Precalculus Examples

$2 x^{4} - x^{3} + 49 x^{2} - 25 x - 25$

Step 1

Regroup terms.

$- x^{3} - 25 x + 2 x^{4} + 49 x^{2} - 25$

Step 2

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

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Step 2.1

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

$- x (x^{2}) - 25 x + 2 x^{4} + 49 x^{2} - 25$

Step 2.2

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

$- x (x^{2}) - x (25) + 2 x^{4} + 49 x^{2} - 25$

Step 2.3

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

$- x (x^{2} + 25) + 2 x^{4} + 49 x^{2} - 25$

Step 3

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

$- x (x^{2} + 25) + 2 {(x^{2})}^{2} + 49 x^{2} - 25$

Step 4

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

$- x (x^{2} + 25) + 2 u^{2} + 49 u - 25$

Step 5

Factor by grouping.

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Step 5.1

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 $b = 49$ .

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Step 5.1.1

Factor $49$ out of $49 u$ .

$- x (x^{2} + 25) + 2 u^{2} + 49 (u) - 25$

Step 5.1.2

Rewrite $49$ as $- 1$ plus $50$

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Factor out the greatest common factor from each group.

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Step 5.2.1

Group the first two terms and the last two terms.

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

Step 5.2.2

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

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

Step 5.3

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

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

Step 6

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

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

Step 7

Factor $x^{2} + 25$ out of $- x (x^{2} + 25) + (2 x^{2} - 1) (x^{2} + 25)$ .

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Step 7.1

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

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

Step 7.2

Factor $x^{2} + 25$ out of $(2 x^{2} - 1) (x^{2} + 25)$ .

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

Step 7.3

Factor $x^{2} + 25$ out of $(x^{2} + 25) (- x) + (x^{2} + 25) (2 x^{2} - 1)$ .

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

Step 8

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

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

Step 9

Factor by grouping.

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Step 9.1

Reorder terms.

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

Step 9.2

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 9.2.1

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

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

Step 9.2.2

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

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

Step 9.2.3

Apply the distributive property.

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

Step 9.2.4

Multiply $u$ by $1$ .

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

Step 9.3

Factor out the greatest common factor from each group.

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Step 9.3.1

Group the first two terms and the last two terms.

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

Step 9.3.2

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

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

Step 9.4

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

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

Step 10

Factor.

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Step 10.1

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

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

Step 10.2

Remove unnecessary parentheses.

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