1
Only a common factor cancels
A factor is joined to the rest by multiplication; a term is joined by addition. (2x)/(2y) cancels to x/y. (2 + x)/(2 + y) cancels to nothing at all.
Lesson · Rational and Radical Expressions · Rational expressions
Simplifying Rational Expressions
One rule does all the work: a factor common to the top and the bottom may be removed. A term may not. The difference between those two words is the most expensive distinction in school algebra.
(PK)/(QK) = P/Q provided K ≠ 0
(x² − 9)/(x² + x − 12) = (x + 3)/(x + 4), x ≠ 3
(3 − x)/(x − 3) = −1 since 3 − x = −(x − 3)
(x + 3)/x is ALREADY lowest terms — 3 is a term
One legal cancellation, one illegal one
Core ideas
What this lesson establishes
Simplifying is a single theorem applied carefully. Nearly every error here is the same error: applying it to something that is not a factor.
2
So factoring is not optional
The rule is stated in terms of factors, so an expression that is not factored cannot be simplified — you cannot see the common factor in (x² − 9)/(x² + x − 12), and it is there.
3
Cancelling changes the formula, not the domain
The simplified form is equal to the original only where the original exists. Carry the exclusions across; they are part of the answer.
The material
Lowest terms, and the one rule that gets you there
Why the rule is true, and exactly what it does not permit.
Theorem · The fundamental principle of fractions
For polynomials P, Q, K with Q ≠ 0 and K ≠ 0, (PK)/(QK) = P/Q. It holds because K/K = 1 and multiplying by 1 changes nothing — which is also why the condition K ≠ 0 cannot be dropped: 0/0 is not 1.
Read the statement carefully. K multiplies P, and K multiplies Q. Nothing in the theorem lets you remove something that is added to P, and there is no companion theorem that does.
Definition · Lowest terms
A rational expression is in lowest terms when its numerator and denominator have no common polynomial factor other than a constant.
The method follows from the theorem. Factor the numerator completely, factor the denominator completely, record the excluded values, then strike out the factors that appear in both.
Example · A cancellation that is legal
(x² − 9)/(x² + x − 12) factors as ((x − 3)(x + 3))/((x + 4)(x − 3)). The denominator is zero at x = 3 and x = −4, so both are excluded. The factor x − 3 is common, and removing it leaves (x + 3)/(x + 4), valid for x ≠ 3 and x ≠ −4.
Notice what happened to x = 3. It is still excluded, because it was excluded from the expression we were given, but (x + 3)/(x + 4) is quite happy there, taking the value 6/7. The two expressions agree everywhere except at that single point, where one exists and the other does not.
Definition · Opposite factors
a − b and b − a differ by a sign: a − b = −(b − a). So a numerator and denominator that look like opposites do share a factor, and their quotient is −1.
Example · The sign case
(3 − x)/(x − 3). Write the top as −(x − 3); the expression becomes (−(x − 3))/(x − 3) = −1, for every x ≠ 3. Checking at x = 0 gives 3/(−3) = −1, which agrees.
Finally, the error the whole lesson is aimed at. In (x + 3)/x the x on the bottom is a factor of the denominator but only a term of the numerator, so nothing cancels and the expression is already in lowest terms.
at x = 1 : (x + 3)/x = 4/1 = 4 3 is not the answer
at x = 3 : (x + 3)/x = 6/3 = 2 3 is not the answer
One numerical test would have caught it. That is worth remembering as a general habit: a proposed simplification is an identity, and an identity that fails at x = 1 is not an identity.
Interactive
Cancelling factors
The common factor found by Euclid, not by pattern-matching
Method
Reducing to lowest terms
The order is fixed: exclusions are recorded before anything is struck out.
1
Factor the numerator and the denominator
Completely, and separately. Greatest common factor first, then the patterns from Polynomials and Factoring — difference of squares, trinomials, grouping.
2
Write down the excluded values
From the factored denominator, before you cancel. This is the only moment at which they are visible.
3
Cancel the common factors
One at a time, and only whole factors. If a factor pair differs by a sign, pull −1 out of one of them first so the match is exact.
4
Check at one convenient value
Pick any x in the domain, usually 0 or 1, and evaluate both forms. They must agree. This catches a cancelled term in seconds.
Factor or term
Score 0 / 0Choose the answer you can justify.
Simplify (x² − 9)/(x² + x − 12)
Two factorisations, one shared factor, and a check that the result really is the same expression.
(x² − 9)/(x² + x − 12)
top: x² − 9 = (x − 3)(x + 3)
bottom: x² + x − 12 = (x + 4)(x − 3)
exclusions FIRST: x ≠ 3, x ≠ −4
= ((x − 3)(x + 3)) / ((x + 4)(x − 3))
= (x + 3)/(x + 4), x ≠ 3, x ≠ −4
check at x = 0 : −9/−12 = 3/4 and 3/4 ✓
check at x = 1 : −8/−10 = 4/5 and 4/5 ✓
at x = 3 : original undefined, (x + 3)/(x + 4) = 6/7
The final line is the reason the exclusion is written twice. The reduced form is defined at x = 3 and gives a perfectly reasonable number there, so nothing about 6/7 announces that it should not exist. Only the condition you carried tells you.
For a faded cancellation, simplify (x² − 16)/(x² + x − 20). The supplied first move is to factor before cancelling and record the original denominator's zeros. Complete both factorisations, cancel one factor and check at x = 0. Compare with (x + 4)/(x + 5), x ≠ 4, −5; if 4 disappeared from your restrictions, you read the domain from the reduced formula.
Common mistakes
1
Cancelling a term because it appears on both lines(x + 3)/(x + 4) does not become 3/4. The x is added, not multiplied, and one substitution — x = 1 gives 4/5 — disposes of it.
2
Dropping the exclusions on the way to the answer(x² − 4)/(x − 2) = x + 2 is only true for x ≠ 2. Without that condition the statement is false at exactly one point, and “Graphs and Asymptotes” draws the hole it leaves.
3
Missing an opposite pair(4 − x²)/(x − 2) looks unfactorable against its denominator until you write 4 − x² = −(x − 2)(x + 2). The answer is −(x + 2), not "no common factor".
Check your understanding
✓
Finish when you can justify each cancellation by naming the factor.
For any expression you reduce, you should be able to point at the polynomial you removed and say that it multiplied both the numerator and the denominator. If you cannot name it, you have not cancelled — you have invented a rule that happens to work on this example.
Note: A numerical check is not a proof, and on this path it is never offered as one: an identity can survive a lucky value. But a single disagreement is conclusive in the other direction, and that is what makes the habit worth having — it cannot confirm your answer, and it will reliably refute a wrong one.