Operations on rational expressions
Operations on Rational Expressions
Operations on Rational Expressions is a topic in Rational Functions in the California Common Core State Standards. It is aligned to Standard A-APR.7, which requires students to understand that rational expressions form a system closed under addition, subtraction, multiplication, and division.
Rational expressions multiply by cancelling, divide by the reciprocal, and add over a common denominator.
Theory
Rational expressions follow the rules for fractions:
- Multiply: factor, cancel, then multiply straight across.
- Divide: multiply by the reciprocal of the second.
- Add / subtract: rewrite over a common denominator.
Division and addition:
How to operate
- Factor every numerator and denominator.
- For \(\times\), cancel then multiply; for \(\div\), flip the second.
- For \(\pm\), build a common denominator.
- Simplify and state restrictions.
Cancel the \(x\), then multiply.
| \(\dfrac{2}{x}\cdot\dfrac{x}{3}\) | \(=\) | \(\dfrac{2}{3}\) |
Use the common denominator \(x(x+1)\).
| \(\dfrac{(x+1)+x}{x(x+1)}\) | \(=\) | \(\dfrac{2x+1}{x(x+1)}\) |
Multiply by the reciprocal.
| \(\dfrac{3}{x+2}\cdot\dfrac{x-1}{6}\) | \(=\) | \(\dfrac{x-1}{2(x+2)}\) |
Same denominator — subtract the numerators.
| \(\dfrac{x-1}{x-1}\) | \(=\) | \(1,\ x\neq1\) |
Common pitfalls
Frequently asked questions
How do you multiply rational expressions?
Factor, cancel common factors, then multiply across.
How do you divide them?
Multiply by the reciprocal of the second expression.
How do you add them?
Rewrite over a common denominator, then add the numerators.
Should you simplify the answer?
Yes — factor and cancel, and state excluded values.