2.3 Zeroes of a Polynomial

Description

Quick Overview

This section introduces the concept of zeroes of a polynomial, explaining how to find them and their significance.

Standard

The section explores the definition of zeroes of a polynomial, how to evaluate polynomial expressions at specific points, and identifies the conditions under which these points become zeroes. Examples illustrate the process of finding and verifying zeroes in different polynomial functions.

Detailed

In this section, we define the zero of a polynomial p(x) as a value 'c' such that p(c) = 0. The section begins by evaluating a polynomial at various points, demonstrating how to compute p(x) for specific values to find its zeroes. Key examples, such as determining whether specific numbers are zeroes of given polynomials, illustrate the concept clearly. Furthermore, it discusses the unique properties of linear polynomials and their zeroes, emphasizing that every linear polynomial has exactly one zero, while non-zero constant polynomials have none. The zero polynomial, by convention, has all real numbers as zeroes. The section concludes with several exercises designed to reinforce understanding of finding and verifying zeroes of polynomials.

Example:

Check whether −1 and 3 are zeros of the polynomial x2−2x−3.

Solution: Let p(x)=x2−2x−3.

Then
p(−1)=(−1)2−2(−1)−3=1+2−3=0
p(3)=(3)2−2(3)−3=9−6−3=0

Therefore, −1 is a zero of the polynomial x2−2x−3, and 3 is also a zero.

Key Concepts

  • Zero of a Polynomial: A zero is a number which, when substituted into the polynomial, yields zero.

  • Linear Polynomial: These polynomials have exactly one zero, which can be found by solving the linear equation.

  • Constant Polynomial: These do not have zeroes unless they are the zero polynomial itself.

  • Verification of Zeroes: To verify if a number is a zero, substitute it into the polynomial and check if it equals zero.

Memory Aids

🎵 Rhymes Time

  • To find a zero, just input and see, if the output is zero, it's meant to be.

📖 Fascinating Stories

  • Imagine a number that unlocks the secret door of a polynomial castle, where it stands as the only key to make things equal zero.

🧠 Other Memory Gems

  • Remember 'Z' for Zero, where p(z) = 0 is the crucial zero definition.

🎯 Super Acronyms

Z.E.R.O

  • Zero Equals Result of Output being zero.

Examples

  • Example 1: For p(x) = 5x^3 - 2x^2 + 3x - 2, find p(1) and p(-1).

  • Example 2: To verify if -2 is a zero of p(x) = x + 2, we check p(-2) = 0.

Glossary of Terms

  • Term: Zero of a Polynomial

    Definition:

    A number c such that p(c) = 0 for a polynomial p(x).

  • Term: Linear Polynomial

    Definition:

    A polynomial of the form p(x) = ax + b where a ≠ 0.

  • Term: Constant Polynomial

    Definition:

    A polynomial with no variable part, such as p(x) = c where c is a constant.

  • Term: Zero Polynomial

    Definition:

    The polynomial p(x) = 0, which has all real numbers as zeroes.