Skill: Determine restrictions on rational expressions

Explanation and Free Practice Resources

A rational expression is defined only for values that make every denominator nonzero; restrictions are found by solving the denominator equation(s), often by factoring polynomial denominators and identifying excluded real values. These restrictions belong to the original expression and remain excluded even when a common factor is canceled during simplification, preserving the expression’s domain and supporting accurate work with rational equations and functions.

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Detailed Explanation: Determine restrictions on rational expressions

A rational expression is defined only when its denominator is not zero. To find the restrictions:

Factor the denominator.
Set each denominator factor equal to zero.
Solve for the excluded values.
Keep these restrictions even if a common factor is canceled later.

Example

Determine the restrictions and simplify:

\frac{x^2-9}{x^2-3x}

Step 1: Factor the denominator.

x^2-3x=x(x-3)

The denominator is zero when either factor is zero:

x=0 \qquad \text{or} \qquad x-3=0

So the restrictions are

x\ne 0 \quad \text{and} \quad x\ne 3.

Step 2: Factor the numerator.

x^2-9=(x-3)(x+3)

Now rewrite the expression:

\frac{(x-3)(x+3)}{x(x-3)}

Step 3: Cancel the common factor.

\frac{(x-3)(x+3)}{x(x-3)} = \frac{x+3}{x}

The simplified expression is

\frac{x+3}{x}, \qquad x\ne 0,\ 3.

Even though the factor $(x-3)$ was canceled, $x=3$ is still excluded because it made the original denominator zero.

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