Precalculus Examples

$6 x^{4} - 11 x^{3} - 51 x^{2} + 99 x - 27$

Step 1

Regroup terms.

$- 11 x^{3} + 99 x + 6 x^{4} - 51 x^{2} - 27$

Step 2

Factor $- 11 x$ out of $- 11 x^{3} + 99 x$ .

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Step 2.1

Factor $- 11 x$ out of $- 11 x^{3}$ .

$- 11 x (x^{2}) + 99 x + 6 x^{4} - 51 x^{2} - 27$

Step 2.2

Factor $- 11 x$ out of $99 x$ .

$- 11 x (x^{2}) - 11 x (- 9) + 6 x^{4} - 51 x^{2} - 27$

Step 2.3

Factor $- 11 x$ out of $- 11 x (x^{2}) - 11 x (- 9)$ .

$- 11 x (x^{2} - 9) + 6 x^{4} - 51 x^{2} - 27$

Step 3

Rewrite $9$ as $3^{2}$ .

$- 11 x (x^{2} - 3^{2}) + 6 x^{4} - 51 x^{2} - 27$

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 $b = 3$ .

$- 11 x ((x + 3) (x - 3)) + 6 x^{4} - 51 x^{2} - 27$

Step 4.2

Remove unnecessary parentheses.

$- 11 x (x + 3) (x - 3) + 6 x^{4} - 51 x^{2} - 27$

Step 5

Factor $3$ out of $6 x^{4} - 51 x^{2} - 27$ .

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Step 5.1

Factor $3$ out of $6 x^{4}$ .

$- 11 x (x + 3) (x - 3) + 3 (2 x^{4}) - 51 x^{2} - 27$

Step 5.2

Factor $3$ out of $- 51 x^{2}$ .

$- 11 x (x + 3) (x - 3) + 3 (2 x^{4}) + 3 (- 17 x^{2}) - 27$

Step 5.3

Factor $3$ out of $- 27$ .

$- 11 x (x + 3) (x - 3) + 3 (2 x^{4}) + 3 (- 17 x^{2}) + 3 (- 9)$

Step 5.4

Factor $3$ out of $3 (2 x^{4}) + 3 (- 17 x^{2})$ .

$- 11 x (x + 3) (x - 3) + 3 (2 x^{4} - 17 x^{2}) + 3 (- 9)$

Step 5.5

Factor $3$ out of $3 (2 x^{4} - 17 x^{2}) + 3 (- 9)$ .

$- 11 x (x + 3) (x - 3) + 3 (2 x^{4} - 17 x^{2} - 9)$

Step 6

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

$- 11 x (x + 3) (x - 3) + 3 (2 {(x^{2})}^{2} - 17 x^{2} - 9)$

Step 7

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

$- 11 x (x + 3) (x - 3) + 3 (2 u^{2} - 17 u - 9)$

Step 8

Factor by grouping.

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Step 8.1

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 $a \cdot c = 2 \cdot - 9 = - 18$ and whose sum is $b = - 17$ .

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Step 8.1.1

Factor $- 17$ out of $- 17 u$ .

$- 11 x (x + 3) (x - 3) + 3 (2 u^{2} - 17 (u) - 9)$

Step 8.1.2

Rewrite $- 17$ as $1$ plus $- 18$

$- 11 x (x + 3) (x - 3) + 3 (2 u^{2} + (1 - 18) u - 9)$

Step 8.1.3

Apply the distributive property.

$- 11 x (x + 3) (x - 3) + 3 (2 u^{2} + 1 u - 18 u - 9)$

Step 8.1.4

Multiply $u$ by $1$ .

$- 11 x (x + 3) (x - 3) + 3 (2 u^{2} + u - 18 u - 9)$

Step 8.2

Factor out the greatest common factor from each group.

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Step 8.2.1

Group the first two terms and the last two terms.

