Find the domain of each rational expression. 3 / x2 - 5x - 6
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1
Identify the rational expression: .
Recognize that the domain of a rational expression is all real numbers except where the denominator is zero.
Set the denominator equal to zero to find the values that are not in the domain: .
Factor the quadratic equation: .
Solve for to find the values that make the denominator zero: and . These values are excluded from the domain.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial because their properties, such as simplification and finding the domain, depend on the behavior of these polynomials. In this case, the expression is 3 divided by a polynomial in the denominator.
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational expressions, the domain is restricted by values that make the denominator zero, as division by zero is undefined. Identifying these restrictions is essential for determining the valid inputs for the expression.
Factoring polynomials involves breaking down a polynomial into simpler components (factors) that, when multiplied together, yield the original polynomial. This process is vital for finding the roots of the denominator in a rational expression, which helps identify the values that must be excluded from the domain due to making the denominator zero.