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David Terr
Ph.D. Math, UC Berkeley

>> 2.5. Dividing Polynomials

2.4. Polynomial Functions and Their Graphs

In the previous section we explained how to graph quadratic functions. In this section we explain how to graph general polynomial functions. As the degrees of polynomials grow, their graphs in general become more and more complicated. A polynomial of degree n can have up to n real zeros, that is, its graph can cross the x-axis up to n times. Finding these zeros is difficult in general; we will defer a general discussion of finding zeros of polynomial functions to the section 2.6. Nevertheless, it is easy to find the zeros of certain polynomials and in any case, by plotting enough points, one can render reasonably accurate graphs of general polynomials. We present several examples below.

• Example 1: Graph the polynomial f(x) = x3 - x.
• Solution: It is easy to factor f(x). First note that f(x) = x(x2 - 1). We also note that by Equation P.7.1 with a = 1, the factor x2 - 1 factors as (x + 1)(x - 1). Thus we have f(x) = x(x + 1)(x - 1), so the zeros of f are at -1, 0, and 1. We also compute the values of f(x) for several nearby points, obtaining the following table. (Values listed are exact.)
x f(x)
-1.4
-1.344
-1.2
-0.528
-1.0
0.000
-0.8
0.288
-0.6
0.384
-0.4
0.336
-0.2
0.192
0.0
0.000
0.2
-0.192
0.4
-0.336
0.6
-0.384
0.8
-0.192
1.0
0.000
1.2
0.528
1.4
1.344

Next we plot these points and join them with a smooth curve, obtaining the following graph. Note that this graph is symmetric about the origin since f is odd.

• Example 2: Graph the polynomial f(x) = x4 - 10x2 + 9.
• Solution: Once again, this function is easy to factor. We have f(x) = (x2 - 1)(x2 - 9) = (x + 1)(x - 1)(x + 3)(x - 3). Thus, f has zeros at ±1 and ±3. The following table lists values of f(x) for x covering this range of zeros. Values of f(x) are rounded to the nearest tenth.
x f(x)
-3.5
42.8
-3.0
0.0
-2.5
-14.4
-2.0
-15.0
-1.5
-8.4
-1.0
0.0
-0.5
6.6
0.0
9.0
0.5
6.6
1.0
0.0
1.5
-8.4
2.0
-15.0
2.5
-14.4
3.0
0.0
3.5
42.8

Here is a graph of f(x). Note that it is symmetric about the y-axis since f is even.

• Example 3: Graph the polynomial f(x) = x3 + x2 + 2x + 4.
• Solution: This polynomial is difficult to factor, so we will not try to do so. Instead, we compute f(x) for some small values of the argument.
x f(x)
-3
-20
-2
-4
-1
2
0
4
1
8
2
20

We obtain the following graph. Note that f(x) is neither even nor odd and that it has just one real zero.

>> 2.5. Dividing Polynomials