Find the domain of each function. f(x) = (2x+7)/(x³ - 5x² - 4x+20)

Verified step by step guidance

1. Step 1: Recall that the domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, the function is undefined when the denominator equals zero. Therefore, we need to find the values of x that make the denominator equal to zero.
2. Step 2: Write the denominator of the function: \( x^3 - 5x^2 - 4x + 20 \). Set it equal to zero to find the values of x that make the denominator undefined: \( x^3 - 5x^2 - 4x + 20 = 0 \).
3. Step 3: Factor the cubic polynomial \( x^3 - 5x^2 - 4x + 20 \). Start by using the Rational Root Theorem to test possible rational roots (e.g., factors of the constant term 20 divided by factors of the leading coefficient 1). Once a root is found, use synthetic division or long division to factor the polynomial further.
4. Step 4: After factoring the cubic polynomial completely, you will have a product of linear and/or quadratic factors. Solve each factor for x to find the roots of the denominator. These roots are the x-values that make the denominator zero.
5. Step 5: Exclude the x-values found in Step 4 from the domain of the function. The domain of \( f(x) \) is all real numbers except the values of x that make the denominator zero.

Key Concepts

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.

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 may require factoring or using numerical methods.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. Understanding the behavior of rational functions, particularly their discontinuities and asymptotes, is crucial for analyzing their domains and overall characteristics.