Consider the functions f(x) = x - 1 and g(x) = (x - 1)(x + 2). Note that g(x) can also be written as . f(x) and g(x) are two examples of polynomial functions.

The general equation for a polynomial function is shown to the right.

In general, a polynomial is an expression that can be written as the sum of power functions whose exponents are nonnegative integers.

The largest exponent in a polynomial is called the degree of the polynomial.

Use the sliders to change the values of p, q, and r. You may check the box under each slider to add or remove that factor from the equation.

So is a polynomial of degree 2 and f(x) is a polynomial of degree 1.

Now consider the function h(x) = (x - 1)(x + 2)(x + 1). We can rewrite the equation by distributing. The largest exponent is 3, so h(x) is a third degree polynomial.

Look at the graph of h(x). Notice that h(x) has three x-intercepts.

How is the number of x-intercepts related to the degree of a polynomial? Let's study the graphs of different polynomial functions.

How does each factor affect the graph? How are the x-intercepts of the polynomial related to the values of p, q, and r? What happens to the graph when two of the factors are the same? What happens when all three factors are the same? Click "Done Exploring" when finished.

The factors of a polynomial function determine the x-intercepts of the graph. If a polynomial has a factor of (x - p), then its graph has an x-intercept at (p, 0).

The number of x-intercepts is equal to the number of distinct factors of the form (x - p).

For a polynomial of degree 3, there could be 1, 2, or 3 x-intercepts. So the number of times the graph of a polynomial function crosses the x-axis is at most equal to the degree of the polynomial.

Consider the function . Let's compare this function to for large values of x. If x is large enough, the value of is much larger than the value of the other terms. So, if x is large enough, the most important contribution to the value of a polynomial is made by the leading term.

For example, if x = 100, f(x) = 1,019,898 and h(x) = 1,000,000. So, if x is large enough, the most important contribution to the value of a polynomial is made by the leading term.

When viewed on a large scale, the graph of f(x) looks like the graph of h(x).

In general, when viewed on a large enough scale, the graph of a polynomial looks like the graph of the power function of the same degree. This behavior is called the long-run behavior of the polynomial.

You have seen how the factors of a polynomial function affect its graph. Now, let's look at the effects of multiplying a polynomial by a constant.

Use the slider to change the value of k in the equation graphed to the right. Notice how changing this constant factor affects the graph.

Think about the following questions as you explore: Does the value of k affect the location of the x-intercepts? Does it affect the number of x-intercepts? Does it affect the y-intercept? What happens when k is negative? Click "Done Exploring" when finished.

Multiplying the equation of a polynomial by a constant causes its graph to be stretched vertically. The x-intercepts remain fixed.

Match each graph to the equation that describes it. Click "Submit" when finished.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education