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All courses Algebra II · A-APR.7 14 of 55
Add, subtract, multiply, and divide rational expressions as a closed system like rational numbers.

Multiply rational expressions and preserve original-domain restrictions

Problem
Multiply and simplify \((x^2~-~1)/(x~+~2)\) times \((x~+~2)/(x~-~1)\). State all restrictions from the original expression.
Your answer
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Hint

Factor every polynomial and list every zero of each original denominator.

Record original denominator zeros before factoring and cancellation. Cancel only identical factors across the overall numerator and denominator.

Solution walkthrough

01

Record original restrictions

\[x+2\ne~0~\text{and}~x-1\ne~0~->~x\ne~-2,1\]

Restrictions come from the original denominators, so x=-2 and x=1 must be excluded before any cancellation.

02

Factor before multiplying

\[x^2-1=(x-1)(x+1)\]

The first numerator is a difference of squares, revealing the factor x-1 that can cancel.

03

Cancel whole common factors

\[[(x-1)(x+1)/(x+2)]*[(x+2)/(x-1)]=x+1\]

The factors x+2 and x-1 each occur once in the overall numerator and denominator, so they cancel for allowed inputs.

04

State the simplified product and domain

\[\text{Factor}~x^2-1=(x-1)(x+1),~\text{cancel}~x+2~\text{and}~x-1,~\text{and}~\text{get}~x+1,~\text{with}~x\ne~-2,1.\]

The product simplifies to x+1, but both zeros of the original denominators remain excluded. The numerator zero x=-1 is allowed and is not an extra restriction, so this is choice A.

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Another way

  1. Multiply straight across first, then factor the combined numerator and denominator and cancel the same two factors while retaining the original restrictions.

!

Common mistake

Do not discard x=-2 and x=1 after cancellation or exclude x=-1 just because it is a numerator zero. Only the original denominator zeros restrict this product.

Solution walkthrough video