PreCalculus 2.3

Zeros of Polynomial Functions

If f is a polynomial function, then the values of x for which f(x) is equal to 0 are called the zeros of f. These values of x are the roots, or solutions, of the polynomial equation f(x) = 0. Each real root of the polynomial equation appears as an x-intercept of the graph of the polynomial function.

Multiplicity and x-Intercepts

If r is a zero of even multiplicity, then the graph touches the x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph crosses the x-axis at r. Regardless of whether the multiplicity of a zero is even or odd, graphs tend to flatten out near zeros with multiplicity greater than one.

The Leading Coefficient Test

As x increases or decreases without bound, the graph of the polynomial function eventually rises or falls. In particular, the sign of the leading coefficient, an, and the degree, n, of the polynomial function reveal its end behavior.

Turning Points of Polynomial Functions

In general, if f is a polynomial function of degree n, then the graph of f has at most n - 1 turning points.

The Intermediate Value Theorem

Let f be a polynomial function with real coefficients. If f(a) and f(b) have opposite signs, then there is at least one value of c between a and b for which f(c) = 0. Equivalently, the equation f(x) = 0 has at least one real root between a and b.

Graphs of Polynomial Functions - Smooth and Continuous

Polynomial functions of degree 2 or higher have graphs that are smooth and continuous. By smooth, we mean that the graphs contain only rounded curves with no sharp corners. By continuous, we mean that the graphs have no breaks and can be drawn without lifting your pencil from the rectangular coordinate system.

A Strategy for Graphing Polynomial Functions

Step 1 Determine end behavior. Step 2 Find x-intercepts (zeros of the function) by setting f(x) = 0. Step 3 Find the y-intercept by computing f(0). Step 4 Use possible symmetry to help draw the graph. Step 5 Use the fact that the maximum number of turning points of the graph is n-1 to check whether it is drawn correctly.

End Behavior of Polynomial Functions

The end behavior of the graph of a function to the far left or the far right is called its end behavior. Although the graph of a polynomial function may have intervals where it increases or decreases, the graph will eventually rise or fall without bound as it moves far to the left or far to the right. The sign of the leading coefficient, an, and the degree, n, of the polynomial function reveal its end behavior.

The Leading Coefficient Test (Steps)

Use the Leading Coefficient Test to determine the end behavior of the graph of The degree of the function is 4, which is even. Even-degree functions have graphs with the same behavior at each end. The leading coefficient, 1, is positive. The graph rises to the left and to the right.