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Richa Nandra, Lovely Professional University Unit 25: Roots of a Polynomial

Unit 25: Roots of a Polynomial Notes

CONTENTS
Objectives

Introduction
25.1 Roots of Polynomials
25.2 Summary
25.3 Keywords
25.4 Review Questions

25.5 Further Readings

Objectives

After studying this unit, you will be able to:
Define roots of polynomials

Discuss examples of roots of polynomial

Introduction

You have seen when we can say that an element in a ring divides another element. Let us recall
the definition in the context of F[x], where F is a field.

25.1 Roots of Polynomials

Definition: Let f(x) and g(x) be in F[x], where F is a field and g(x)  0. We say that g(x) divides
f(x)(or g(x) is a factor of f(x), or f(x) is divisible by gi(x)) if there-exists q(x)  F[x] such that
f(x) = q(x) g(x).

We write g(x) | f(x) for ‘g(x) divides f(x)’, and g(x) | f(x) for ‘g(x) does not divide f(x)’.
Now, if f(x)  F[x] and g(x)  F[x], where g(x)  0, when g(x) | f(x)? We find that g(x) | f(x) if
r(x) = 0.
Definition: Let F be a field and f(x)  F[x]. We say that an element a  F is a root (or zero) of f(x)
if f(n) = 0.
For example, 1 is a root of x – 1  R[x], since 1 – 1 = 0.
2
2
1 1
2
3
Similarly, –1 is a root of f(x) = x  x  x   Q[x], since
2 2
1 1
f(–1) = –1 +1    0.
2 2

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