In many applications of mathematics, using analytical methods to find the exact zeros of a polynomial is either not possible, or too tedious to employ. One method, called the “bisection method,” is used to approximate the zeros of a polynomial based on the intermediate value theorem. Using the outlined steps (a) to (e) below, find zero of the polynomial p(x)=x^3-4x^2-2x-3. Use a calculator or computer to work on the following. Consider the interval [4, 5]. Evaluate p(4) and p(5). Round to four decimal places if necessary. What do you notice about the signs of p(4) and p(5)? Does p(x) have a real zero on the interval [4, 5]? From part (b), we know that p(x) has at least one zero somewhere between 4 and 5. As such, we can take the midpoint 4.5 as an approximation of a zero on that interval. To get a better approximation, we can repeat this process using a smaller interval. To find a smaller interval, first evaluate p(4.5). Round to four decimal places if necessary. Based on the signs of p(4), p(5), and p(4.5), which subinterval, [4, 4.5] or [4.5, 5], is guaranteed to have a zero of p(x)? What is the midpoint of this interval? Thus, what is the new approximation of a zero on this interval? Repeat steps (a) – (d) by filling in the table. In each row, the midpoint of the interval [a, b] is the new approximation of a zero of p(x). Repeat this process until two consecutive values of the midpoint differ by less than 0.1. Interval [a, b] P(a) P(b) midpoint (approximation of zero) P(midpoint) New Interval [4, 5] -11 12 4.5 -1.875 [4.5, 5] [4.5, 5]

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

In many applications of mathematics, using analytical methods to find the exact zeros of a polynomial is either not possible, or too tedious to employ. One method, called the “bisection method,” is used to approximate the zeros of a polynomial based on the intermediate value theorem. Using the outlined steps (a) to (e) below, find zero of the polynomial p(x)=x^3-4x^2-2x-3. Use a calculator or computer to work on the following.

Consider the interval [4, 5]. Evaluate p(4) and p(5). Round to four decimal places if necessary.

What do you notice about the signs of p(4) and p(5)? Does p(x) have a real zero on the interval [4, 5]?

From part (b), we know that p(x) has at least one zero somewhere between 4 and 5. As such, we can take the midpoint 4.5 as an approximation of a zero on that interval. To get a better approximation, we can repeat this process using a smaller interval. To find a smaller interval, first evaluate p(4.5). Round to four decimal places if necessary.

Based on the signs of p(4), p(5), and p(4.5), which subinterval, [4, 4.5] or [4.5, 5], is guaranteed to have a zero of p(x)? What is the midpoint of this interval? Thus, what is the new approximation of a zero on this interval?

Repeat steps (a) – (d) by filling in the table. In each row, the midpoint of the interval [a, b] is the new approximation of a zero of p(x). Repeat this process until two consecutive values of the midpoint differ by less than 0.1.