Rational Expression Generator
Simplifying, multiplying, dividing, and combining rational expressions are covered in full in the Rational Expressions lesson; this generator is where that skill turns into speed. Every problem asks you to evaluate a rational expression at a given \(x\) — genuine polynomial arithmetic on both the numerator and denominator, with a single checkable answer at the end.
How the difficulty levels work
| Difficulty | What it drills | Example |
|---|---|---|
| Easy | Evaluate a linear-over-linear expression | \(\dfrac{7x+7}{x+4}\) at \(x=5\) |
| Medium | Evaluate a quadratic-over-linear expression | \(\dfrac{2x^{2}+3x+8}{x-5}\) at \(x=0\) |
| Hard | Evaluate a quadratic-over-quadratic expression, or a sum of two fractions | \(\dfrac{8}{x} + \dfrac{4}{x+2}\) at \(x=-6\) |
Easy evaluates a simple linear-over-linear expression, keeping the arithmetic light while the fraction structure is still new.
Medium puts a quadratic in the numerator, so evaluating it correctly requires the order of operations to be exactly right before dividing.
Hard alternates between a full quadratic-over-quadratic evaluation and a sum of two separate rational expressions with different denominators — the exact skill the lesson’s LCD section covers, just checked numerically instead of symbolically.
Using the generator
Pick a difficulty and a question count, and a fresh set appears instantly. Whole-number answers go in as-is; fraction answers are accepted either as a fraction (like 7/3) or a decimal.
- 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 |
|---|---|
| Multiplying | \(\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd}\) |
| Dividing | \(\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c}\) |
| Adding, different denominators | Find the LCD, rewrite each fraction, then combine |
See the Rational Expressions lesson for the full derivation of each rule, plus simplifying and domain-restriction techniques this generator doesn’t test directly.
Common mistakes to watch for
Forgetting the order of operations in the numerator. Evaluate the whole polynomial numerator first, then divide by the whole denominator — not term by term.
Mixing up the sign of a negative x when squaring. \(x^{2}\) at \(x=-5\) means \((-5)^{2}=25\), a positive number.
Adding numerators of two fractions without a common denominator. \(\dfrac{8}{x} + \dfrac{4}{x+2}\) has to be rewritten over a shared denominator before the numerators combine — the same LCD process as plain fraction addition.
Entering a decimal that’s rounded too early. If the exact answer is a repeating decimal, enter it as a fraction instead to avoid a rounding mismatch.
Where to go next
Once these feel automatic, the Factoring Generator builds the skill this generator doesn’t test directly — recognizing and factoring the patterns that simplify a rational expression before you ever evaluate it. For the full reasoning behind every rule drilled here, see the Rational Expressions lesson.