Combining with unlike denominators
Add and subtract rational expressions with unlike denominators: find the LCD, build equivalent fractions, combine numerators, and simplify the result by factoring and canceling.
Key Concepts
To add or subtract rational expressions with different denominators, you must first find the least common denominator (LCD). Next, rewrite each fraction as an equivalent fraction that has the LCD as its new denominator. Once the denominators are the same, you can add or subtract the numerators as usual and simplify the result if possible.
$\frac{3}{4x^2} + \frac{5}{6x} = \frac{3 \cdot 3}{12x^2} + \frac{5 \cdot 2x}{12x^2} = \frac{9+10x}{12x^2}$ $\frac{2}{x^2 1} \frac{3}{x^2 x 2} = \frac{2}{(x 1)(x+1)} \frac{3}{(x 2)(x+1)} = \frac{2(x 2) 3(x 1)}{(x 1)(x+1)(x 2)} = \frac{ x 1}{(x 1)(x+1)(x 2)}$.
This is the big leagues of fraction operations! You can't just mash them together. It’s a three step dance: first, find the common ground (the LCD). Second, give each fraction a makeover by multiplying its top and bottom by whatever factors are missing. Third, with their new matching denominators, you can finally combine the numerators.
Common Questions
What is the LCD and why do you need it to combine unlike rational expressions?
The LCD (Least Common Denominator) is the smallest polynomial expression divisible by each individual denominator. You need it to convert all fractions into equivalent forms with a common denominator, making addition and subtraction of numerators valid.
What are the steps for adding rational expressions with unlike denominators?
Factor each denominator. Find the LCD by taking each unique factor to its highest power. Multiply each fraction's numerator and denominator by whatever is missing from the LCD. Add or subtract the numerators over the LCD. Factor and simplify the result by canceling common factors.
How do you simplify after combining rational expressions?
Factor the resulting numerator fully. Check whether any factor in the numerator matches a factor in the denominator. Cancel any common factors. State the domain restrictions (values excluded because they make any original denominator zero) since those remain invalid even after simplification.