Factor the numerator and the denominator
Rewrite each polynomial as a product: x² + 5x factors into x(x + 5), and x² − 9 factors into (x + 3)(x − 3). Cancellation works on factors, never on separated terms.
Algebraic fractions
Simplifying Rational Expressions
A rational expression is a fraction with polynomials on top and bottom — and it simplifies exactly the way 6/9 does: factor both sides, cancel the factors they share, and keep the bar honest. The one new rule: cancel factors, never terms.
- factor first
- cancel common factors
- one fraction bar, two polynomials
- note the excluded values
Algebra addition tool
Add fractions that carry a variable
Type two terms with the same variable — the tool finds the common denominator and combines the numerators step by step.
Type the full expression
Examples: x/3 + 1/6, x/4 + x/6, 2x/5 + x/3, or 1/2 + x/8.
Derivation
x/3 + 1/6
= 2x/6 + 1/6
= (2x + 1)/6
Quick answer
Factor, cancel, note the exclusions
(x² + 5x)/x simplifies by factoring the numerator: x(x + 5)/x. The shared factor x cancels, leaving x + 5 for x ≠ 0. The factoring step is what makes the cancellation legal — the x in x(x + 5) is part of a product, so it can divide the whole numerator.
Without factoring, the same expression looks like a sum with an x in each term — and no legal cancellation exists there. Factor first is the whole method.
The method
Four steps: factor, cancel, stop, exclude
Cancel the shared factors
Any factor that appears on both the top and the bottom divides out: (x(x + 5))/x loses one x and leaves x + 5. Numeric factors cancel the same way — 6/9 cancels to 2/3.
Stop when nothing cancels
A result like (x + 5)/x is fully simplified even though it still looks busy: x + 5 is a sum, not a product, so no factor can cross the bar.
Track the excluded values
Any value that makes an original denominator zero is excluded from the answer: (x² − 9)/(x + 3) simplifies to x − 3 only for x ≠ −3, because the original bar forbids x = −3.
Worked example
Numeric factors cancel: 6x/9
6x/9 = (3 · 2x)/(3 · 3) = 2x/3
The 3 sitting inside both 6 and 9 is a shared factor of the whole numerator and the whole denominator, so it cancels. The x stays — nothing on the bottom pairs with it.
Worked example
Variable factors cancel: 5x²/10x
5x²/10x = (5x · x)/(5x · 2) = x/2
Factor each side completely: the numerator is 5x · x and the denominator is 5x · 2. The shared factor 5x crosses out, leaving x/2. The shortcut — subtract the powers of x and halve the coefficients — is this cancellation written quickly.
Worked example
Factor a difference of squares: (x² − 9)/(x + 3)
(x² − 9)/(x + 3) = (x + 3)(x − 3)/(x + 3) = x − 3, x ≠ −3
The numerator is not two loose terms — it is the product (x + 3)(x − 3). Once factored, the (x + 3) cancels with the denominator. The excluded value x = −3 stays attached to the answer, because the original bar divided by zero there.
Worked example
Pull out a common factor: (x² + 5x)/x
(x² + 5x)/x = x(x + 5)/x = x + 5, x ≠ 0
The numerator factors as x(x + 5), and the shared x cancels. The result x + 5 cannot simplify further — it is a sum, and only factors cancel across a fraction bar.
Cancel factors, never terms
The single rule that separates correct simplification from plausible nonsense: only complete factors — quantities joined by multiplication — may cancel across the bar. In (x² + 5x)/x, the x inside the factored product x(x + 5) cancels; in x² + 5x un-factored, no cancellation is legal, even though an x appears in both terms.
The arithmetic version of the same trap: (3 + 2)/3 is not 2, because the 3s are terms joined by addition, not factors joined by multiplication. (3 · 2)/3 is 2, because there the 3 is a factor of the whole numerator. Factoring is the step that converts sums into products so the legal cancellation becomes visible.
Why rational expressions are fractions with polynomial parts
Everything on this page is ordinary fraction behaviour wearing algebra. A rational expression is one polynomial divided by another; simplifying it is simplifying a fraction like 6/9, just with letters that can carry more structure. The denominator still cannot be zero, the bar still means division, and the same-factor cancellation still follows from multiplying by 1.
That continuity runs in both directions. Students who trust fraction rules already own the rational-expression method — the new work is factoring. And the fraction intuition guards against the classic error: a sum over a bar does not simplify term by term, exactly as (3 + 2)/3 does not collapse.
Where subtracting algebraic fractions fits
Subtracting algebraic fractions uses the simplification from this page at both ends: simplify each fraction before combining, and simplify the result after. The combination step itself is the familiar one — match denominators, subtract the numerators — with variables riding in the numerators.
For example, x/3 − 1/4 needs the common denominator 12: 4x/12 − 3/12 = (4x − 3)/12. The numerator 4x − 3 is a sum that will not factor in general, so the answer is finished. The adding-variable-fractions guide walks the matching addition step by step, and the method is identical for subtraction.
Common mistakes when simplifying rational expressions
- Cancelling a term inside a sum: (x² + 5x)/x does not become x + 5 by deleting the x from each term — the x in x² is part of the product x·x, and the plus sign blocks cancellation. Factor first: x(x + 5)/x = x + 5. Same bar, legal move, only after factoring.
- Cancelling across a plus or minus: In (x + 2)/(x² + 2x), the (x + 2) cancels only because the denominator factors as x(x + 2). Writing the denominator as x² + 2x and crossing out the 2s cancels nothing — the bar separates whole products, not matching digits.
- Forgetting the excluded values: Simplifying (x² − 9)/(x + 3) to x − 3 changes the graph of the expression unless x = −3 is excluded: the original expression is undefined there. State the exclusion so the simplified and original forms match everywhere they are both defined.
- Expanding instead of factoring: Multiplying x(x + 5) out to x² + 5x before simplifying destroys the product structure the cancellation needs. Factor as far as possible, cancel, and only then — if asked — expand.
Keep learning
The algebraic fractions topic
Each page below takes one piece of working with fractions that carry variables — the operations, the equations, and the addition walkthrough.
Fractions with Variables
The overview: how every fraction operation treats a variable riding in the numerator.
Fraction Equations
Solve algebraic equations with fractions: x/3 + 1/4 = 5/6 gives x = 7/4, step by step.
Find the Product
Multiply rational expressions and cancel: 2x/3 × 6/x² = 4/x.
Adding with a Variable
The full walk of the common-denominator rewrite with variables in the numerators.
Common questions
Simplifying Rational Expressions FAQ
A rational expression is one polynomial divided by another — a fraction whose numerator and denominator contain variables, such as (x² + 5x)/x or (x + 3)/(x − 2). Like any fraction, its denominator cannot be zero, which is why simplification carries a note about excluded values.