Precalculus Examples Factoring Polynomials

Factor

$27 x^{4} - x$

Step 1

Factor $x$ out of $27 x^{4} - x$ .

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Step 1.1

Factor $x$ out of $27 x^{4}$ .

$x (27 x^{3}) - x$

Step 1.2

Factor $x$ out of $- x$ .

$x (27 x^{3}) + x \cdot - 1$

Step 1.3

Factor $x$ out of $x (27 x^{3}) + x \cdot - 1$ .

$x (27 x^{3} - 1)$

Step 2

Rewrite $27 x^{3}$ as ${(3 x)}^{3}$ .

$x ({(3 x)}^{3} - 1)$

Step 3

Rewrite $1$ as $1^{3}$ .

$x ({(3 x)}^{3} - 1^{3})$

Step 4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 3 x$ and $b = 1$ .

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

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

Apply the product rule to $3 x$ .

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

Step 5.1.2

Raise $3$ to the power of $2$ .

$x ((3 x - 1) (9 x^{2} + 3 x \cdot 1 + 1^{2}))$

Step 5.1.3

Multiply $3$ by $1$ .

$x ((3 x - 1) (9 x^{2} + 3 x + 1^{2}))$

Step 5.1.4

One to any power is one.

$x ((3 x - 1) (9 x^{2} + 3 x + 1))$

Step 5.2

Remove unnecessary parentheses.

$x (3 x - 1) (9 x^{2} + 3 x + 1)$

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