Math101 · Independent Ontario learning library
Math101learn.math101.caRational Expressions
Rational expressions are polynomial fractions whose algebra is governed by factoring, common denominators, and domain restrictions.
Treat a rational expression like a numerical fraction—but record every value that would make an original denominator zero.
Definition and restrictions
A rational expression is a quotient of polynomials,
Values making $Q(x)=0$ are excluded from the domain. State restrictions from the original expression before simplifying, because cancellation does not restore an input that was never permitted.
Simplifying by factoring
Factor numerator and denominator completely, then cancel common factors:
The original restrictions are $x\ne-3,2$. The simplified formula still represents a hole at $x=-3$.
Factors versus terms
Cancellation applies across multiplication, not across addition. For example,
cannot lose the $x$ because $x+3$ is a sum, not a product containing an $x$ factor. Factoring first reveals whether a genuine common factor exists.
Multiplying rational expressions
Factor everything, cancel factors across any numerator and denominator, then multiply what remains.
Dividing rational expressions
Multiply by the reciprocal of the second expression:
In addition to original denominator restrictions, the entire divisor $C/D$ cannot equal zero, so values making its numerator $C=0$ must also be excluded.
Adding and subtracting
Rational expressions need a least common denominator (LCD). Rewrite each fraction with that denominator, combine numerators, and simplify.
For example,
with $x\ne0,-1$.
Do not add denominators; fractions describe parts of differently sized wholes until a common denominator is built.
A subtraction sign
When subtracting, place the entire second numerator in parentheses:
The leading subtraction changes every sign in the second product after expansion.
Complex rational expressions
A complex fraction contains fractions in its numerator, denominator, or both. Multiply the complete numerator and denominator by the LCD of all smaller fractions.
This clears nested denominators without changing the value, provided all restrictions are retained. Simplify only after the structure is clear.
Equivalent forms and domain
Two rational formulas can agree for every shared allowed input while having different written domains. For instance,
only for $x\ne1$. The original graph has a removable hole at $x=1$, whereas $y=x+4$ alone includes that point.
Common mistakes
Cancelling terms instead of factors. Factor before cancelling.
Forgetting original restrictions. Cancelled denominator factors still exclude values.
Adding denominators. Build an LCD and adjust numerators.
Failing to distribute subtraction. Parenthesize the complete second numerator.
Ignoring that a divisor cannot be zero. Division creates an extra restriction from the reciprocal's denominator.
Quick self-check
- What values make any original denominator zero?
- Is every polynomial factored completely?
- Am I cancelling only factors?
- For addition or subtraction, is the LCD complete?
- For division, have I excluded values making the divisor zero?
- Does the final form retain all restrictions?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Simplify (x² − 9)/(x² + x − 6) and retain the original restrictions.
- x² − 9 = (x − 3)(x + 3).
- x² + x − 6 = (x + 3)(x − 2).
- Cancel x + 3 but retain x ≠ −3 and x ≠ 2.
End of lesson
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