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Rational expressions: simplify, multiply, divide, add and subtract

What rational expressions are, their excluded values, and how to simplify, multiply, divide, add and subtract them — then solve rational equations.

2 分で読めます2026年11月2日 更新執筆：Sumzo teamレビュー：Sumzo team

Rational expressions are fractions whose top and bottom are polynomials, such as x2−9x+3\frac{x^2 - 9}{x + 3} or 2x−1\frac{2}{x - 1}. They follow the same rules as number fractions — simplify by cancelling common factors, multiply straight across, add over a common denominator — with one extra rule: the denominator can never be zero.

Excluded values

An excluded value is a value of xx that makes the denominator zero. For x+1x2−4\frac{x + 1}{x^2 - 4}, the denominator x2−4=(x−2)(x+2)x^2 - 4 = (x - 2)(x + 2) is zero at x=2x = 2 and x=−2x = -2, so both are excluded. Find them before you simplify — they stay excluded even if the factor cancels.

How to simplify rational expressions

解き方付き例題

Easy: factor and cancel

Simplify x2−9x2+5x+6\frac{x^2 - 9}{x^2 + 5x + 6}.

解き方を表示
  1. Factor the top and bottom.
    (x−3)(x+3)(x+2)(x+3)\frac{(x - 3)(x + 3)}{(x + 2)(x + 3)}
  2. Cancel the common factor (x+3)(x + 3).
    x−3x+2,x≠−3,−2\frac{x - 3}{x + 2}, \quad x \neq -3, -2

答え： x−3x+2\frac{x - 3}{x + 2}, with x≠−3x \neq -3 and x≠−2x \neq -2

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Multiplying and dividing rational expressions

To multiply, factor everything, multiply the numerators and the denominators, and cancel. To divide, multiply by the reciprocal of the second fraction.

解き方付き例題

Medium: dividing

Simplify x2−4x+3÷x−2x2+3x\frac{x^2 - 4}{x + 3} \div \frac{x - 2}{x^2 + 3x}.

解き方を表示
  1. Multiply by the reciprocal.
    x2−4x+3×x2+3xx−2\frac{x^2 - 4}{x + 3} \times \frac{x^2 + 3x}{x - 2}
  2. Factor.
    (x−2)(x+2)x+3×x(x+3)x−2\frac{(x - 2)(x + 2)}{x + 3} \times \frac{x(x + 3)}{x - 2}
  3. Cancel (x−2)(x - 2) and (x+3)(x + 3).
    x(x+2)x(x + 2)

答え： x(x+2)x(x + 2), with x≠−3x \neq -3, x≠0x \neq 0 and x≠2x \neq 2

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Adding and subtracting rational expressions

解き方付き例題

Medium: adding with different denominators

Add 1x+1+2x−1\frac{1}{x + 1} + \frac{2}{x - 1}.

解き方を表示
  1. The common denominator is (x+1)(x−1)(x + 1)(x - 1).
    1(x−1)+2(x+1)(x+1)(x−1)\frac{1(x - 1) + 2(x + 1)}{(x + 1)(x - 1)}
  2. Simplify the numerator.
    3x+1(x+1)(x−1)\frac{3x + 1}{(x + 1)(x - 1)}

答え： 3x+1(x+1)(x−1)\frac{3x + 1}{(x + 1)(x - 1)}

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How to solve rational expressions in equations

An expression is simplified; an equation is solved. When a rational expression is set equal to something — the same skill as solving rational functions for their zeros — multiply every term by the common denominator to clear the fractions, solve, and reject any answer that is an excluded value.

解き方付き例題

Hard: an equation with fractions

Solve 2x−1=3x+2\frac{2}{x - 1} = \frac{3}{x + 2}.

解き方を表示
  1. Excluded values: x≠1x \neq 1 and x≠−2x \neq -2.
  2. Multiply both sides by (x−1)(x+2)(x - 1)(x + 2).
    2(x+2)=3(x−1)2(x + 2) = 3(x - 1)
  3. Solve.
    2x+4=3x−3⇒x=72x + 4 = 3x - 3 \Rightarrow x = 7
  4. Check: 26=39=13\frac{2}{6} = \frac{3}{9} = \frac{1}{3} ✓, and 7 is not excluded.

答え： x=7x = 7

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Common mistakes

  • Cancelling terms instead of factors: in x+3x+5\frac{x + 3}{x + 5} nothing cancels.
  • Adding the denominators: 1x+1y=x+yxy\frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy}, not 2x+y\frac{2}{x + y}.
  • Forgetting excluded values once a factor has cancelled.
  • Keeping a solution of a rational equation that makes a denominator zero.

よくある質問

What is a rational expression?

A fraction with a polynomial in the numerator and a non-zero polynomial in the denominator, such as x+1x2−4\frac{x + 1}{x^2 - 4}.

How do you simplify rational expressions?

Factor the numerator and denominator completely, cancel factors that appear in both, and state the excluded values from the original denominator.

How do I solve rational expressions?

An expression on its own is simplified, not solved. If it is part of an equation, clear the fractions by multiplying by the common denominator, solve, and check that no answer is an excluded value.

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