Algebra Examples

$f (x) = x^{4} + 9 x^{3} + 15 x^{2} + 9 x + 14$

Step 1

Regroup terms.

$f (x) = 9 x^{3} + 9 x + x^{4} + 15 x^{2} + 14$

Step 2

Factor $9 x$ out of $9 x^{3} + 9 x$ .

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Step 2.1

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

$f (x) = 9 x (x^{2}) + 9 x + x^{4} + 15 x^{2} + 14$

Step 2.2

Factor $9 x$ out of $9 x$ .

$f (x) = 9 x (x^{2}) + 9 x (1) + x^{4} + 15 x^{2} + 14$

Step 2.3

Factor $9 x$ out of $9 x (x^{2}) + 9 x (1)$ .

$f (x) = 9 x (x^{2} + 1) + x^{4} + 15 x^{2} + 14$

Step 3

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

$f (x) = 9 x (x^{2} + 1) + {(x^{2})}^{2} + 15 x^{2} + 14$

Step 4

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

$f (x) = 9 x (x^{2} + 1) + u^{2} + 15 u + 14$

Step 5

Factor $u^{2} + 15 u + 14$ 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 $14$ and whose sum is $15$ .

$1, 14$

Step 5.2

Write the factored form using these integers.

$f (x) = 9 x (x^{2} + 1) + (u + 1) (u + 14)$

Step 6

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

$f (x) = 9 x (x^{2} + 1) + (x^{2} + 1) (x^{2} + 14)$

Step 7

Factor $x^{2} + 1$ out of $9 x (x^{2} + 1) + (x^{2} + 1) (x^{2} + 14)$ .

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Step 7.1

Factor $x^{2} + 1$ out of $9 x (x^{2} + 1)$ .

$f (x) = (x^{2} + 1) (9 x) + (x^{2} + 1) (x^{2} + 14)$

Step 7.2

Factor $x^{2} + 1$ out of $(x^{2} + 1) (9 x) + (x^{2} + 1) (x^{2} + 14)$ .

$f (x) = (x^{2} + 1) (9 x + x^{2} + 14)$

Step 8

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

$f (x) = (x^{2} + 1) (9 u + u^{2} + 14)$

Step 9

Factor $9 u + u^{2} + 14$ 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 $14$ and whose sum is $9$ .

$2, 7$

Step 9.2

Write the factored form using these integers.

$f (x) = (x^{2} + 1) ((u + 2) (u + 7))$

Step 10

Factor.

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Step 10.1

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

$f (x) = (x^{2} + 1) ((x + 2) (x + 7))$

Step 10.2

Remove unnecessary parentheses.

$f (x) = (x^{2} + 1) (x + 2) (x + 7)$