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All courses Math III · A-APR.7 7 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 (x²-1)/(x+2) · (x+2)/(x-1). Enter the fully factored product, whole canceled factors, simplified product, original denominator zeros, and all exclusions.
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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

Factor the numerator

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

Rewrite the difference of squares so the common factor is visible.

02

Multiply the factored forms

\[\frac{(x-1)(x+1)}{x+2}~\cdot~\frac{x+2}{x-1}\]

Now the product shows matching factors in numerators and denominators.

03

Cancel common factors

\[x+1\]

Cancel \(x+2\) and \(x-1\) because they appear as common factors.

04

State the restrictions

\[x\ne~-2,\ 1\]

The original denominators were \(x+2\) and \(x-1\), so those values stay excluded.

05

State the final answer

\[\text{factored}~\text{product}=[(x-1)(x+1)/(x+2)]·[(x+2)/(x-1)];~\text{canceled}~\text{factors}=x+2,x-1;~\text{simplified}~\text{product}=x+1;~\text{original}~\text{denominator}~\text{zeros}=-2,1;~\text{exclusions}=x≠-2,1\]

This combines the simplified expression with the two restrictions from the original denominators.

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Another way

  1. You can multiply straight across first, then factor the resulting numerator and denominator and cancel common factors.

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Common mistake

A common mistake is to cancel before factoring \(x^2-1\), or to forget that \(x=-2\) and \(x=1\) are still excluded even after simplification.