Rational Expression Simplifier Notes

Published on: September 29, 2026

This Rational Expression Simplifier Calculator reduces a fraction made of polynomials, and shows the working line by line. The top and the bottom are factored first, because only whole factors may be cancelled and terms never may. The values that would make an original denominator zero are written down before any cancelling, since those stay banned even once the factor causing them has gone. Following the same pattern every time keeps the restrictions from being lost.

Step-by-step method

  1. Set up the problem.
  2. If it is a complex fraction, rewrite it by multiplying by the reciprocal.
  3. Factor the numerator and denominator (only if something factors).
  4. State the restrictions (values that make any original denominator equal 0).
  5. Cancel common factors (factors only, not individual terms).
  6. Write the final simplified form with the restrictions.
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. If it is a complex fraction, rewrite it by multiplying by the reciprocal.
  3. Factor the numerator and denominator (only if something factors).
  4. State the restrictions (values that make any original denominator equal 0).
  5. Cancel common factors (factors only, not individual terms).
  6. Write the final simplified form with the restrictions.

Step 1

Step 1 - Set up the problem.

In this problem: The expression to simplify is \(\frac{x^{2} - 1}{x + 1}\), and simplifying means cancelling whatever the top and the bottom share.

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

Step 2

Step 2 - Factor the numerator and denominator (only if something factors).

In this problem: Factor the numerator and denominator before cancelling.

\(\frac{\left(x - 1\right) \left(x + 1\right)}{x + 1}\)

Step 3

Step 3 - State the restrictions (values that make any original denominator equal 0).

In this problem: Find the values that make any original denominator equal 0. These values are not allowed.

\(\begin{gathered} x + 1 \neq 0 \\ x \neq -1 \end{gathered}\)

Step 4

Step 4 - Cancel common factors (factors only, not individual terms).

In this problem: Cancel only identical factors that appear in both the numerator and denominator.

\(\frac{\left(x - 1\right) \left(x + 1\right)}{x + 1} = x - 1\)

Step 5

Step 5 - Write the final simplified form with the restrictions.

In this problem: Write the final simplified form together with the restrictions.

\(x - 1,\; x \neq -1\)

Final Answer

\(x - 1,\; x \neq -1\)
See Example 2 Hide Example 2

Problem

\(\frac{3 x^{2} - 12}{x^{2} + 4 x + 4}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. If it is a complex fraction, rewrite it by multiplying by the reciprocal.
  3. Factor the numerator and denominator (only if something factors).
  4. State the restrictions (values that make any original denominator equal 0).
  5. Cancel common factors (factors only, not individual terms).
  6. Write the final simplified form with the restrictions.

Step 1

Step 1 - Set up the problem.

In this problem: The expression to simplify is \(\frac{3 x^{2} - 12}{x^{2} + 4 x + 4}\), and simplifying means cancelling whatever the top and the bottom share.

\(\frac{3 x^{2} - 12}{x^{2} + 4 x + 4}\)

Step 2

Step 2 - Factor the numerator and denominator (only if something factors).

In this problem: Factor the numerator and denominator before cancelling.

\(\frac{3 \left(x - 2\right) \left(x + 2\right)}{\left(x + 2\right)^{2}}\)

Step 3

Step 3 - State the restrictions (values that make any original denominator equal 0).

In this problem: Find the values that make any original denominator equal 0. These values are not allowed.

\(\begin{gathered} \left(x + 2\right)^{2} \neq 0 \\ x \neq -2 \end{gathered}\)

Step 4

Step 4 - Cancel common factors (factors only, not individual terms).

In this problem: Cancel only identical factors that appear in both the numerator and denominator.

\(\frac{3 \left(x - 2\right) \left(x + 2\right)}{\left(x + 2\right)^{2}} = \frac{3 x - 6}{x + 2}\)

Step 5

Step 5 - Write the final simplified form with the restrictions.

In this problem: Write the final simplified form together with the restrictions.

\(\frac{3 x - 6}{x + 2},\; x \neq -2\)

Final Answer

\(\frac{3 x - 6}{x + 2},\; x \neq -2\)
See Example 3 Hide Example 3

Problem

\(\frac{x^{2} - 25}{x - 5}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. If it is a complex fraction, rewrite it by multiplying by the reciprocal.
  3. Factor the numerator and denominator (only if something factors).
  4. State the restrictions (values that make any original denominator equal 0).
  5. Cancel common factors (factors only, not individual terms).
  6. Write the final simplified form with the restrictions.

Step 1

Step 1 - Set up the problem.

In this problem: The expression to simplify is \(\frac{x^{2} - 25}{x - 5}\), and simplifying means cancelling whatever the top and the bottom share.

\(\frac{x^{2} - 25}{x - 5}\)

Step 2

Step 2 - Factor the numerator and denominator (only if something factors).

In this problem: Factor the numerator and denominator before cancelling.

\(\frac{\left(x - 5\right) \left(x + 5\right)}{x - 5}\)

Step 3

Step 3 - State the restrictions (values that make any original denominator equal 0).

In this problem: Find the values that make any original denominator equal 0. These values are not allowed.

\(\begin{gathered} x - 5 \neq 0 \\ x \neq 5 \end{gathered}\)

Step 4

Step 4 - Cancel common factors (factors only, not individual terms).

In this problem: Cancel only identical factors that appear in both the numerator and denominator.

\(\frac{\left(x - 5\right) \left(x + 5\right)}{x - 5} = x + 5\)

Step 5

Step 5 - Write the final simplified form with the restrictions.

In this problem: Write the final simplified form together with the restrictions.

\(x + 5,\; x \neq 5\)

Final Answer

\(x + 5,\; x \neq 5\)
See Example 4 Hide Example 4

Problem

\(\frac{x^{2} + 3 x}{x^{2} + 6 x + 9}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. If it is a complex fraction, rewrite it by multiplying by the reciprocal.
  3. Factor the numerator and denominator (only if something factors).
  4. State the restrictions (values that make any original denominator equal 0).
  5. Cancel common factors (factors only, not individual terms).
  6. Write the final simplified form with the restrictions.

Step 1

Step 1 - Set up the problem.

In this problem: The expression to simplify is \(\frac{x^{2} + 3 x}{x^{2} + 6 x + 9}\), and simplifying means cancelling whatever the top and the bottom share.

\(\frac{x^{2} + 3 x}{x^{2} + 6 x + 9}\)

Step 2

Step 2 - Factor the numerator and denominator (only if something factors).

In this problem: Factor the numerator and denominator before cancelling.

\(\frac{x \left(x + 3\right)}{\left(x + 3\right)^{2}}\)

Step 3

Step 3 - State the restrictions (values that make any original denominator equal 0).

In this problem: Find the values that make any original denominator equal 0. These values are not allowed.

\(\begin{gathered} \left(x + 3\right)^{2} \neq 0 \\ x \neq -3 \end{gathered}\)

Step 4

Step 4 - Cancel common factors (factors only, not individual terms).

In this problem: Cancel only identical factors that appear in both the numerator and denominator.

\(\frac{x \left(x + 3\right)}{\left(x + 3\right)^{2}} = \frac{x}{x + 3}\)

Step 5

Step 5 - Write the final simplified form with the restrictions.

In this problem: Write the final simplified form together with the restrictions.

\(\frac{x}{x + 3},\; x \neq -3\)

Final Answer

\(\frac{x}{x + 3},\; x \neq -3\)