Approximating the Real Zeros of a Polynomial

Example

Approximate the real zeros of $f\left(x\right)=x^4-6x^{^3}+8x^2+2x-1$

Solution

The dominating term is $x^4$, so the graph will have end behavior similar to the graph of $f\left(x\right)=x^4$, which is positive for all values of x with large absolute values. That is, the end behavior is up at the left and at the right, 

There are at most four real zeros, since the polynomial is fourth-degree.

Since f(0) = −1, the y-intercept is −1. Because the end behavior is positive on the left and the right, by the intermediate value theorem f has at least one zero on either side of x = 0. To approximate the zeros, we use a graphing calculator.

The graph below shows that there are four real zeros, and the table indicates that they are between −1 and 0, 0 and 1, 2 and 3, and 3 and 4 because there is a sign change in f(x) in each case.

Using the capability of the calculator, we can find the zeros to a great degree of accuracy. The graph shown here shows that the negative zero is approximately −0.4142136. Similarly, we find that the other three zeros are approximately 0.26794919, 2.4142136, and 3.7320508.

Please complete Homework 3.4 on the next page.