Precalculus Examples

$x^{4} - 2 x^{3} + 13 x^{2} - 32 x - 48$

Step 1

Regroup terms.

$- 2 x^{3} - 32 x + x^{4} + 13 x^{2} - 48$

Step 2

Factor $- 2 x$ out of $- 2 x^{3} - 32 x$ .

Tap for more steps...

Step 2.1

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

$- 2 x (x^{2}) - 32 x + x^{4} + 13 x^{2} - 48$

Step 2.2

Factor $- 2 x$ out of $- 32 x$ .

$- 2 x (x^{2}) - 2 x (16) + x^{4} + 13 x^{2} - 48$

Step 2.3

Factor $- 2 x$ out of $- 2 x (x^{2}) - 2 x (16)$ .

$- 2 x (x^{2} + 16) + x^{4} + 13 x^{2} - 48$

Step 3

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

$- 2 x (x^{2} + 16) + {(x^{2})}^{2} + 13 x^{2} - 48$

Step 4

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

$- 2 x (x^{2} + 16) + u^{2} + 13 u - 48$

Step 5

Factor $u^{2} + 13 u - 48$ using the AC method.

Tap for more steps...

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 $- 48$ and whose sum is $13$ .

$- 3, 16$

Step 5.2

Write the factored form using these integers.

$- 2 x (x^{2} + 16) + (u - 3) (u + 16)$

Step 6

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

$- 2 x (x^{2} + 16) + (x^{2} - 3) (x^{2} + 16)$

Step 7

Factor $x^{2} + 16$ out of $- 2 x (x^{2} + 16) + (x^{2} - 3) (x^{2} + 16)$ .

Tap for more steps...

Step 7.1

Factor $x^{2} + 16$ out of $- 2 x (x^{2} + 16)$ .

$(x^{2} + 16) (- 2 x) + (x^{2} - 3) (x^{2} + 16)$

Step 7.2

Factor $x^{2} + 16$ out of $(x^{2} - 3) (x^{2} + 16)$ .

$(x^{2} + 16) (- 2 x) + (x^{2} + 16) (x^{2} - 3)$

Step 7.3

Factor $x^{2} + 16$ out of $(x^{2} + 16) (- 2 x) + (x^{2} + 16) (x^{2} - 3)$ .

$(x^{2} + 16) (- 2 x + x^{2} - 3)$

Step 8

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

$(x^{2} + 16) (- 2 u + u^{2} - 3)$

Step 9

Factor $- 2 u + u^{2} - 3$ using the AC method.

Tap for more steps...

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 9.2

Write the factored form using these integers.

$(x^{2} + 16) ((u - 3) (u + 1))$

Step 10

Factor.

Tap for more steps...

Step 10.1

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

$(x^{2} + 16) ((x - 3) (x + 1))$

Step 10.2

Remove unnecessary parentheses.

$(x^{2} + 16) (x - 3) (x + 1)$