Add Rational Expressions Calculator
Add rational expressions by finding LCD and combining numerators with step-by-step solutions
First Rational Expression
Second Rational Expression
How to Add Rational Expressions
Overview
Adding rational expressions is similar to adding numerical fractions. You need a common denominator before you can add the numerators.
Step 1: Find the LCD
The Least Common Denominator is the smallest expression that both denominators divide into evenly.
- If denominators are the same, that's your LCD
- If different, factor each denominator completely
- LCD includes each unique factor to its highest power
Step 2: Convert Each Fraction
Multiply numerator and denominator of each fraction by what's needed to create the LCD.
Step 3: Add Numerators
With common denominators, add the numerators and keep the common denominator.
Step 4: Simplify
Combine like terms in the numerator, factor if possible, and cancel common factors.
Example
Add: 2/(x + 3) + 3/(x - 2)
- LCD = (x + 3)(x - 2)
- Convert: [2(x-2)]/[(x+3)(x-2)] + [3(x+3)]/[(x+3)(x-2)]
- Add: [2x - 4 + 3x + 9]/[(x + 3)(x - 2)]
- Simplify: [5x + 5]/[(x + 3)(x - 2)] = 5(x + 1)/[(x + 3)(x - 2)]
- Restrictions: x ≠ -3, 2
Frequently Asked Questions
What if the denominators are already the same?
Great! Simply add the numerators and keep the common denominator. Then simplify if possible.
How do I find the LCD with polynomial denominators?
Factor each denominator completely, then take each unique factor to its highest power. Multiply these together to get the LCD.
What's the difference between LCD and LCM?
They're the same concept! LCD (Least Common Denominator) is just LCM (Least Common Multiple) applied to denominators.
Can I just multiply the denominators together?
You can, but it's not always the LCD. (x)(x) = x², but the LCD of x and x is just x. Using the true LCD makes simplification easier.
What if one denominator is a factor of the other?
The larger expression is the LCD. For example, if denominators are x and x², the LCD is x².
Do I need to expand the final numerator?
Not necessarily. If the numerator factors nicely and shares factors with the denominator, leave it factored for easier cancellation.
What if my numerators have different signs?
That's fine! Add them algebraically: (+3x) + (-2x) = x. Be careful with distributing negative signs.
How is adding different from subtracting rational expressions?
The process is identical except you subtract the numerators instead of adding them. Be sure to distribute the negative sign!