Simplifying Rational Expressions Practice

Simplifying Rational Expressions Practice Problems

The Simplifying Rational Expressions lesson covers factoring, cancelling, and excluded values in full; this practice set 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	, excluded value
Medium	The larger of two excluded values, quadratic denominator	, excluded values and
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.

Worked examples

One example at each difficulty level, solved the way the practice set expects.

Easy. Find the excluded value of .

The denominator can’t be zero: gives .

Medium. Find the excluded values of , then simplify.

Factor the denominator: , so and . Cancelling gives , but stays excluded even though its factor cancelled.

Hard. Simplify and evaluate it at .

, with . At : . The original gives too, which confirms the cancelling.

Using the practice problems

Select your difficulty and count, and a fresh set drops in immediately. Whole-number answers go in as-is; the cancel-then-evaluate problems accept a fraction or its decimal equivalent.

Check Answers scores every problem at once and calls out what still needs correcting
Show Answers shows every answer in one pass, handy when reviewing offline work
Print Worksheet formats a clean page for printing, suited for offline practice or classroom handouts

The rules these practice problems drill

Rule	Statement
Excluded value	Any input that makes the original denominator equal
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 Practice Problems. For the full factor-then-cancel technique and every excluded-value example, see the Simplifying Rational Expressions lesson.

Frequently Asked Questions

Why does this practice set focus on excluded values instead of the simplified expression?+

A simplified rational expression is itself an expression, which can't be automatically checked as a single number. The values excluded from its domain — wherever the original denominator equals zero — are always checkable numbers, so every problem is built around finding those directly.

How do I find an excluded value?+

Set the original, unsimplified denominator equal to zero and solve — every solution is a value the expression can't be evaluated at. A linear denominator has one excluded value; a quadratic denominator usually has two.

Why does cancelling a shared factor not remove an excluded value?+

The excluded values come from the original denominator, before any simplifying happens — cancelling a factor can make a restriction disappear from the written expression, but the expression was still genuinely undefined there to begin with.

What does the cancel-then-evaluate problem test?+

It gives a rational expression that's already factored with a shared binomial in the numerator and denominator, and asks you to evaluate the simplified version at a chosen x — testing that cancelling correctly still gives the right numeric value.

How is this different from the Rational Expression Practice Problems?+

The Rational Expression practice set evaluates a rational expression directly, without any simplifying involved. This practice set is specifically about the domain-restriction and cancelling side of the topic, which that set doesn't test.

a/bSimplifying Rational Expressions Practice Problems