Step by Step
1
Factor everything completely
Fully factor both the numerator and denominator before attempting to simplify anything.
2
State restrictions from the ORIGINAL denominator
Before canceling anything, identify every value that would make the original, unfactored (or fully factored, pre-cancellation) denominator equal zero — those values are excluded from the domain.
3
Cancel common factors
Only after restrictions are stated, cancel any factors that appear in both the numerator and denominator.
4
Write the simplified expression, restrictions attached
Present your simplified expression along with the restriction(s) you identified — the restriction doesn't disappear just because the factor got canceled.
Applied Walkthrough
1
Simplify (x² − 4)/(x² − 2x), factoring first: (x+2)(x−2) / x(x−2).
2
Before canceling, state restrictions from the original denominator x(x−2): x cannot equal 0, and x cannot equal 2 (since either would make the original denominator zero).
3
Now cancel the common factor of (x−2) from numerator and denominator: (x+2)/x.
4
The simplified answer is (x+2)/x, with the restrictions x ≠ 0 and x ≠ 2 still applying — even though (x−2) no longer visibly appears in the simplified expression, the original restriction still holds.
Exam Application
Exams test specifically whether you state restrictions BEFORE canceling common factors, since a restriction from a canceled factor doesn't disappear just because it's no longer visible in the simplified expression.
⚠ Common Trap
The most common and most heavily tested trap: canceling factors first and then trying to state restrictions from the already-simplified expression — this misses restrictions that existed in the original but are no longer visible after cancellation.
✓ Quick Self-Check
1. What must you do before canceling common factors in a rational expression?
State all restrictions from the original (fully factored, pre-cancellation) denominator.
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2. Simplify (x²−4)/(x²−2x) and state the restrictions.
(x+2)/x, with restrictions x ≠ 0 and x ≠ 2.
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3. Why do restrictions from canceled factors still matter?
Because the original expression was undefined at those values, even though the factor causing that is no longer visible after simplifying.
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4. What is the first step in simplifying any rational expression?
Factor the numerator and denominator completely.
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5. What's the biggest trap when working with rational expressions?
Canceling first and stating restrictions afterward from the simplified expression — this misses restrictions from factors that got canceled.
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