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# Polynomial Functions Help (page 2)

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By McGraw-Hill Professional
Updated on Sep 6, 2011

## Finding the x-Intercepts of a Polynomial Function

Finding the x -intercepts (if any) for the graph of a polynomial function is very important. The x -intercept of any graph is where the graph touches the x -axis. This happens when the y -coordinate of the point is 0. We found the x -intercepts for some quadratic functions by factoring and setting each factor equal to zero. This is how we will find the x -intercepts for polynomial functions. It is not always easy to do. In fact, some polynomials are so hard to factor that the best we can do is approximate the x -intercepts (using graphing calculators or calculus). This will not be the case for the polynomials in this book, however. Every polynomial here will be factorable using techniques covered later.

### Zeros of a Polynomial

Because an x -intercept for f(x) = a n x n + a n −1 x n −1 + ... + a 1 x + a 0 is a solution to the equation 0 = a n x n + a n −1 x n −1 + ... + a 1 x + a 0 , x -intercepts are also called zeros of the polynomial. All of the following statements have the same meaning for a polynomial. Let c be a real number, and let P(x) be a polynomial function.

1. c is an x -intercept of the graph of P (x).
2. c is a zero for P(x) .
3. xc is a factor of P(x) .

#### Examples

• x − 1 is a factor means that 1 is an x -intercept and a zero.
• x + 5 is a factor means that −5 is an x -intercept and a zero.
• x is a factor means that 0 is an x -intercept and a zero.
• 3 is a zero means that x − 3 is a factor and 3 is an x -intercept.

We can find the zeros of a function (or at least the approximate zeros) by looking at its graph.

The x -intercepts of the graph in Figure 7.5 are 2 and −2. Now we know that x − 2 and x + 2 (which is x − (−2)) are factors of the polynomial.

The graph of the polynomial function in Figure 7.6 has x -intercepts of −1, 1, and 2. This means that x − 1, x − 2, and x + 1 (as x − (−1)) are factors of the polynomial.

Fig. 7.5

Fig. 7.6

Fig. 7.7

The x -intercepts for the graph in Figure 7.7 are −3, 0, and 2, making x + 3, x (as x − 0), and x − 2 factors of the polynomial.

## Matching a Graph to its Polynomial Function Examples

Now that we know about the end behavior of the graphs of polynomial functions and the relationship between x -intercepts and factors, we can look at a polynomial and have a pretty good idea of what its graph looks like. In the next set of examples, we will match the graphs from the previous section with their polynomial functions.

#### Examples

Match the functions with the graphs in Figures 7.5-7.7.

Fig. 7.5

Fig. 7.6

Fig. 7.7

• Because f(x) is a polynomial whose degree is even and whose leading coefficient is positive, we will look for a graph that goes up on the left and up on the right. Because the factors are x 2 , x + 3, and x − 2, we will also look for a graph with x -intercepts 0, −3, and 2. The graph in Figure 7.7 satisfies these conditions.

Fig. 7.7
• Because g(x) is a polynomial whose degree is odd and whose leading coefficient is negative, we will look for a graph that goes up on the left and down on the right. The factors are x − 1, x − 2, and x + 1, we will also look for a graph with 1, 2, and −1 as x -intercepts. The graph in Figure 7.6 satisfies these conditions.

Fig. 7.6

• Because P(x) is a polynomial whose degree is odd and whose leading term is positive, we will look for a graph that goes down on the left and up on the right. The x -intercepts are 2 and −2. The graph in Figure 7.5 satisfies these conditions.

Fig. 7.5

Practice problems for this concept can be found at: Polynomial Functions Practice Test.

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