# 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

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*) = 0.9*x*^{4} + 0.4*x*^{3} − 6.49*x*^{2} + 7.244*x* − 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.

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

*p*(*x*) = 0.9*x*^{4} + 0.4*x*^{3} − 6.49*x*^{2} + 7.244*x* − 2.112 = 0

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.

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