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Determining the Values for Which a Rational Expression is Undefined
Before performing any operation with a rational expression, it is essential to identify the variable values that would make the denominator equal zero — because a zero denominator makes the expression undefined. While the numerator of a rational expression is allowed to be zero, the denominator must never be zero. Identifying these restricted values ahead of time is especially important when solving rational equations, since it reveals whether the algebraic solutions obtained are valid or must be rejected.
To find the values for which a rational expression is undefined, follow a two-step procedure:
- Set the denominator equal to zero. Write an equation in which the denominator polynomial is set equal to .
- Solve the equation in the set of real numbers, if possible. Use appropriate techniques — such as isolating the variable, factoring, or applying the Zero Product Property — to find the value(s) of the variable that satisfy the equation.
Each solution to that equation is a value at which the original rational expression is undefined.
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