Rational Expression Operations: Step-by-Step Guide
You add, subtract, multiply, and divide rational expressions (fractions with polynomials) the same way you do with number fractions, but factoring is a big part of it. To multiply, factor and cancel. To divide, flip the second fraction and multiply. To add or subtract, find a common denominator (the LCD) and combine the tops.
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Factor everything. Factor every top and bottom completely before doing anything else. This shows you what can cancel and what the LCD is.\[\frac{P(x)}{Q(x)} = \frac{f_{1}(x)\,f_{2}(x)}{g_{1}(x)\,g_{2}(x)}\]
- Do the operation. Multiply: multiply the tops and multiply the bottoms. Divide: flip the second fraction and multiply. Add or subtract: give each fraction the LCD, then add or subtract the tops.\[\frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c},\quad \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\]
- Cancel common factors. After multiplying, cancel any factor that is on both the top and the bottom.\[\frac{f(x)\,h(x)}{g(x)\,h(x)} = \frac{f(x)}{g(x)}\]
- List the values that are not allowed. Find every value that makes ANY of the original bottoms zero. These stay off limits even if the factor cancels.\[x \ne \{\text{zeros of all original denominators}\}\]
- Write the final answer. Write the answer in simplest form with the restrictions.\[\frac{N(x)}{D(x)},\quad x\ne\ldots\]
Worked example
Compute and simplify: (x^2 - 1)/(x + 2) * (x + 2)/(x - 1)
Step 1: Factor: x^2 - 1 = (x - 1)(x + 1). The problem becomes [(x - 1)(x + 1) / (x + 2)] x [(x + 2) / (x - 1)].
Step 2: Multiply the tops and the bottoms: [(x - 1)(x + 1)(x + 2)] / [(x + 2)(x - 1)].
Step 3: Cancel (x - 1) and (x + 2) from the top and bottom. What is left is x + 1.
Step 4: Restrictions from the originals: x + 2 cannot be 0, so x is not -2. x - 1 cannot be 0, so x is not 1.
Step 5: The answer is x + 1, where x is not -2 and x is not 1.
Check: Try x = 0. Original: (-1)(1) / 2 x 2 / (-1) = (-1/2) x (-2) = 1. Simplified: 0 + 1 = 1. They match.
Tips and common mistakes
- Factor first, then multiply. Multiplying first hides what could cancel.
- To divide, flip only the second fraction, never the first.
- To add or subtract, never add the bottoms. Find the LCD first.
- The LCD uses each different factor, at the highest power it shows up in any bottom.
- Cancel factors, not terms. (x + 1)/(x + 2) does not simplify.
- Restrictions come from every original bottom, even the ones that cancel.
- When subtracting, subtract the whole top of the second fraction. Distribute the minus sign.
- Check with a number that is allowed.