Simplifying Rational Expressions Generator
The Simplifying Rational Expressions lesson covers factoring, cancelling, and excluded values in full; this generator focuses on the one piece of simplifying that always stays a single checkable number — where the expression is undefined.
How the difficulty levels work
| Difficulty | What it drills | Example |
|---|---|---|
| Easy | The single excluded value of a linear denominator | \(\dfrac{x-5}{x+7}\), excluded value \(-7\) |
| Medium | The larger of two excluded values, quadratic denominator | \(\dfrac{x+2}{x^{2}-x-6}\), excluded values \(3\) and \(-2\) |
| Hard | The smaller excluded value with bigger numbers, or a cancel-then-evaluate check | Factor, cancel, then evaluate |
Easy finds the single value excluded by a simple linear denominator.
Medium finds the larger of two values excluded by a quadratic denominator, which factors into two linear pieces.
Hard alternates between the smaller of two excluded values with bigger numbers, and a genuinely different check: a rational expression already factored with a shared binomial, evaluated after cancelling.
Using the generator
Pick a difficulty and a question count, and a fresh set appears instantly. Whole-number answers go in as-is; the cancel-then-evaluate problems accept a fraction or its decimal equivalent.
- Check Answers scores the set and flags exactly which problems need another look
- Show Answers reveals every solution, useful for reviewing a paper attempt
- Print Worksheet outputs a clean page for offline practice or classroom handouts
The rules this generator drills
| Rule | Statement |
|---|---|
| Excluded value | Any input that makes the original denominator equal \(0\) |
| Cancelling | Only a shared factor — never a term — can be cancelled |
| After simplifying | Excluded values still apply, even if the simplified denominator no longer shows them |
See the Simplifying Rational Expressions lesson for the full factor-then-cancel technique and the opposite-factors pattern.
Common mistakes to watch for
Losing an excluded value that came from a cancelled factor. It’s still a restriction, even once the simplified expression no longer shows it.
Cancelling a term instead of a factor. Only something multiplying the entire numerator or denominator can be cancelled — an added or subtracted term never can.
Forgetting to factor a quadratic denominator before finding its excluded values. Both roots of the factored form are excluded, not just the more obvious one.
Assuming every rational expression simplifies. Many genuinely have no shared factor, and “already in lowest terms” is a valid answer.
Where to go next
For evaluating a rational expression directly, without any simplifying involved, see the Rational Expression Generator. For the full factor-then-cancel technique and every excluded-value example, see the Simplifying Rational Expressions lesson.