Question 3.4.7: Approximating Real Zeros of a Polynomial Function Approximat...
Approximating Real Zeros of a Polynomial Function
Approximate the real zeros of ƒ(x) = x^{4} – 6x³ + 8x² + 2x – 1.
The dominating term is x^{4}, so the graph will have end behavior similar to the graph of ƒ(x) = 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 the right,⮬⮭ . There are at most four real zeros because the polynomial is fourth-degree.
Since ƒ(0) = -1, the y-intercept is (0, -1). Because the end behavior is
positive on the left and the right, by the intermediate value theorem ƒ has at least one real zero on either side of x = 0. To approximate the zeros, we use a graphing calculator. The graph in Figure 36 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 ƒ(x) = y_{1} in each case.
Using a calculator, we can find zeros to a great degree of accuracy.
Figure 37 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.
