Rational Expressions Practice Test

Factor before canceling, preserve restrictions, combine fractions with LCDs, and simplify complex rational expressions.

Rational Expressions Practice Test

This test has 20 questions

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Rational expression domain passport

Simplify the fraction. Keep the original restrictions.

Rational expressions require two kinds of bookkeeping at once. Algebraically, factors can cancel and fractions can combine. Logically, every value that made an original denominator zero must remain excluded even if the factor causing that restriction disappears during simplification.

Anchor 01Factor first; cancel factors, not terms.

Anchor 02Keep excluded values from the original expression.

Anchor 03Invert only the divisor when dividing.

Anchor 04Add and subtract with the least common denominator.

1. Treat simplification and domain restrictions as separate records

A canceled factor disappears from the simplified formula, but it does not erase the value that was forbidden in the original denominator. Record restrictions before canceling.

Factor numerator and denominator

→

Record denominator zeros

→

Cancel common factors and simplify

Expression record

Simplified form

The final algebraic expression may be shorter after common factors are removed.

Domain record

Excluded values stay

Any value that made an original denominator zero remains excluded.

Final answer

Report both

A complete simplification includes the simplified expression together with its original restrictions.

2. Cancel factors, never pieces of a sum

Cancellation is division by a common multiplicative factor. Terms separated by addition or subtraction are not independent factors and cannot be crossed out.

Factor gate

Only complete matching factors may pass through the cancellation gate.

Original expression

\frac{x^{2} + 2 x - 15}{x^{2} - 5 x + 6}

Factor completely

\frac{(x + 5) (x - 3)}{(x - 2) (x - 3)}

Cancel common factor

\frac{x + 5}{x - 2}

Legal cancellationA matching factor can cancel because it multiplies the entire remaining expression.

Illegal cancellationIn

\frac{x + 4}{x}

, the variable is not a factor of the entire numerator sum.

Factor firstMany valid cancellations are invisible until quadratics or common factors are factored.

Restrictions firstRecord forbidden denominator values before removing a common factor.

3. A canceled denominator factor still leaves an excluded value

Domain restrictions belong to the original expression. Simplification changes the formula, not the historical fact that certain inputs originally caused division by zero.

\frac{(x - 3) (x + 5)}{(x - 3) (x - 2)} = \frac{x + 5}{x - 2}

Original denominator

(x - 3) (x - 2)

Forbidden values

x \neq 3, x \neq 2

After cancellationThe value

3

is still excluded even though its factor no longer appears in the simplified denominator.

4. Multiplication and division are factor problems

For multiplication, factor first and cancel across the product. For division, invert only the divisor, then treat the result as multiplication.

Multiply

\frac{x - 2}{x + 3} \cdot \frac{x + 3}{x + 5} = \frac{x - 2}{x + 5}

Cancel matching factors across the multiplication.

Divide

\frac{a}{b} \div \frac{x + 1}{x - 4} = \frac{a}{b} \cdot \frac{x - 4}{x + 1}

Only the divisor fraction is inverted.

Restriction alert

After inversion, values that make the new denominator zero may add restrictions because the original divisor itself must also be nonzero.

5. Addition and subtraction require a least common denominator

Unlike multiplication, rational-expression addition does not allow direct cancellation across a plus sign. Build a common denominator first, then combine numerators.

Illustrative sum

\frac{2}{x - 1} + \frac{3}{x + 2}

The denominators are different, so each fraction must be rewritten using the common product denominator.

Factor denominators

→

Build the LCD

→

Combine numerators

Common denominator result

\frac{2 (x + 2) + 3 (x - 1)}{(x - 1) (x + 2)}

Simplify the numerator only after both fractions have the same denominator, and retain restrictions from all original denominator factors.

6. Subtracting rational expressions requires sign control in the numerator

After creating the LCD, distribute the subtraction sign across the entire second numerator before combining like terms.

