Find a polynomial function f of least degree having the graph shown. (Hint: See the NOTE following Example 4.)

Verified step by step guidance

1. Identify the x-intercepts of the graph. These are the points where the graph crosses the x-axis. From the graph, the x-intercepts are approximately at x = -5, x = 1, and x = 4.
2. Determine the multiplicity of each root. Since the graph crosses the x-axis at each intercept, each root has a multiplicity of 1.
3. Write the polynomial function in factored form using the roots and their multiplicities: f(x) = a(x + 5)(x - 1)(x - 4).
4. Use the given point (0, 20) to find the value of the leading coefficient 'a'. Substitute x = 0 and f(x) = 20 into the factored form: 20 = a(0 + 5)(0 - 1)(0 - 4).
5. Solve for 'a' by simplifying the equation: 20 = a(-20).

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where n is a non-negative integer. Understanding the degree of the polynomial is crucial, as it determines the function's behavior and the number of roots it can have.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It influences the shape and number of turning points in the graph. For example, a polynomial of degree n can have up to n real roots and n-1 turning points, which is essential for analyzing the graph's behavior and finding the least degree polynomial that fits a given graph.

Graphing Polynomial Functions

Graphing polynomial functions involves plotting points based on the function's values and understanding its key features, such as intercepts, turning points, and end behavior. The graph's shape can provide insights into the polynomial's degree and the nature of its roots. In this case, the graph shows a point at (0, 20), indicating that the polynomial must pass through this point, which is critical for determining its equation.