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Restrictions for Rational Expressions Solutions

• A    rational expression    is a ratio between two polynomial expressions in the form $\frac{p}{q}$ where p and q are polynomials and    q ≠ 0   .
• Notice that q cannot equal zero. A denominator of zero makes the expression undefined. This is called a    restriction    or excluded value.
• Therefore, any values that result in a rational expression with a denominator of zero are    excluded values    or restricted from the expression.
• To find the restrictions for a rational expression:

Note

Finding the restrictions for rational expressions are also referred to as: determining undefined values, naming values that will make the expression undefined, naming the excluded values.

• Use the    not-equal-to ≠    symbol to name restrictions on the denominator.

Example 1

Determine the restrictions for the denominator.
$\frac{20x^2+13}{3x^2–12}$

Plan
Factor the denominator

Set each expression in the denominator equal to zero

Solve for the restrictions of the denominator

$\begin{array}{ccc}x–2=0& & x+2=0\\ x=2& & x=–2\end{array}$

$x\ne \pm 2$

Note

Recall that the restrictions are any values that will make the denominator undefined. This means the value of the denominator is zero.


 

See Lesson 3 More to Explore for ways to use technology to check answers and determine restrictions.

Example 2

State all numbers that make the given expression undefined.
$\frac{11}{3x^3–19x^2+6x}$