5. Roots of Polynomial Equations using Graphs

by M. Bourne

Background

In the previous sections in this chapter we learned how to find the roots of polynomial equations using algebra, in the sections:

1. Polynomial Functions and Equations

2. Remainder and Factor Theorems

3. How to Factor polynomials

4. Roots of a Polynomial Function

The algebraic approach is OK as long as the numbers are "nice" and we only go up to degree 3 or maybe 4 (the degree is the highest power of x appearing in the polynomial). However, in most real situations, the numbers involved in solving polynomial equations are far from "nice", and we can come across higher degree cases.

Solving Polynomial Roots Using a Graph

The important thing in this work is the concept that the x-axis intersections represent the "roots" of the equation. That is, the values of x that give us zero when subtituted into the polynomial.

In the following graph, you can zoom in and pan the graph left-right, up-down to easily find the roots. The polynomial equation represented by the graph is not "nice", but within seconds we can zoom in on the x-axis intersects, and we have our roots. That would be next to impossible using the Factor and Remainder Theorems.

Things to do

This is the graph of the polynomial p(x) = 0.9x⁴ + 0.4x³ − 6.49x² + 7.244x − 2.112.

We aim to find the "roots", which are the x-values that give us 0 when substituted. They are represented by the x-axis intersects.

1. Zoom in on the x-axis intersect near x = −3.5. The further you go in, the greater the accuracy of the root.
2. Zoom out and then investigate what's happening near x = 1. It looks like there's one root there, but is it so?

In the "real world", where the coefficients of the polynomials we work with are rarely "nice", we usually find the roots numerically (using techniques like Newton's Method, and see also the interactive Newton's Method explanation), or graphically as above.