Factor x^3-2x+x^5

$x^{3} - 2 x + x^{5}$

Step 1

Factor $x$ out of $x^{3} - 2 x + x^{5}$ .

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

$x \cdot x^{2} - 2 x + x^{5}$

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

$x \cdot x^{2} + x \cdot - 2 + x^{5}$

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

$x \cdot x^{2} + x \cdot - 2 + x \cdot x^{4}$

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

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

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

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

Step 2

Factor $x^{2} - 2 + x^{4}$ 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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

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

Step 3

Rewrite $1$ as $1^{2}$ .

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

Step 4

Factor.

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 $b = 1$ .

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

Remove unnecessary parentheses.

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

Factor x^3-2x+x^5

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