Algebra Examples

$f (x) = x^{4} + x^{3} - 51 x^{2} - 9 x + 378$

Step 1

Regroup terms.

$f (x) = x^{3} - 9 x + x^{4} - 51 x^{2} + 378$

Step 2

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

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Step 2.1

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

$f (x) = x \cdot x^{2} - 9 x + x^{4} - 51 x^{2} + 378$

Step 2.2

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

$f (x) = x \cdot x^{2} + x \cdot - 9 + x^{4} - 51 x^{2} + 378$

Step 2.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 9$ .

$f (x) = x (x^{2} - 9) + x^{4} - 51 x^{2} + 378$

Step 3

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

$f (x) = x (x^{2} - 3^{2}) + x^{4} - 51 x^{2} + 378$

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 $b = 3$ .

$f (x) = x ((x + 3) (x - 3)) + x^{4} - 51 x^{2} + 378$

Step 4.2

Remove unnecessary parentheses.

$f (x) = x (x + 3) (x - 3) + x^{4} - 51 x^{2} + 378$

Step 5

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

$f (x) = x (x + 3) (x - 3) + {(x^{2})}^{2} - 51 x^{2} + 378$

Step 6

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

$f (x) = x (x + 3) (x - 3) + u^{2} - 51 u + 378$

Step 7

Factor $u^{2} - 51 u + 378$ 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 $378$ and whose sum is $- 51$ .

$- 42, - 9$

Step 7.2

Write the factored form using these integers.

$f (x) = x (x + 3) (x - 3) + (u - 42) (u - 9)$

Step 8

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

$f (x) = x (x + 3) (x - 3) + (x^{2} - 42) (x^{2} - 9)$

Step 9

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

$f (x) = x (x + 3) (x - 3) + (x^{2} - 42) (x^{2} - 3^{2})$

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 $b = 3$ .

$f (x) = x (x + 3) (x - 3) + (x^{2} - 42) ((x + 3) (x - 3))$

Step 10.2

Remove unnecessary parentheses.

$f (x) = x (x + 3) (x - 3) + (x^{2} - 42) (x + 3) (x - 3)$

Step 11

Factor $(x + 3) (x - 3)$ out of $x (x + 3) (x - 3) + (x^{2} - 42) (x + 3) (x - 3)$ .

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Step 11.1

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

$f (x) = (x + 3) (x - 3) (x) + (x^{2} - 42) (x + 3) (x - 3)$

Step 11.2

Factor $(x + 3) (x - 3)$ out of $(x^{2} - 42) (x + 3) (x - 3)$ .

$f (x) = (x + 3) (x - 3) (x) + (x + 3) (x - 3) (x^{2} - 42)$

Step 11.3

Factor $(x + 3) (x - 3)$ out of $(x + 3) (x - 3) (x) + (x + 3) (x - 3) (x^{2} - 42)$ .

$f (x) = (x + 3) (x - 3) (x + x^{2} - 42)$

Step 12

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

$f (x) = (x + 3) (x - 3) (u + u^{2} - 42)$

Step 13

Factor $u + u^{2} - 42$ 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 $- 42$ and whose sum is $1$ .

$- 6, 7$

Step 13.2

Write the factored form using these integers.

$f (x) = (x + 3) (x - 3) ((u - 6) (u + 7))$

Step 14

Factor.

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Step 14.1

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

$f (x) = (x + 3) (x - 3) ((x - 6) (x + 7))$

Step 14.2

Remove unnecessary parentheses.

$f (x) = (x + 3) (x - 3) (x - 6) (x + 7)$