Simplifying Rational Expressions
Simplify rational expressions in Grade 9 algebra by factoring numerator and denominator completely, then canceling common factors, while identifying excluded values where the denominator equals zero.
Key Concepts
Property To simplify a rational expression, find the greatest common factor (GCF) for the numerator and denominator. Factor both completely, then divide out any common factors.
Examples $\frac{2x^2 10x}{4x^2+6x} = \frac{2x(x 5)}{2x(2x+3)} = \frac{x 5}{2x+3}$ $\frac{2x^2 14x}{4x 28} = \frac{2x(x 7)}{4(x 7)} = \frac{2x}{4} = \frac{x}{2}$ $\frac{4x+28}{3x^2+21x} = \frac{4(x+7)}{3x(x+7)} = \frac{4}{3x}$.
Explanation Simplifying these expressions is like a cleanup mission! Your goal is to find identical pieces (factors) on the top and bottom and cancel them out. By factoring everything first, you can easily spot the matching parts. Once they’re gone, you're left with a much neater and simpler expression that is easier to work with.
Common Questions
What are the steps to simplify a rational expression?
Factor both the numerator and denominator completely. Identify any common factors that appear in both. Cancel (divide out) those common factors. State excluded values for any x that makes the denominator zero.
Why must you factor before canceling in rational expressions?
You can only cancel entire factors, not individual terms. For (x² - 4)/(x + 2), factor the numerator to (x+2)(x-2), then cancel the (x+2) factor to get x - 2. Canceling without factoring leads to incorrect results.
What is an excluded value and why does it matter when simplifying?
An excluded value is any value of x that makes the original denominator zero. Even after simplification, these values remain undefined for the original expression and must be stated. For (x²-4)/(x+2) simplified to x-2, x ≠ -2.