Simplify (27x^4-x)/(x^4+x^3+x)

$\frac{27 x^{4} - x}{x^{4} + x^{3} + x}$

Simplify the numerator.

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

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

$\frac{x (27 x^{3}) - x}{x^{4} + x^{3} + x}$

Factor $x$ out of $- x$ .

$\frac{x (27 x^{3}) + x \cdot - 1}{x^{4} + x^{3} + x}$

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

$\frac{x (27 x^{3} - 1)}{x^{4} + x^{3} + x}$

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

$\frac{x ({(3 x)}^{3} - 1)}{x^{4} + x^{3} + x}$

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

$\frac{x ({(3 x)}^{3} - 1^{3})}{x^{4} + x^{3} + x}$

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 $b = 1$ .

$\frac{x ((3 x - 1) ({(3 x)}^{2} + 3 x \cdot 1 + 1^{2}))}{x^{4} + x^{3} + x}$

Simplify.

Apply the product rule to $3 x$ .

$\frac{x ((3 x - 1) (3^{2} x^{2} + 3 x \cdot 1 + 1^{2}))}{x^{4} + x^{3} + x}$

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

$\frac{x ((3 x - 1) (9 x^{2} + 3 x \cdot 1 + 1^{2}))}{x^{4} + x^{3} + x}$

Multiply $3$ by $1$ .

$\frac{x ((3 x - 1) (9 x^{2} + 3 x + 1^{2}))}{x^{4} + x^{3} + x}$

One to any power is one.

$\frac{x ((3 x - 1) (9 x^{2} + 3 x + 1))}{x^{4} + x^{3} + x}$

$\frac{x (3 x - 1) (9 x^{2} + 3 x + 1)}{x^{4} + x^{3} + x}$

Simplify terms.

Factor $x$ out of $x^{4} + x^{3} + x$ .

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

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

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

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

Raise $x$ to the power of $1$ .

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

Factor $x$ out of $x^{1}$ .

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

Factor $x$ out of $x \cdot x^{3} + x \cdot x^{2}$ .

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

Factor $x$ out of $x (x^{3} + x^{2}) + x \cdot 1$ .

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (27x^4-x)/(x^4+x^3+x)

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