Math101 · Independent Ontario learning library
Math101Dividing Rational Expressions
Dividing rational expressions uses the reciprocal of the divisor: $A/B\div C/D=A/B\cdot D/C$.
Division of algebraic fractions appears in compound rates and solving rational equations. Restriction tracking preserves equivalence and prevents hidden undefined values.
Intuition and core definition
Dividing rational expressions uses the reciprocal of the divisor: $A/B\div C/D=A/B\cdot D/C$. Every original denominator must be nonzero, and the divisor $C/D$ must itself be nonzero, so both $C$ and $D$ carry restrictions. Factoring reveals cancellable factors after the reciprocal step.
Notation, language, and conditions
A rational expression is a quotient of polynomials. Cancellation is valid only for nonzero common factors in a product. Domain restrictions are collected from the original expressions before any cancellation; a simplified formula does not redefine missing input values.
Why this idea matters
Dividing algebraic fractions combines reciprocal multiplication with a domain audit that also excludes zeros of the divisor's numerator.
A dependable method
- Factor all numerators and denominators and state original restrictions.
- Confirm the divisor is not zero.
- Change division to multiplication and invert only the divisor.
- Cancel common nonzero factors across the product.
- Multiply remaining factors and retain every original restriction.
