Dividing
Part of the Polynomials & Rational Expressions section of Coddy's Algebra 2 journey. Lesson 30 of 54.
Dividing by a fraction is multiplying by its reciprocal, the fraction turned upside down: (2/3)/(4/9) is (2/3) × (9/4), which is 3/2. The reason: a fraction times its reciprocal is 1, so multiplying by 9/4 undoes multiplying by 4/9, which is exactly what dividing by it does.
Rational expressions follow the same rule. In ((x + 1)/(x + 5))/((x + 1)/(x^2 - 25)), keep the first fraction and flip the second: ((x + 1)/(x + 5)) × ((x^2 - 25)/(x + 1)). Now it is a multiplication. Factor x^2 - 25 as (x + 5)(x - 5), multiply across, and cancel x + 1 and x + 5. What is left is x - 5.
Only the second fraction flips. The excluded values grow here: 5 and -5 from the bottoms, and also -1, because at -1 the fraction you divide by is 0, and nothing can be divided by 0.
Challenge
Divide and simplify ((x + 3)/(x - 2))/((x + 3)/(x^2 - 4)).
Try it yourself
Write it as one fraction
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Polynomials & Rational Expressions
1Polynomial Long Division
How Long Division WorksOne Term at a TimeA Remainder Left OverA Missing TermDividing by a QuadraticRecap: Long Division7Adding and Subtracting
Same BottomDifferent BottomsSubtractingFactor the Bottoms FirstRecap: Adding Fractions2Remainders and Factors
The Remainder TheoremThe Factor TheoremFinding a Missing NumberUsing a Known FactorRecap: Remainders and Factors5Sign Charts
Between the RootsOutside the RootsA Negative Leading TermReading a Sign ChartRecap: Sign Charts3Factoring Higher Degree
Factor by GroupingGrouping All the WayDifference of CubesSum of CubesA Quadratic in DisguiseRecap: Factoring6Rational Expressions
Cancelling FactorsExcluded ValuesMultiplyingDividingRecap: Rational Expressions