$- 11 x (x + 3) (x - 3) + 3 ((2 u^{2} + u) - 18 u - 9)$

Step 8.2.2

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

$- 11 x (x + 3) (x - 3) + 3 (u (2 u + 1) - 9 (2 u + 1))$

Step 8.3

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

$- 11 x (x + 3) (x - 3) + 3 ((2 u + 1) (u - 9))$

Step 9

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

$- 11 x (x + 3) (x - 3) + 3 ((2 x^{2} + 1) (x^{2} - 9))$

Step 10

Rewrite $9$ as $3^{2}$ .

$- 11 x (x + 3) (x - 3) + 3 ((2 x^{2} + 1) (x^{2} - 3^{2}))$

Step 11

Factor.

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Step 11.1

Factor.

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Step 11.1.1

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 $b = 3$ .

$- 11 x (x + 3) (x - 3) + 3 ((2 x^{2} + 1) ((x + 3) (x - 3)))$

Step 11.1.2

Remove unnecessary parentheses.

$- 11 x (x + 3) (x - 3) + 3 ((2 x^{2} + 1) (x + 3) (x - 3))$

Step 11.2

Remove unnecessary parentheses.

$- 11 x (x + 3) (x - 3) + 3 (2 x^{2} + 1) (x + 3) (x - 3)$

Step 12

Factor $(x + 3) (x - 3)$ out of $- 11 x (x + 3) (x - 3) + 3 (2 x^{2} + 1) (x + 3) (x - 3)$ .

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Step 12.1

Factor $(x + 3) (x - 3)$ out of $- 11 x (x + 3) (x - 3)$ .

$(x + 3) (x - 3) (- 11 x) + 3 (2 x^{2} + 1) (x + 3) (x - 3)$

Step 12.2

Factor $(x + 3) (x - 3)$ out of $3 (2 x^{2} + 1) (x + 3) (x - 3)$ .

$(x + 3) (x - 3) (- 11 x) + (x + 3) (x - 3) (3 (2 x^{2} + 1))$

Step 12.3

Factor $(x + 3) (x - 3)$ out of $(x + 3) (x - 3) (- 11 x) + (x + 3) (x - 3) (3 (2 x^{2} + 1))$ .

$(x + 3) (x - 3) (- 11 x + 3 (2 x^{2} + 1))$

Step 13

Apply the distributive property.

$(x + 3) (x - 3) (- 11 x + 3 (2 x^{2}) + 3 \cdot 1)$

Step 14

Multiply $2$ by $3$ .

$(x + 3) (x - 3) (- 11 x + 6 x^{2} + 3 \cdot 1)$

Step 15

Multiply $3$ by $1$ .

$(x + 3) (x - 3) (- 11 x + 6 x^{2} + 3)$

Step 16

Reorder terms.

$(x + 3) (x - 3) (6 x^{2} - 11 x + 3)$

Step 17

Factor.

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Step 17.1

Factor by grouping.

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Step 17.1.1

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 $a \cdot c = 6 \cdot 3 = 18$ and whose sum is $b = - 11$ .

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Step 17.1.1.1

Factor $- 11$ out of $- 11 x$ .

$(x + 3) (x - 3) (6 x^{2} - 11 (x) + 3)$

Step 17.1.1.2

Rewrite $- 11$ as $- 2$ plus $- 9$

$(x + 3) (x - 3) (6 x^{2} + (- 2 - 9) x + 3)$

Step 17.1.1.3

Apply the distributive property.

$(x + 3) (x - 3) (6 x^{2} - 2 x - 9 x + 3)$

Step 17.1.2

Factor out the greatest common factor from each group.

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Step 17.1.2.1

Group the first two terms and the last two terms.

$(x + 3) (x - 3) ((6 x^{2} - 2 x) - 9 x + 3)$

Step 17.1.2.2

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

$(x + 3) (x - 3) (2 x (3 x - 1) - 3 (3 x - 1))$

Step 17.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(x + 3) (x - 3) ((3 x - 1) (2 x - 3))$

Step 17.2

Remove unnecessary parentheses.

$(x + 3) (x - 3) (3 x - 1) (2 x - 3)$