Subtract Rational Expressions with a Common Denominator
To subtract rational expressions with a common denominator, keep the denominator unchanged and subtract the numerators, making sure to distribute the minus sign to every term in the second numerator. For example, (5x − 3)/(x + 2) − (2x + 1)/(x + 2) = (5x − 3 − 2x − 1)/(x + 2) = (3x − 4)/(x + 2). This skill is foundational in Chapter 8 of OpenStax Elementary Algebra 2E, where students build toward adding and subtracting rational expressions with unlike denominators. Missing the sign distribution step is the single most common source of errors in rational expression arithmetic.
Key Concepts
Property If $p$, $q$, and $r$ are polynomials where $r \neq 0$, then $$ \frac{p}{r} \frac{q}{r} = \frac{p q}{r} $$ To subtract rational expressions, subtract the numerators and place the difference over the common denominator. Be very careful to distribute the negative sign to every term in the second numerator.
Examples Subtract: $\frac{y^2}{y 4} \frac{16}{y 4}$. This equals $\frac{y^2 16}{y 4}$. Factoring the numerator as a difference of squares gives $\frac{(y 4)(y+4)}{y 4}$, which simplifies to $y+4$.
Subtract: $\frac{x^2}{x+5} \frac{3x+10}{x+5}$. This equals $\frac{x^2 (3x+10)}{x+5} = \frac{x^2 3x 10}{x+5}$. Factoring the numerator gives $\frac{(x 5)(x+2)}{x+5}$. This cannot be simplified further.
Common Questions
How do you subtract rational expressions with a common denominator?
Keep the denominator and subtract the numerators. Write the result as (numerator 1 − numerator 2) over the common denominator. Be sure to distribute the minus sign to every term in the second numerator before combining like terms.
Why do I have to distribute the minus sign when subtracting rational expressions?
Because the minus sign applies to the entire second numerator, not just its first term. Forgetting this turns (5x − 3) − (2x + 1) into 5x − 3 − 2x + 1 incorrectly; the correct result is 5x − 3 − 2x − 1.
What is a rational expression?
A rational expression is a fraction whose numerator and denominator are polynomials. Examples include (3x + 1)/(x − 2) and x²/(x + 5).
What should I do after subtracting rational expressions?
After combining numerators, always factor the result and cancel any common factors with the denominator to write the expression in simplest form.
When do students learn to subtract rational expressions?
This topic is covered in algebra 1 and early algebra 2, appearing in OpenStax Elementary Algebra 2E Chapter 8.
What is the most common mistake when subtracting rational expressions?
Not distributing the subtraction sign across the entire second numerator. Always put parentheses around the second numerator before subtracting.
How does subtracting rational expressions differ from adding them?
Addition simply combines numerators. Subtraction requires flipping the sign of every term in the second numerator, making careful parentheses essential.