Simplifying Rational Expressions

Overview

This lesson covers how to simplify rational expressions by factoring, canceling common factors, and using properties of exponents.

To simplify, factor both numerator and denominator and cancel common terms.
When dividing terms with the same base, subtract exponents (e.g., ( x^5 / x^2 = x^{3} )).
Always look for the greatest common factor (GCF) before canceling.

( \frac{35x^5}{49x^2} ) simplifies to ( \frac{5x^3}{7} ) after canceling the common factors of 7 and ( x^2 ).
( \frac{4x^2 + 8x}{3x + 6} ) factors to ( \frac{4x(x+2)}{3(x+2)} ), then simplifies to ( \frac{4x}{3} ).
( \frac{x^2 - 16}{x^2 + 9x + 20} ) factors to ( \frac{(x+4)(x-4)}{(x+4)(x+5)} ), then simplifies to ( \frac{x-4}{x+5} ).
If expressions look similar but reversed (e.g., ( \frac{5-x}{x-5} )), factor out -1 and reorder to allow cancellation, resulting in ( -1 ).

( \frac{72x^8y^7}{64x^5y^4} ) simplifies to ( \frac{9x^3y^3}{8} ).
( \frac{5x^2 - 15x}{8x - 24} ) becomes ( \frac{5x(x-3)}{8(x-3)} = \frac{5x}{8} ).
( \frac{42 - 6x}{3x - 21} ) becomes ( \frac{7-x}{x-7} ), factor out -1 to get ( -2 ).
( \frac{x^2-8x+15}{2x^2-18} ) factors and cancels to ( \frac{x-5}{2(x+3)} ).
( \frac{2x^2-5x-3}{4x^2-1} ) factors to ( \frac{(x-3)(2x+1)}{(2x+1)(2x-1)} ), cancels to ( \frac{x-3}{2x-1} ).

Rational Expression — A ratio of two polynomials.
Greatest Common Factor (GCF) — The largest factor shared by terms in an expression.
Difference of Squares — Factoring technique for ( a^2 - b^2 = (a+b)(a-b) ).
Trinomial Factoring — Breaking a quadratic expression into two binomials.

Practice factoring and simplifying additional rational expressions.
Review homework problems involving GCF, difference of squares, and trinomial factoring.