Precalculus Examples

$7 x^{4} + 6 x^{3} - 92 x^{2} - 78 x + 13$

Step 1

Regroup terms.

$6 x^{3} - 78 x + 7 x^{4} - 92 x^{2} + 13$

Step 2

Factor $6 x$ out of $6 x^{3} - 78 x$ .

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Step 2.1

Factor $6 x$ out of $6 x^{3}$ .

$6 x (x^{2}) - 78 x + 7 x^{4} - 92 x^{2} + 13$

Step 2.2

Factor $6 x$ out of $- 78 x$ .

$6 x (x^{2}) + 6 x (- 13) + 7 x^{4} - 92 x^{2} + 13$

Step 2.3

Factor $6 x$ out of $6 x (x^{2}) + 6 x (- 13)$ .

$6 x (x^{2} - 13) + 7 x^{4} - 92 x^{2} + 13$

Step 3

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

$6 x (x^{2} - 13) + 7 {(x^{2})}^{2} - 92 x^{2} + 13$

Step 4

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

$6 x (x^{2} - 13) + 7 u^{2} - 92 u + 13$

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 $a \cdot c = 7 \cdot 13 = 91$ and whose sum is $b = - 92$ .

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Step 5.1.1

Factor $- 92$ out of $- 92 u$ .

$6 x (x^{2} - 13) + 7 u^{2} - 92 (u) + 13$

Step 5.1.2

Rewrite $- 92$ as $- 1$ plus $- 91$

$6 x (x^{2} - 13) + 7 u^{2} + (- 1 - 91) u + 13$

Step 5.1.3

Apply the distributive property.

$6 x (x^{2} - 13) + 7 u^{2} - 1 u - 91 u + 13$

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.

$6 x (x^{2} - 13) + (7 u^{2} - 1 u) - 91 u + 13$

Step 5.2.2

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

$6 x (x^{2} - 13) + u (7 u - 1) - 13 (7 u - 1)$

Step 5.3

Factor the polynomial by factoring out the greatest common factor, $7 u - 1$ .

$6 x (x^{2} - 13) + (7 u - 1) (u - 13)$

Step 6

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

$6 x (x^{2} - 13) + (7 x^{2} - 1) (x^{2} - 13)$

Step 7

Factor $x^{2} - 13$ out of $6 x (x^{2} - 13) + (7 x^{2} - 1) (x^{2} - 13)$ .

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Step 7.1

Factor $x^{2} - 13$ out of $6 x (x^{2} - 13)$ .

$(x^{2} - 13) (6 x) + (7 x^{2} - 1) (x^{2} - 13)$

Step 7.2

Factor $x^{2} - 13$ out of $(7 x^{2} - 1) (x^{2} - 13)$ .

$(x^{2} - 13) (6 x) + (x^{2} - 13) (7 x^{2} - 1)$

Step 7.3

Factor $x^{2} - 13$ out of $(x^{2} - 13) (6 x) + (x^{2} - 13) (7 x^{2} - 1)$ .

$(x^{2} - 13) (6 x + 7 x^{2} - 1)$

Step 8

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

$(x^{2} - 13) (6 u + 7 u^{2} - 1)$

Step 9

Factor by grouping.

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Step 9.1

Reorder terms.

$(x^{2} - 13) (7 u^{2} + 6 u - 1)$

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 $a \cdot c = 7 \cdot - 1 = - 7$ and whose sum is $b = 6$ .

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Step 9.2.1

Factor $6$ out of $6 u$ .

$(x^{2} - 13) (7 u^{2} + 6 (u) - 1)$

Step 9.2.2

Rewrite $6$ as $- 1$ plus $7$

$(x^{2} - 13) (7 u^{2} + (- 1 + 7) u - 1)$

Step 9.2.3

Apply the distributive property.

$(x^{2} - 13) (7 u^{2} - 1 u + 7 u - 1)$

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.

$(x^{2} - 13) ((7 u^{2} - 1 u) + 7 u - 1)$

Step 9.3.2

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

$(x^{2} - 13) (u (7 u - 1) + 1 (7 u - 1))$

Step 9.4

Factor the polynomial by factoring out the greatest common factor, $7 u - 1$ .

$(x^{2} - 13) ((7 u - 1) (u + 1))$

Step 10

Factor.

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Step 10.1

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

$(x^{2} - 13) ((7 x - 1) (x + 1))$

Step 10.2

Remove unnecessary parentheses.

$(x^{2} - 13) (7 x - 1) (x + 1)$