Precalculus Examples

$3 x^{4} + 7 x^{3} - 10 x^{2} - 28 x - 8$

Step 1

Regroup terms.

$7 x^{3} - 28 x + 3 x^{4} - 10 x^{2} - 8$

Step 2

Factor $7 x$ out of $7 x^{3} - 28 x$ .

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Step 2.1

Factor $7 x$ out of $7 x^{3}$ .

$7 x (x^{2}) - 28 x + 3 x^{4} - 10 x^{2} - 8$

Step 2.2

Factor $7 x$ out of $- 28 x$ .

$7 x (x^{2}) + 7 x (- 4) + 3 x^{4} - 10 x^{2} - 8$

Step 2.3

Factor $7 x$ out of $7 x (x^{2}) + 7 x (- 4)$ .

$7 x (x^{2} - 4) + 3 x^{4} - 10 x^{2} - 8$

Step 3

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

$7 x (x^{2} - 2^{2}) + 3 x^{4} - 10 x^{2} - 8$

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 $b = 2$ .

$7 x ((x + 2) (x - 2)) + 3 x^{4} - 10 x^{2} - 8$

Step 4.2

Remove unnecessary parentheses.

$7 x (x + 2) (x - 2) + 3 x^{4} - 10 x^{2} - 8$

Step 5

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

$7 x (x + 2) (x - 2) + 3 {(x^{2})}^{2} - 10 x^{2} - 8$

Step 6

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

$7 x (x + 2) (x - 2) + 3 u^{2} - 10 u - 8$

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 $a \cdot c = 3 \cdot - 8 = - 24$ and whose sum is $b = - 10$ .

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Step 7.1.1

Factor $- 10$ out of $- 10 u$ .

$7 x (x + 2) (x - 2) + 3 u^{2} - 10 (u) - 8$

Step 7.1.2

Rewrite $- 10$ as $2$ plus $- 12$

$7 x (x + 2) (x - 2) + 3 u^{2} + (2 - 12) u - 8$

Step 7.1.3

Apply the distributive property.

$7 x (x + 2) (x - 2) + 3 u^{2} + 2 u - 12 u - 8$

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.

$7 x (x + 2) (x - 2) + (3 u^{2} + 2 u) - 12 u - 8$

Step 7.2.2

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

$7 x (x + 2) (x - 2) + u (3 u + 2) - 4 (3 u + 2)$

Step 7.3

Factor the polynomial by factoring out the greatest common factor, $3 u + 2$ .

$7 x (x + 2) (x - 2) + (3 u + 2) (u - 4)$

Step 8

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

$7 x (x + 2) (x - 2) + (3 x^{2} + 2) (x^{2} - 4)$

Step 9

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

$7 x (x + 2) (x - 2) + (3 x^{2} + 2) (x^{2} - 2^{2})$

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 $b = 2$ .

$7 x (x + 2) (x - 2) + (3 x^{2} + 2) ((x + 2) (x - 2))$

Step 10.2

Remove unnecessary parentheses.

$7 x (x + 2) (x - 2) + (3 x^{2} + 2) (x + 2) (x - 2)$

Step 11

Factor $(x + 2) (x - 2)$ out of $7 x (x + 2) (x - 2) + (3 x^{2} + 2) (x + 2) (x - 2)$ .

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Step 11.1

Factor $(x + 2) (x - 2)$ out of $7 x (x + 2) (x - 2)$ .

$(x + 2) (x - 2) (7 x) + (3 x^{2} + 2) (x + 2) (x - 2)$

Step 11.2

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

$(x + 2) (x - 2) (7 x) + (x + 2) (x - 2) (3 x^{2} + 2)$

Step 11.3

Factor $(x + 2) (x - 2)$ out of $(x + 2) (x - 2) (7 x) + (x + 2) (x - 2) (3 x^{2} + 2)$ .

$(x + 2) (x - 2) (7 x + 3 x^{2} + 2)$

Step 12

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

$(x + 2) (x - 2) (7 u + 3 u^{2} + 2)$

Step 13

Factor by grouping.

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Step 13.1

Reorder terms.

$(x + 2) (x - 2) (3 u^{2} + 7 u + 2)$

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 $a \cdot c = 3 \cdot 2 = 6$ and whose sum is $b = 7$ .

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Step 13.2.1

Factor $7$ out of $7 u$ .

$(x + 2) (x - 2) (3 u^{2} + 7 (u) + 2)$

Step 13.2.2

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

$(x + 2) (x - 2) (3 u^{2} + (1 + 6) u + 2)$

Step 13.2.3

Apply the distributive property.

$(x + 2) (x - 2) (3 u^{2} + 1 u + 6 u + 2)$

Step 13.2.4

Multiply $u$ by $1$ .

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

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.

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

Step 13.3.2

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

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

Step 13.4

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

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

Step 14

Factor.

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Step 14.1

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

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

Step 14.2

Remove unnecessary parentheses.

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

Step 15

Combine exponents.

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Step 15.1

Raise $x + 2$ to the power of $1$ .

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

Step 15.2

Raise $x + 2$ to the power of $1$ .

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

Step 15.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 15.4

Add $1$ and $1$ .

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