$

Step 7.1.3

Apply the distributive property.

$- x (x + 5) (x - 5) + 2 u^{2} - 1 u - 50 u + 25$

Step 7.2

Factor out the greatest common factor from each group.

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Step 7.2.1

Group the first two terms and the last two terms.

$- x (x + 5) (x - 5) + (2 u^{2} - 1 u) - 50 u + 25$

Step 7.2.2

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

$- x (x + 5) (x - 5) + u (2 u - 1) - 25 (2 u - 1)$

Step 7.3

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

$- x (x + 5) (x - 5) + (2 u - 1) (u - 25)$

Step 8

Replace all occurrences of $u$ with $x^{2}$ .

$- x (x + 5) (x - 5) + (2 x^{2} - 1) (x^{2} - 25)$

Step 9

Rewrite $25$ as $5^{2}$ .

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

Step 10

Factor.

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Step 10.1

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

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

Step 10.2

Remove unnecessary parentheses.

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

Step 11

Factor $(x + 5) (x - 5)$ out of $- x (x + 5) (x - 5) + (2 x^{2} - 1) (x + 5) (x - 5)$ .

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Step 11.1

Factor $(x + 5) (x - 5)$ out of $- x (x + 5) (x - 5)$ .

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

Step 11.2

Factor $(x + 5) (x - 5)$ out of $(2 x^{2} - 1) (x + 5) (x - 5)$ .

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

Step 11.3

Factor $(x + 5) (x - 5)$ out of $(x + 5) (x - 5) (- x) + (x + 5) (x - 5) (2 x^{2} - 1)$ .

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

Step 12

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

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

Step 13

Factor by grouping.

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Step 13.1

Reorder terms.

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

Step 13.2

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 $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = - 1$ .

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Step 13.2.1

Factor $- 1$ out of $- u$ .

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

Step 13.2.2

Rewrite $- 1$ as $1$ plus $- 2$

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

Step 13.2.3

Apply the distributive property.

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

Step 13.2.4

Multiply $u$ by $1$ .

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

Step 13.3

Factor out the greatest common factor from each group.

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Step 13.3.1

Group the first two terms and the last two terms.

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

Step 13.3.2

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

$(x + 5) (x - 5) (u (2 u + 1) - (2 u + 1))$

Step 13.4

Factor the polynomial by factoring out the greatest common factor, $2 u + 1$ .

$(x + 5) (x - 5) ((2 u + 1) (u - 1))$

Step 14

Factor.

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Step 14.1

Replace all occurrences of $u$ with $x$ .

$(x + 5) (x - 5) ((2 x + 1) (x - 1))$

Step 14.2

Remove unnecessary parentheses.

$(x + 5) (x - 5) (2 x + 1) (x - 1)$