Here are the essential concepts you must grasp in order to answer the question correctly.
Domain of a Function
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is typically all real numbers except where the denominator equals zero, as division by zero is undefined.
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Finding Zeros of a Polynomial
To determine the domain of the function f(x) = (2x+7)/(x^3 - 5x^2 - 4x+20), it is essential to find the values of x that make the denominator zero. This involves solving the polynomial equation x^3 - 5x^2 - 4x + 20 = 0, which can be done using methods such as factoring, synthetic division, or the Rational Root Theorem.
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Exclusion of Undefined Points
Once the zeros of the denominator are identified, these values must be excluded from the domain of the function. The domain will then consist of all real numbers except for these specific x-values, ensuring that the function remains defined and avoids division by zero.
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