How to Simplify a Rational Expression
Simplifying is the same idea as reducing 6/8 to 3/4: factor the top and bottom, then cancel factors that appear on both. The word that matters is factors. You can cancel (x + 2) from top and bottom, but you can never cancel the 2 inside it.
(x² − 4)/(x² − x − 6)
- Factor the top as a difference of squares: x² − 4 = (x − 2)(x + 2).
- Factor the bottom: x² − x − 6 = (x − 3)(x + 2).
- Write the fraction as (x − 2)(x + 2) ÷ (x − 3)(x + 2).
- Cancel the common factor (x + 2).
- Simplified: (x − 2) ÷ (x − 3), with x ≠ 3 and x ≠ −2.
Answer: (x − 2)/(x − 3), x ≠ 3, −2
Excluded Values: Holes or Vertical Asymptotes?
Excluded values are the x-values that make a denominator zero, found before cancelling. After cancelling, a factor that vanished leaves a hole in the graph, and a factor that stayed in the denominator leaves a vertical asymptote.
In the example above, x = −2 cancelled, so there is a hole there. Plug −2 into the simplified form: (−2 − 2) ÷ (−2 − 3) = −4 ÷ −5 = 0.8, so the hole sits at (−2, 0.8). The factor x − 3 stayed, so x = 3 is a vertical asymptote.
Multiplying and Dividing Rational Expressions
Multiply tops, multiply bottoms, and cancel before you expand because it keeps the algebra small. To divide, flip the second fraction and multiply.
(x+3)/(x²−4) · (x−2)/(x²−9)
- Factor every piece: x² − 4 = (x − 2)(x + 2) and x² − 9 = (x − 3)(x + 3).
- Write one fraction: (x + 3)(x − 2) ÷ (x − 2)(x + 2)(x − 3)(x + 3).
- Cancel (x + 3) and (x − 2).
- What is left on the bottom is (x + 2)(x − 3).
Answer: 1/((x + 2)(x − 3)), x ≠ ±2, ±3
Adding and Subtracting: Find the Common Denominator
Different denominators mean you need a common one before you add, exactly as with numeric fractions. Multiply each fraction by whatever the other denominator has that it lacks.
1/x + 1/(x+1)
- The denominators x and x + 1 share no factor, so the common denominator is x(x + 1).
- Rewrite: 1/x = (x + 1)/(x(x + 1)) and 1/(x + 1) = x/(x(x + 1)).
- Add the numerators: (x + 1) + x = 2x + 1.
Answer: (2x + 1)/(x(x + 1)), x ≠ 0, −1
Solving Rational Equations and Spotting Extraneous Solutions
Multiply every term by the common denominator to clear the fractions, solve the resulting equation, then substitute back. Any answer that makes an original denominator zero is extraneous and has to go.
x/(x−3) = 3/(x−3) + 3
- Multiply every term by x − 3: x = 3 + 3(x − 3).
- Expand the right side: x = 3 + 3x − 9 = 3x − 6.
- Move terms: −2x = −6, so x = 3.
- Check: x = 3 makes the denominator x − 3 equal to zero.
- So x = 3 is extraneous and the equation has no solution.
Answer: No solution
A Worked Example: Hole or Asymptote?
Simplify (x² − 4) ÷ (x² − x − 2). Factor: (x − 2)(x + 2) ÷ [(x − 2)(x + 1)]. Cancel (x − 2) to get (x + 2) ÷ (x + 1). Two restrictions remain from the original: x ≠ 2 and x ≠ −1. At x = −1 the denominator of the simplified form is also zero, giving a vertical asymptote. At x = 2 the factor cancelled, giving a hole at (2, 4/3).
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: cancelling as soon as possible. Without factoring first you will cancel terms, not factors.
Underrated: writing the restrictions before simplifying. They come from the original denominator and survive the cancellation.
Plain wrong: the claim that a cancelled factor “disappears”. It leaves a hole in the graph, and the function is still undefined there.
Horizontal Asymptotes and Mistakes to Avoid
| Degree of top vs bottom | Horizontal asymptote | Example |
|---|---|---|
| Top smaller | y = 0 | 1/(x + 2) |
| Equal | y = ratio of leading coefficients | (3x + 1)/(x − 5) → y = 3 |
| Top bigger | none (slant if exactly one bigger) | x²/(x − 1) |
- Cancelling terms instead of factors, such as (x + 3) ÷ 3.
- Finding excluded values after cancelling, which hides the holes.
- Skipping the final check when solving a rational equation.
Why Students Get Stuck on Rational Expressions, and What a Good Calculator Should Do
Searching for help with rational expressions usually starts late at night with one problem that will not work out. Factoring and cancelling are easy to misapply, and restrictions are easy to forget. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to factor first, cancel factors and note the restrictions, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Cancelling terms instead of factorsSimplify (x^2 - 9)/(x - 3)↗
- Forgetting the restricted valuesSimplify (x^2 + 5x + 6)/(x^2 - 4)↗
- Not finding a common denominatorSolve 1/x + 1/(x + 2) = 1↗
- Not factoring completely first
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants the factoring shown first
- Wants restrictions listed
- Wants addition and division shown
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Simplify (x^2 - 9)/(x - 3)↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.