Before combining

Preserve the grouped numerator

\frac{3 (x + 1) - (2 x - 5)}{(x - 2) (x + 1)}

The minus sign applies to every term in the second numerator.

After sign distribution

Then simplify the numerator

\frac{3 x + 3 - 2 x + 5}{(x - 2) (x + 1)}

Only after the sign is handled correctly should like numerator terms be combined.

7. Complex fractions become simpler when every small denominator is cleared

A complex rational expression contains fractions inside a larger fraction. One reliable method is to multiply the entire numerator and denominator by the LCD of all inner denominators.

Starting structure

\frac{\frac{1}{x} + \frac{1}{2}}{\frac{3}{x}}

Identify the inner denominators before changing anything.

Clear inner fractions

Multiply the large numerator and large denominator by the same LCD so the value of the whole expression is unchanged.

Then simplify

After the inner denominators disappear, factor and cancel using ordinary rational-expression rules.

8. Reciprocals can create new denominator restrictions

When a rational expression is inverted, its original numerator becomes a denominator. Values that make that numerator zero must therefore be excluded from the reciprocal.

Original expression

\frac{x - 4}{x + 2}

Original restriction: $x \neq - 2$ .

Reciprocal

\frac{x + 2}{x - 4}

The reciprocal also requires $x \neq 4$ , because the original numerator is now in the denominator.

9. Evaluate only after checking excluded values

Before substituting a number into a rational expression, verify that the input is allowed by the original domain restrictions.

Step 1

Check the denominator

Determine whether the proposed input makes any original denominator factor zero.

Step 2

Reject excluded inputs

If the input violates a restriction, the expression is undefined there even if a simplified formula appears to accept it.

Step 3

Substitute allowed inputs

Only after the domain check should numerical evaluation begin.

\frac{(x - 3) (x + 1)}{(x - 3) (x + 4)} = \frac{x + 1}{x + 4}, x \neq 3, x \neq - 4

10. Error analysis: most mistakes confuse factors with terms or formulas with domains

The algebra may look shorter after a cancellation, but that does not mean the original expression had a larger domain.

Canceled terms inside a sum

Cancellation requires common multiplicative factors, not matching pieces separated by addition.

Restriction erased after cancellation

An excluded value from an original denominator remains excluded in the simplified result.

Both fractions inverted in division

Keep the first expression unchanged and invert only the divisor.

Numerators and denominators added separately

Addition and subtraction require an LCD before numerators can combine.

Subtraction sign lost

When combining over an LCD, distribute the minus sign through the entire second numerator.

Reciprocal restrictions missed

After inversion, zeros of the original numerator can become new excluded values.

11. Worked mini-set: identify the governing rule first

These examples are illustrative teaching examples, not questions copied from the test.

Example A

Factor and cancel

$\frac{x^{2} - 9}{x^{2} - x - 6} = \frac{x + 3}{x + 2}$

Example B

Preserve restrictions

Even after simplification, values that made the original denominator zero remain excluded.

Example C

Multiply

Factor both numerators and denominators before canceling across the product.

Example D

Divide

Rewrite division as multiplication by the reciprocal of the divisor only.

Example E

Add with LCD

Build the least common denominator, rewrite both fractions, and then combine numerators.

Example F

Complex fraction

Multiply the large numerator and denominator by the inner LCD to clear nested denominators.

Final rational-expression checklist

Before accepting a simplified result, verify both the algebraic form and the original domain restrictions.

Did I factor before canceling?Cancellation applies to factors, never to isolated terms in a sum.

Did I record original denominator zeros?Those excluded values remain even after factors cancel.

If dividing, did I invert only the divisor?The first rational expression stays unchanged.

If adding or subtracting, did I use the LCD?Rewrite both fractions over the same denominator before combining numerators.

Did a reciprocal create new restrictions?Zeros of the original numerator may become denominator exclusions after inversion.

Did I report the simplified expression together with its restrictions?Both pieces are part of a complete answer.