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Algebra 2 practice

How to Simplify Rational Expressions

Factor the numerator and denominator, cancel the common factors — whole factors only, never individual terms — and carry the original domain restrictions. Below: a worked example, the illegal cancels that cost the most marks, and practice with the reasoning on every question.

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What you'll practice

A rational expression is a fraction of polynomials. Simplifying means factoring top and bottom, then canceling common FACTORS — never individual terms. These questions cover factoring both parts, spotting the illegal term-cancel, and the domain restrictions that canceling hides.

Worked example

Simplify x2−9x2+5x+6\frac{x^2 - 9}{x^2 + 5x + 6}.

  1. Factor the numerator: x2−9=(x−3)(x+3)x^2 - 9 = (x-3)(x+3) (difference of squares).
  2. Factor the denominator: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x+2)(x+3).
  3. Cancel the common factor (x+3)(x+3): (x−3)(x+3)(x+2)(x+3)=x−3x+2\frac{(x-3)(x+3)}{(x+2)(x+3)} = \frac{x-3}{x+2}.
  4. Note the restrictions from the ORIGINAL denominator: x≠−2,−3x \neq -2, -3.

Common mistakes

  • Canceling terms instead of factors: in x2−9x2+5x+6\frac{x^2-9}{x^2+5x+6} you cannot cancel the x2x^2's — only whole factors cancel.
  • Losing a restriction: after canceling (x+3)(x+3), the value x=−3x = -3 is still excluded — the original expression was undefined there.
  • Canceling across addition: x+33\frac{x+3}{3} is not x+1x + 1 — the 3's aren't factors of the whole numerator.

Try a few

1. Simplify x2−4x+2\frac{x^2 - 4}{x + 2}.

  • x−2x - 2 (for x≠−2x \neq -2)
  • x+2x + 2
  • x2−2x^2 - 2
  • It cannot be simplified

2. Simplify x2+3xx2−9\frac{x^2 + 3x}{x^2 - 9}.

  • xx−3\frac{x}{x-3}
  • xx+3\frac{x}{x+3}
  • 3x−9\frac{3x}{-9}
  • 1x−3\frac{1}{x-3}

3. Simplify 2x2+6x4x\frac{2x^2 + 6x}{4x}.

  • x+32\frac{x+3}{2}
  • 2x+64\frac{2x+6}{4}
  • x2+3x2\frac{x^2+3x}{2}
  • 2x+32x + 3

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FAQ

What is a rational expression?
A fraction whose numerator and denominator are polynomials, like (x² − 9)/(x² + 5x + 6). It's defined everywhere except where the denominator equals zero.
What can be canceled in a rational expression?
Only common factors of the entire numerator and entire denominator — never individual terms. Factor both completely first; then anything shared cancels.
Why do restrictions matter after canceling?
The original expression is undefined wherever its denominator was zero — and canceling can hide that. The restriction survives even after the factor is gone.
Is Cramzoo free?
Yes — this skill is one of the free ones. Every unit opens with free skills, no account needed. Premium ($6/month or $30/year) unlocks every skill plus the parent progress dashboard.
How is this different from IXL?
No punishing score. A wrong answer gives you a step-by-step explanation, not a number that drops — and every question is double-checked for accuracy before it reaches you.

Related Algebra 2 skills

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