To simplify an algebraic fraction, factor the top and the bottom, cancel only the factors that repeat and keep the values of x that made the original denominator zero. So (x² − 4)/(x − 2) = x + 2, but only for x ≠ 2. If you want the calculator to do it for you, use algebraic fractions. The reason a denominator cannot be zero is in the guide on the zero denominator.
Four steps and the mistake each one avoids
An algebraic fraction is the quotient of two expressions with letters, and if both are polynomials it is called rational. The table sums up the protocol.
| Step | What you do | Mistake it avoids |
|---|---|---|
| 1. Domain | Set each original denominator equal to zero and solve | Losing a restriction when canceling |
| 2. Factor | Break the numerator and denominator down completely | Canceling without seeing the factors |
| 3. Cancel | Remove equal factors on top and bottom | Crossing out terms |
| 4. Write | Give the result together with the restrictions from step 1 | Widening the domain |
Step 1 comes before factoring because, once a factor is canceled, no trace of it is left in the expression. Steps 2 and 3 are the ones most confused with the rule of “crossing out what repeats,” which only holds for factors.
Worked example: (x² − 4)/(x − 2)
Simplify the fraction and state for which values of x the result is valid.
- Domain. The denominator x − 2 is zero when x = 2, so x = 2 is excluded.
- Factor. x² − 4 is a difference of squares: (x − 2)(x + 2).
- Cancel. The factor x − 2 multiplies the whole numerator and the whole denominator, and x + 2 is left.
- Write. The equality holds for x ≠ 2.
Result: x + 2, with x ≠ 2.
Check: with x = 4, the original gives 12/2 = 6 and x + 2 also gives 6. With x = 2, the original is 0/0, while x + 2 would be 4.
That is why the graph of the fraction is the line y = x + 2 with a hole at the point (2, 4). A single test value does not prove the equality, but it is enough to rule out many canceling mistakes.
When a restriction stays in sight
Not all restrictions disappear when canceling. In (x² − 9)/(x² − x − 6), the denominator factors as (x − 3)(x + 2), and the numerator as (x − 3)(x + 3). The x − 3 cancels and (x + 3)/(x + 2) is left. The exclusions from the original denominator are x ≠ 3 and x ≠ −2, even though only the second one shows in the result. With x = 4, both forms equal 7/6.
Why terms are not canceled
Canceling a factor is dividing top and bottom by the same number. When the x’s are added, they are not a factor of the whole numerator, and crossing them out changes the value.
An expression can be split when the denominator is a single term: (2x + 4)/(2x) = 2(x + 2)/(2x) = (x + 2)/x, with x ≠ 0. Here the factor that cancels is the 2, which does multiply both terms of the numerator.
An expression is not an equation
Simplifying transforms an expression and has nothing to solve. An equation, on the other hand, asks for the values of x that satisfy it, and when you clear denominators, candidates appear that have to be checked in the original equation.
| Equation | Candidate | Verdict |
|---|---|---|
| (x² − 9)/(x − 3) = 6 | x + 3 = 6 gives x = 3 | Makes x − 3 zero: no solution |
| (x² − 9)/(x − 3) = 5 | x + 3 = 5 gives x = 2 | Allowed, and (4 − 9)/(2 − 3) = 5: a solution |
The table shows that the candidate in the first row matches the excluded value, and the one in the second does not. Always substitute into the original denominators, not into the already simplified equation, because it lost the restrictions. The equations with fractions calculator discards the excluded candidates. If, when checking with a value of x, the two results do not match, the guide to fraction mistakes gathers the most common slips. If you need to split a quotient into simpler fractions, the path is another one and it is covered by partial fraction decomposition.
Practice
- Simplify (x² − 16)/(x − 4) and state the restriction.
- Simplify (3x + 6)/(x + 2).
- Simplify (x² − 25)/(x² − 5x) and write the two restrictions.
- Solve (x² − 36)/(x − 6) = 12.
- Is it correct to reduce (x + 5)/(x + 7) to 5/7?
Show the answers
- x + 4, with x ≠ 4.
- 3(x + 2)/(x + 2) = 3, with x ≠ −2.
- (x − 5)(x + 5)/(x(x − 5)) = (x + 5)/x, with x ≠ 0 and x ≠ 5.
- You get x + 6 = 12, which gives x = 6. That value makes x − 6 zero, so there is no solution.
- No. With x = 1, the fraction equals 6/8 = 3/4, which is not 5/7.
Sources
The English Wikipedia article Algebraic fraction defines an algebraic fraction as one whose numerator and denominator are algebraic expressions, and calls it a rational algebraic fraction when both are polynomials. The article Rational function defines the domain as the values of x for which the denominator is not zero, and notes that a reduced form may have a larger domain than the original (consulted October 17, 2026). Both are open encyclopedias, so it is worth checking them against your textbook.