Scotch Egg Home
Unit 5: Exponents, Roots, and Rational Expressions

Topic 3

Simplifying Rational Expressions

A rational expression is a fraction in which the numerator and denominator are polynomials. Simplifying a rational expression follows the same logic as simplifying numerical fractions — divide out common factors, not terms.
For example:
x2+3xx≠x+3\frac{x^2 + 3x}{x} \ne x + 3
because x2+3xx^2 + 3x can be factored, while xx in the denominator only cancels with matching factors:
x2+3xx=x(x+3)x=x+3.\frac{x^2 + 3x}{x} = \frac{x(x + 3)}{x} = x + 3.
Here, cancellation was valid because both numerator and denominator shared a factor of xx.
Always factor completely before simplifying, and remember that any value making the denominator zero must be excluded from the domain.

Core Skills

  • Factor numerators and denominators completely.
  • Identify and cancel common factors (not terms).
  • Simplify using exponent laws where possible.
  • Rewrite complex fractions into single rational expressions.
  • Recognize and note restrictions where denominators are zero.

Example 1: Basic Common Factor Cancellation

Simplify 6x23x\dfrac{6x^2}{3x}.
6x23x=63⋅x2x=2x\dfrac{6x^2}{3x} = \dfrac{6}{3} \cdot \dfrac{x^2}{x} = 2x
Final Answer: 2x\boxed{2x}, where x≠0.x \ne 0.

Example 2: Factoring Before Cancelling

Simplify x2+5xx\dfrac{x^2 + 5x}{x}.
x2+5xx=x(x+5)x=x+5\dfrac{x^2 + 5x}{x} = \dfrac{x(x + 5)}{x} = x + 5
Final Answer: x+5\boxed{x + 5}, where x≠0.x \ne 0.

Example 3: Polynomial with Common Binomial Factor

Simplify (x+2)(x−3)(x−3)\dfrac{(x + 2)(x - 3)}{(x - 3)}.
(x−3) cancels out.x+2(x - 3)\text{ cancels out.} \quad \boxed{x + 2}
Restriction: x≠3.x \ne 3.

Example 4: Quadratic Factorization

Simplify x2−9x2−3x\dfrac{x^2 - 9}{x^2 - 3x}.
(x−3)(x+3)x(x−3)=x+3x\dfrac{(x - 3)(x + 3)}{x(x - 3)} = \dfrac{x + 3}{x}
Final Answer: x+3x\boxed{\dfrac{x + 3}{x}}, where x≠0,3.x \ne 0, 3.

Example 5: Simplifying with Exponents

Simplify x5x2\dfrac{x^5}{x^2}.
x5−2=x3x^{5 - 2} = x^3
Final Answer: x3\boxed{x^3}, where x≠0.x \ne 0.

Example 6: Complex Rational Expression

Simplify xx+1x2x+1\dfrac{\frac{x}{x+1}}{\frac{x^2}{x+1}}.
xx+1x2x+1=xx+1⋅x+1x2=1x\dfrac{\frac{x}{x+1}}{\frac{x^2}{x+1}} = \frac{x}{x+1} \cdot \frac{x+1}{x^2} = \frac{1}{x}
Final Answer: 1x\boxed{\dfrac{1}{x}}, where x≠0,−1.x \ne 0, -1.

Key Takeaways

  • Always factor completely before cancelling.
  • Cancel only common factors, never terms connected by addition or subtraction.
  • Record any values that make the denominator zero as restrictions.
  • Simplifying rational expressions mirrors numerical fraction simplification.
  • Use exponent rules to handle variable powers efficiently.