Simplifying Rational Expressions: Step-by-Step Guide
A rational expression is a fraction with polynomials on the top and bottom. To simplify one, factor both the top and the bottom completely and cancel the factors they share. Always write down the values of x that make the original bottom equal 0, because those values are not allowed.
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Factor the top and the bottom completely. Factor each polynomial fully. Do not multiply anything out.\[\frac{P(x)}{Q(x)} = \frac{f_{1}(x)\,f_{2}(x)\cdots}{g_{1}(x)\,g_{2}(x)\cdots}\]
- Find the values that are not allowed. Set the ORIGINAL bottom equal to 0 and list every x that makes it 0.\[Q(x)\ne 0\]
- Cancel common factors. Cross out factors that appear on both the top and the bottom. You can only cancel factors (things that are multiplied), not terms (things that are added).\[\frac{f(x)\,h(x)}{g(x)\,h(x)} = \frac{f(x)}{g(x)}\]
- Write the answer with the restrictions. Write the simplified expression and the values of x that are not allowed.\[\frac{f(x)}{g(x)},\quad x \ne \text{excluded values}\]
Worked example
Simplify: (x^2 - 9) / (x^2 + 5x + 6)
Step 1: Factor the top: x^2 - 9 = (x - 3)(x + 3). Factor the bottom: x^2 + 5x + 6 = (x + 2)(x + 3).
Step 2: Find the restrictions from the ORIGINAL bottom: (x + 2)(x + 3) = 0 gives x = -2 and x = -3.
Step 3: Cancel the common factor (x + 3): [(x - 3)(x + 3)] / [(x + 2)(x + 3)] = (x - 3) / (x + 2).
Step 4: The answer is (x - 3) / (x + 2), where x cannot be -2 or -3.
Check: Try x = 1. Original: (1 - 9) / (1 + 5 + 6) = -8/12 = -2/3. Simplified: (1 - 3) / (1 + 2) = -2/3. They match.
Tips and common mistakes
- Only cancel factors, never terms. (x + 2)/(x + 3) does not simplify to 2/3.
- Always list the restrictions from the original bottom. A factor you cancel still counts.
- Factor everything first. Partly factored expressions hide things you could cancel.
- Watch for opposite factors. (a - b) = -(b - a), so (a - b)/(b - a) = -1.
- Do not try to cancel before you factor. That leads to wrong cancellations.
- Check by putting a number (that is allowed) into the original and the simplified version. They should match.