Factor -13x^3+36x+x^5

$- 13 x^{3} + 36 x + x^{5}$

Step 1

Factor $x$ out of $- 13 x^{3} + 36 x + x^{5}$ .

Factor $x$ out of $- 13 x^{3}$ .

$x (- 13 x^{2}) + 36 x + x^{5}$

Factor $x$ out of $36 x$ .

$x (- 13 x^{2}) + x \cdot 36 + x^{5}$

Factor $x$ out of $x^{5}$ .

$x (- 13 x^{2}) + x \cdot 36 + x \cdot x^{4}$

Factor $x$ out of $x (- 13 x^{2}) + x \cdot 36$ .

$x (- 13 x^{2} + 36) + x \cdot x^{4}$

Factor $x$ out of $x (- 13 x^{2} + 36) + x \cdot x^{4}$ .

$x (- 13 x^{2} + 36 + x^{4})$

Step 2

Rewrite the middle term.

$x (- 13 x^{2} + 2 \cdot 6 x^{2} - 25 x^{2} + x^{4})$

Step 3

Rearrange terms.

$x (- 13 x^{2} + 2 \cdot 6 x^{2} + x^{4} - 25 x^{2})$

Step 4

Factor first three terms by perfect square rule.

$x ({(6 + x^{2})}^{2} - 25 x^{2})$

Step 5

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

$x ({(6 + x^{2})}^{2} - {(5 x)}^{2})$

Step 6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6 + x^{2}$ and $b = 5 x$ .

$x ((6 + x^{2} + 5 x) (6 + x^{2} - (5 x)))$

Step 7

Factor.

Simplify.

Factor $6 + x^{2} + 5 x$ using the AC method.

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 $6$ and whose sum is $5$ .

$2, 3$

Write the factored form using these integers.

$x ((x + 2) (x + 3) (6 + x^{2} - (5 x)))$

Multiply $5$ by $- 1$ .

$x ((x + 2) (x + 3) (6 + x^{2} - 5 x))$

Remove unnecessary parentheses.

$x (x + 2) (x + 3) (6 + x^{2} - 5 x)$

Step 8

Factor.

Factor $6 + x^{2} - 5 x$ using the AC method.

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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Write the factored form using these integers.

$x (x + 2) (x + 3) ((x - 3) (x - 2))$

Remove unnecessary parentheses.

$x (x + 2) (x + 3) (x - 3) (x - 2)$

Factor -13x^3+36x+x^5

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