Add and subtract with unlike denominators
Adding and subtracting rational expressions with unlike denominators requires finding the least common denominator (LCD), rewriting each expression with the LCD, then combining the numerators. This essential algebra skill is taught in Openstax Intermediate Algebra 2E, Chapter 7: Rational Expressions and Functions. The result should always be simplified to lowest terms.
Key Concepts
Property To add or subtract rational expressions with unlike denominators: Step 1. Determine if the expressions have a common denominator. If not, find the LCD. Then, rewrite each rational expression as an equivalent expression with the LCD. Step 2. Add or subtract the numerators and place the result over the common denominator. Step 3. Simplify the resulting rational expression, if possible.
Examples Add: $\frac{5}{x 4} + \frac{3}{x+2}$. The LCD is $(x 4)(x+2)$. This becomes $\frac{5(x+2)}{(x 4)(x+2)} + \frac{3(x 4)}{(x 4)(x+2)} = \frac{5x+10+3x 12}{(x 4)(x+2)} = \frac{8x 2}{(x 4)(x+2)}$.
Subtract: $\frac{7y}{y^2 25} \frac{2}{y 5}$. The LCD is $(y 5)(y+5)$. This becomes $\frac{7y}{(y 5)(y+5)} \frac{2(y+5)}{(y 5)(y+5)} = \frac{7y 2y 10}{(y 5)(y+5)} = \frac{5y 10}{(y 5)(y+5)}$.
Common Questions
How do you add rational expressions with unlike denominators?
Find the LCD of all denominators, rewrite each expression with the LCD, then add the numerators and simplify.
How do you find the LCD of rational expressions?
Factor all denominators and take the product of the highest power of each distinct factor.
Where is adding rational expressions with unlike denominators taught in Openstax?
This is in Openstax Intermediate Algebra 2E, Chapter 7: Rational Expressions and Functions.
What is the difference between adding fractions and rational expressions?
The process is the same: find the LCD and combine numerators. Rational expressions simply have polynomial denominators instead of integers.
How do you subtract rational expressions?
Write both with the LCD, then subtract the numerators, distributing the negative sign carefully.