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Learn Index · Patterns and Algebra Grade 11–12

Operations with rational expressions

Adding, subtracting, multiplying and dividing rational expressions.

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Same rules as fractions

Operations with rational expressions follow the same rules as numerical fractions. The extra steps: factor everything, and track restrictions.

Multiplying

  1. Factor all numerators and denominators.
  2. State restrictions.
  3. Cancel common factors.
  4. Multiply what’s left.

Example: (x² − 4)/(x + 3) × (x + 3)/(x − 2)

  • = (x + 2)(x − 2)/(x + 3) × (x + 3)/(x − 2)
  • Cancel (x − 2) and (x + 3)
  • = x + 2, x ≠ −3, 2

Dividing

Multiply by the reciprocal of the second expression.

Example: (x + 1)/(x − 5) ÷ (x² − 1)/(2x − 10)

  • = (x + 1)/(x − 5) × 2(x − 5)/[(x + 1)(x − 1)]
  • = 2/(x − 1)
  • Restrictions: x ≠ 5, x ≠ ±1 (include values that make the divisor’s numerator zero, since you flipped it)

Adding and subtracting

You need a common denominator.

Same denominators:

3/(x + 2) + x/(x + 2) = (x + 3)/(x + 2), x ≠ −2

Different denominators: find the lowest common denominator (LCD).

Example: 2/x + 3/(x + 1)

  • LCD = x(x + 1)
  • = 2(x + 1)/[x(x + 1)] + 3x/[x(x + 1)]
  • = (2x + 2 + 3x)/[x(x + 1)]
  • = (5x + 2)/[x(x + 1)], x ≠ 0, −1

Example: x/(x² − 9) − 1/(x − 3)

  • x² − 9 = (x + 3)(x − 3); LCD = (x + 3)(x − 3)
  • = [x − (x + 3)]/[(x + 3)(x − 3)]
  • = −3/[(x + 3)(x − 3)], x ≠ ±3

Use brackets when subtracting a numerator!

Strategy summary

Operation Key step
Multiply factor, cancel, multiply
Divide flip the second, then multiply
Add/Subtract factor denominators, find LCD, combine numerators

Worked examples

Example 1: (3x/4) × (8/x²) = 24x/(4x²) = 6/x, x ≠ 0.

Example 2: 5/(2x) − 1/x = 5/(2x) − 2/(2x) = 3/(2x), x ≠ 0.

Example 3: (x² + 3x)/(x − 1) ÷ x/(x² − 1) = x(x + 3)/(x − 1) × (x − 1)(x + 1)/x = (x + 3)(x + 1), x ≠ 0, ±1.

Common mistakes

  • Adding denominators when adding fractions.
  • Forgetting to distribute a minus sign when subtracting.
  • Missing restrictions from the flipped divisor.
  • Cancelling before factoring.

Try it yourself

  1. (x/(x + 4)) × ((x + 4)/x²)
  2. ((x − 2)/6) ÷ ((x − 2)/3)
  3. 4/(x − 1) + 2/(x − 1)
  4. 1/x + 1/(2x)
  5. 3/(x + 2) − 2/(x − 2)

Answers: 1) 1/x, x ≠ 0, −4 2) 1/2, x ≠ 2 3) 6/(x − 1), x ≠ 1 4) 3/(2x), x ≠ 0 5) (x − 10)/[(x + 2)(x − 2)], x ≠ ±2

Quick recap

  • Multiply: factor and cancel. Divide: multiply by the reciprocal.
  • Add or subtract with a lowest common denominator.
  • Track all restrictions.