simplify (x^3-x)/x — step-by-step solution

The original denominators must be nonzero:

x \neq 0

Let's reduce:

\frac{x^{3} - x}{x}

Let's factorize in

ℝ

x^{3} - x

This expression can be factorized into:

x \cdot (x^{2} - 1)

⇥Let's factor in

ℝ

(x^{2} - 1)

⇥For

x^{2} - 1

, we recognize a difference of two squares:

x^{2} - 1^{2}

. Apply

a^{2} - b^{2} = (a - b) \cdot (a + b)

, with

a = x

and

b = 1

⇥ⓘ The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor:

\frac{x^{3} - x}{x}

(x + 1) \cdot (x - 1)

⇥Factors identified:

(x + 1) \cdot (x - 1)

ⓘ Expand by multiplying each term inside the parentheses:

(x + 1) \cdot (x - 1)

x \cdot x

+ x \cdot (-1)

+ 1 \cdot x

+ 1 \cdot (-1)

x \cdot x

+ x \cdot (-1)

+ 1 \cdot x

+ 1 \cdot (-1)

ⓘ Multiplying powers with the same base adds the exponents:

x \cdot x

x^{2}

■Multiplying by -1 gives the opposite:

x \cdot (-1)

- x

x^{2}

- x

+ x

- 1

ⓘ Like terms are grouped by adding their coefficients:

- x + x

0

x^{2} + 0 - 1

ⓘ Adding 0 changes nothing:

0 - 1

-1

▶Simplified form:

x^{2} - 1