Subtract Rational Expressions Calculator
Subtract rational expressions by finding LCD and properly distributing negative signs
First Rational Expression (Minuend)
Second Rational Expression (Subtrahend)
How to Subtract Rational Expressions
Overview
Subtracting rational expressions follows the same process as addition, with one critical difference: you must distribute the negative sign to all terms in the second numerator.
Step-by-Step Process
- Find the LCD of the denominators
- Convert each fraction to have the LCD
- Subtract the numerators
- CRITICAL: Distribute the negative sign to ALL terms in the second numerator
- Combine like terms
- Factor and simplify if possible
- State domain restrictions
Common Mistake Warning!
WRONG: (3x + 5) - (2x + 1) = 3x + 5 - 2x + 1 = x + 6
RIGHT: (3x + 5) - (2x + 1) = 3x + 5 - 2x - 1 = x + 4
Always distribute the negative to every term in parentheses!
Example
Subtract: 5/(x - 1) - 2/(x + 1)
- LCD = (x - 1)(x + 1)
- Convert: [5(x+1)]/[(x-1)(x+1)] - [2(x-1)]/[(x-1)(x+1)]
- Expand: [5x + 5]/[...] - [2x - 2]/[...]
- Subtract: [5x + 5 - (2x - 2)]/[(x - 1)(x + 1)]
- Distribute: [5x + 5 - 2x + 2]/[(x - 1)(x + 1)]
- Simplify: [3x + 7]/[(x - 1)(x + 1)]
- Restrictions: x ≠ 1, -1
Frequently Asked Questions
What's the most common mistake when subtracting rational expressions?
Forgetting to distribute the negative sign to all terms in the second numerator. When you see -(2x + 3), it must become -2x - 3, not -2x + 3.
Is subtraction the same as addition of a negative?
Yes! a/b - c/d is the same as a/b + (-c)/d. You can convert subtraction to addition of the opposite if that's easier for you.
How is subtracting different from adding rational expressions?
The only difference is distributing the negative sign. All other steps (finding LCD, converting, simplifying) are identical.
Do I subtract denominators too?
No! Never subtract denominators. You need a common denominator, then only subtract the numerators.
What if the second numerator has multiple terms?
Use parentheses and distribute the negative to every term: (a + b) - (c + d + e) = a + b - c - d - e
Can the answer have a negative numerator?
Yes! Negative numerators are perfectly valid. You can factor out -1 if you prefer: -x + 3 = -(x - 3)
What if I get zero in the numerator?
If the numerator simplifies to zero, the entire rational expression equals zero (as long as the denominator isn't zero).
Should I use parentheses when subtracting?
Yes! Always put parentheses around the numerator you're subtracting: (num1) - (num2). This helps prevent sign errors.