Precalculus Examples

$x^{4} + 6 x^{3} + 2 x^{2} + 54 x - 63$

Step 1

Regroup terms.

$6 x^{3} + 54 x + x^{4} + 2 x^{2} - 63$

Step 2

Factor $6 x$ out of $6 x^{3} + 54 x$ .

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Step 2.1

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

$6 x (x^{2}) + 54 x + x^{4} + 2 x^{2} - 63$

Step 2.2

Factor $6 x$ out of $54 x$ .

$6 x (x^{2}) + 6 x (9) + x^{4} + 2 x^{2} - 63$

Step 2.3

Factor $6 x$ out of $6 x (x^{2}) + 6 x (9)$ .

$6 x (x^{2} + 9) + x^{4} + 2 x^{2} - 63$

Step 3

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

$6 x (x^{2} + 9) + {(x^{2})}^{2} + 2 x^{2} - 63$

Step 4

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

$6 x (x^{2} + 9) + u^{2} + 2 u - 63$

Step 5

Factor $u^{2} + 2 u - 63$ using the AC method.

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Step 5.1

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 $- 63$ and whose sum is $2$ .

$- 7, 9$

Step 5.2

Write the factored form using these integers.

$6 x (x^{2} + 9) + (u - 7) (u + 9)$

Step 6

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

$6 x (x^{2} + 9) + (x^{2} - 7) (x^{2} + 9)$

Step 7

Factor $x^{2} + 9$ out of $6 x (x^{2} + 9) + (x^{2} - 7) (x^{2} + 9)$ .

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Step 7.1

Factor $x^{2} + 9$ out of $6 x (x^{2} + 9)$ .

$(x^{2} + 9) (6 x) + (x^{2} - 7) (x^{2} + 9)$

Step 7.2

Factor $x^{2} + 9$ out of $(x^{2} - 7) (x^{2} + 9)$ .

$(x^{2} + 9) (6 x) + (x^{2} + 9) (x^{2} - 7)$

Step 7.3

Factor $x^{2} + 9$ out of $(x^{2} + 9) (6 x) + (x^{2} + 9) (x^{2} - 7)$ .

$(x^{2} + 9) (6 x + x^{2} - 7)$

Step 8

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

$(x^{2} + 9) (6 u + u^{2} - 7)$

Step 9

Factor $6 u + u^{2} - 7$ using the AC method.

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Step 9.1

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 $- 7$ and whose sum is $6$ .

$- 1, 7$

Step 9.2

Write the factored form using these integers.

$(x^{2} + 9) ((u - 1) (u + 7))$

Step 10

Factor.

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Step 10.1

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

$(x^{2} + 9) ((x - 1) (x + 7))$

Step 10.2

Remove unnecessary parentheses.

$(x^{2} + 9) (x - 1) (x + 7)$