Precalculus Examples

$3 x^{4} - 10 x^{3} - 9 x^{2} + 40 x - 12$

Step 1

Regroup terms.

$- 10 x^{3} + 40 x + 3 x^{4} - 9 x^{2} - 12$

Step 2

Factor $- 10 x$ out of $- 10 x^{3} + 40 x$ .

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Step 2.1

Factor $- 10 x$ out of $- 10 x^{3}$ .

$- 10 x (x^{2}) + 40 x + 3 x^{4} - 9 x^{2} - 12$

Step 2.2

Factor $- 10 x$ out of $40 x$ .

$- 10 x (x^{2}) - 10 x (- 4) + 3 x^{4} - 9 x^{2} - 12$

Step 2.3

Factor $- 10 x$ out of $- 10 x (x^{2}) - 10 x (- 4)$ .

$- 10 x (x^{2} - 4) + 3 x^{4} - 9 x^{2} - 12$

Step 3

Rewrite $4$ as $2^{2}$ .

$- 10 x (x^{2} - 2^{2}) + 3 x^{4} - 9 x^{2} - 12$

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 $b = 2$ .

$- 10 x ((x + 2) (x - 2)) + 3 x^{4} - 9 x^{2} - 12$

Step 4.2

Remove unnecessary parentheses.

$- 10 x (x + 2) (x - 2) + 3 x^{4} - 9 x^{2} - 12$

Step 5

Factor $3$ out of $3 x^{4} - 9 x^{2} - 12$ .

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Step 5.1

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

$- 10 x (x + 2) (x - 2) + 3 (x^{4}) - 9 x^{2} - 12$

Step 5.2

Factor $3$ out of $- 9 x^{2}$ .

$- 10 x (x + 2) (x - 2) + 3 (x^{4}) + 3 (- 3 x^{2}) - 12$

Step 5.3

Factor $3$ out of $- 12$ .

$- 10 x (x + 2) (x - 2) + 3 x^{4} + 3 (- 3 x^{2}) + 3 \cdot - 4$

Step 5.4

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

$- 10 x (x + 2) (x - 2) + 3 (x^{4} - 3 x^{2}) + 3 \cdot - 4$

Step 5.5

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

$- 10 x (x + 2) (x - 2) + 3 (x^{4} - 3 x^{2} - 4)$

Step 6

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

$- 10 x (x + 2) (x - 2) + 3 ({(x^{2})}^{2} - 3 x^{2} - 4)$

Step 7

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

$- 10 x (x + 2) (x - 2) + 3 (u^{2} - 3 u - 4)$

Step 8

Factor $u^{2} - 3 u - 4$ using the AC method.

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Step 8.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 8.2

Write the factored form using these integers.

$- 10 x (x + 2) (x - 2) + 3 ((u - 4) (u + 1))$

Step 9

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

$- 10 x (x + 2) (x - 2) + 3 ((x^{2} - 4) (x^{2} + 1))$

Step 10

Rewrite $4$ as $2^{2}$ .

$- 10 x (x + 2) (x - 2) + 3 ((x^{2} - 2^{2}) (x^{2} + 1))$

Step 11

Factor.

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Step 11.1

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 $b = 2$ .

$- 10 x (x + 2) (x - 2) + 3 ((x + 2) (x - 2) (x^{2} + 1))$

Step 11.2

Remove unnecessary parentheses.

$- 10 x (x + 2) (x - 2) + 3 (x + 2) (x - 2) (x^{2} + 1)$

Step 12

Factor $(x + 2) (x - 2)$ out of $- 10 x (x + 2) (x - 2) + 3 (x + 2) (x - 2) (x^{2} + 1)$ .

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Step 12.1

Factor $(x + 2) (x - 2)$ out of $- 10 x (x + 2) (x - 2)$ .

$(x + 2) (x - 2) (- 10 x) + 3 (x + 2) (x - 2) (x^{2} + 1)$

Step 12.2

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

$(x + 2) (x - 2) (- 10 x) + (x + 2) (x - 2) ((3) (x^{2} + 1))$

Step 12.3

Factor $(x + 2) (x - 2)$ out of $(x + 2) (x - 2) (- 10 x) + (x + 2) (x - 2) ((3) (x^{2} + 1))$ .

$(x + 2) (x - 2) (- 10 x + (3) (x^{2} + 1))$

Step 13

Apply the distributive property.

$(x + 2) (x - 2) (- 10 x + 3 x^{2} + 3 \cdot 1)$

Step 14

Multiply $3$ by $1$ .

$(x + 2) (x - 2) (- 10 x + 3 x^{2} + 3)$

Step 15

Reorder terms.

$(x + 2) (x - 2) (3 x^{2} - 10 x + 3)$

Step 16

Factor.

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Step 16.1

Factor by grouping.

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Step 16.1.1

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 $a \cdot c = 3 \cdot 3 = 9$ and whose sum is $b = - 10$ .

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Step 16.1.1.1

Factor $- 10$ out of $- 10 x$ .

$(x + 2) (x - 2) (3 x^{2} - 10 (x) + 3)$

Step 16.1.1.2

Rewrite $- 10$ as $- 1$ plus $- 9$

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

Step 16.1.1.3

Apply the distributive property.

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

Step 16.1.2

Factor out the greatest common factor from each group.

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Step 16.1.2.1

Group the first two terms and the last two terms.

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

Step 16.1.2.2

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

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

Step 16.1.3

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

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

Step 16.2

Remove unnecessary parentheses.

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