Precalculus Examples

$x^{4} + 10 x^{3} - 20 x^{2} - 90 x + 99$

Step 1

Regroup terms.

$10 x^{3} - 90 x + x^{4} - 20 x^{2} + 99$

Step 2

Factor $10 x$ out of $10 x^{3} - 90 x$ .

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Step 2.1

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

$10 x (x^{2}) - 90 x + x^{4} - 20 x^{2} + 99$

Step 2.2

Factor $10 x$ out of $- 90 x$ .

$10 x (x^{2}) + 10 x (- 9) + x^{4} - 20 x^{2} + 99$

Step 2.3

Factor $10 x$ out of $10 x (x^{2}) + 10 x (- 9)$ .

$10 x (x^{2} - 9) + x^{4} - 20 x^{2} + 99$

Step 3

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

$10 x (x^{2} - 3^{2}) + x^{4} - 20 x^{2} + 99$

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 $b = 3$ .

$10 x ((x + 3) (x - 3)) + x^{4} - 20 x^{2} + 99$

Step 4.2

Remove unnecessary parentheses.

$10 x (x + 3) (x - 3) + x^{4} - 20 x^{2} + 99$

Step 5

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

$10 x (x + 3) (x - 3) + {(x^{2})}^{2} - 20 x^{2} + 99$

Step 6

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

$10 x (x + 3) (x - 3) + u^{2} - 20 u + 99$

Step 7

Factor $u^{2} - 20 u + 99$ using the AC method.

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Step 7.1

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 $99$ and whose sum is $- 20$ .

$- 11, - 9$

Step 7.2

Write the factored form using these integers.

$10 x (x + 3) (x - 3) + (u - 11) (u - 9)$

Step 8

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

$10 x (x + 3) (x - 3) + (x^{2} - 11) (x^{2} - 9)$

Step 9

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

$10 x (x + 3) (x - 3) + (x^{2} - 11) (x^{2} - 3^{2})$

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 $b = 3$ .

$10 x (x + 3) (x - 3) + (x^{2} - 11) ((x + 3) (x - 3))$

Step 10.2

Remove unnecessary parentheses.

$10 x (x + 3) (x - 3) + (x^{2} - 11) (x + 3) (x - 3)$

Step 11

Factor $(x + 3) (x - 3)$ out of $10 x (x + 3) (x - 3) + (x^{2} - 11) (x + 3) (x - 3)$ .

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Step 11.1

Factor $(x + 3) (x - 3)$ out of $10 x (x + 3) (x - 3)$ .

$(x + 3) (x - 3) (10 x) + (x^{2} - 11) (x + 3) (x - 3)$

Step 11.2

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

$(x + 3) (x - 3) (10 x) + (x + 3) (x - 3) (x^{2} - 11)$

Step 11.3

Factor $(x + 3) (x - 3)$ out of $(x + 3) (x - 3) (10 x) + (x + 3) (x - 3) (x^{2} - 11)$ .

$(x + 3) (x - 3) (10 x + x^{2} - 11)$

Step 12

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

$(x + 3) (x - 3) (10 u + u^{2} - 11)$

Step 13

Factor $10 u + u^{2} - 11$ using the AC method.

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Step 13.1

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 $- 11$ and whose sum is $10$ .

$- 1, 11$

Step 13.2

Write the factored form using these integers.

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

Step 14

Factor.

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Step 14.1

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

$(x + 3) (x - 3) ((x - 1) (x + 11))$

Step 14.2

Remove unnecessary parentheses.

$(x + 3) (x - 3) (x - 1) (x + 11)$