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# Polynomial Functions Help

based on 3 ratings
By McGraw-Hill Professional
Updated on Sep 6, 2011

## Introduction to Polynomial Functions

A polynomial function is a function in the form f(x) = a n x n + a n −1 x n −1 + ... + a 1 x + a 0 , where each a i is a real number and the powers on x are whole numbers. There is no x under a root sign and no x in a denominator. The number a i is called a coefficient . For example, in the polynomial function f(x) = −2 x 3 + 5 x 2 − 4 x + 8, the coefficients are −2, 5, −4, and 8. The constant term (the term with no variable) is 8. The powers on x are 3, 2, and 1. The degree of the polynomial (and polynomial function) is the highest power on x . In this example, the degree is 3. Quadratic functions are degree 2. Linear functions of the form f(x) = mx + b (if m ≠ 0) are degree 1. Constant functions of the form f(x) = b are degree 0 (this is because x 0 = 1, making f(x) = bx 0 ).

### Leading Term and Leading Coefficient - Determining the Behavior of a Graph

The leading term of a polynomial (and polynomial function) is the term having x to the highest power. Usually, but not always, the leading term is written first. The leading coefficient is the coefficient on the leading term. In our example, the leading term is −2 x 3 , and the leading coefficient is −2. By looking at the leading term only, we can tell roughly what the graph looks like. The graph of any polynomial will either go up on both ends, go down on both ends, or go up on one end and down on the other. This is called the end behavior of the graph. The figures below illustrate the end behavior of polynomial functions. The shape of the dashed part of the graph depends on the individual function.

Fig. 7.1

This graph goes up on both ends.

Fig. 7.2

This graph goes down on both ends.

Fig. 7.3

This graph goes down on the left and up on the right.

Fig. 7.4

This graph goes up on the left and down on the right.

If the degree of the polynomial is an even number, the graph will look like the graph in Figure 7.1 or in Figure 7.2. If the leading coefficient is a positive number, the graph will look like the graph in Figure 7.1. If the leading coefficient is a negative number, the graph will look like the graph in Figure 7.2. If the degree of the polynomial is an odd number, the graph will look like the one in Figure 7.3 or in Figure 7.4. If the leading coefficient is a positive number, the graph will look like the graph in Figure 7.3. If the leading coefficient is a negative number, the graph will look like the graph in Figure 7.4.

How can one term in a polynomial function give us this information? For polynomial functions, the leading term dominates all of the other terms. For x -values large enough (both large positive numbers and large negative numbers), the other terms don’t contribute much to the size of the y -values.

#### Examples

Match the graph of the given function with one of the graphs in Figures 7.1-7.4.

• f(x) = 4 x 5 + 6 x 3 −2 x 2 + 8 x + 11
• We only need to look at the leading term, 4 x 5 . The degree, 5, is odd, and the leading coefficient, 4, is positive. The graph of this function looks like the one in Figure 7.3.

• P(x ) = 5 + 2 x −6 x 2
• The leading term is −6 x 2 . The degree, 2, is even, and the leading coefficient, −6, is negative. The graph of this function looks like the one in Figure 7.2.

• h(x) = −2 x 3 + 4 x 2 −7 x + 9
• The leading term is −2 x 3 . The degree, 3, is odd, and the leading coefficient, −2, is negative. The graph of this function looks like the one in Figure 7.4.

• g(x) = x 4 + 4 x 3 − 8 x 2 + 3 x − 5
• The leading term is x 4 . The degree, 4, is even, and the leading coefficient, 1, is positive. The graph of this function looks like the one in Figure 7.1 .

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