Relationship between Zeroes and Coefficients of a Polynomial

2.3 Relationship between Zeroes and Coefficients of a Polynomial

Description

Quick Overview

The section explains the relationship between the zeroes of a polynomial and its coefficients, particularly focusing on quadratic and cubic polynomials.

Standard

This section delves into how the zeroes of quadratic and cubic polynomials are influenced by their coefficients, establishing important formulas for their sums and products. Examples illustrate these relationships through practical calculations.

Detailed

Relationship between Zeroes and Coefficients of a Polynomial

In this section, we explore the profound relationship between the zeroes of polynomial functions and their coefficients. We start with quadratic polynomials, represented in the general form as p(x) = ax^2 + bx + c, where a, b, and c are real coefficients with a ≠ 0.

Key Concepts:

  • Zeroes of Quadratic Polynomials: The zeroes can be found through factoring, and specific relationships define how they relate to coefficients:
  • Sum of Zeroes:
    • Formula: α + β = -b/a
  • Product of Zeroes:
    • Formula: α * β = c/a

We examine several examples:
- Example 1: For p(x) = 2x^2 - 8x + 6, the zeroes 1 and 3 can be derived; their sum and product verify the aforementioned formulas.
- Example 2: For p(x) = 3x^2 + 5x - 2, zeroes derived (1/3 and -2) also fit the patterns established.

Next, we extend to cubic polynomials of the general form p(x) = ax^3 + bx^2 + cx + d and establish relationships:
- Sum of the Zeroes:
- α + β + γ = -b/a
- Sum of Products of Zeroes Taken Two at a Time:
- αβ + βγ + γα = c/a
- Product of Zeroes:
- αβγ = -d/a

Example 3 demonstrates these principles in action for a cubic polynomial p(x) = 3x^3 - 5x^2 - 11x - 3 where the zeroes 3, -1, -1/3 are verified, showing these relationships hold true.

This section serves as a foundation for understanding polynomials, vital for further algebraic exploration such as polynomial division methods and complex functions.

Key Concepts

  • Zeroes of Quadratic Polynomials: The zeroes can be found through factoring, and specific relationships define how they relate to coefficients:

  • Sum of Zeroes:

  • Formula: α + β = -b/a

  • Product of Zeroes:

  • Formula: α * β = c/a

  • We examine several examples:

  • Example 1: For p(x) = 2x^2 - 8x + 6, the zeroes 1 and 3 can be derived; their sum and product verify the aforementioned formulas.

  • Example 2: For p(x) = 3x^2 + 5x - 2, zeroes derived (1/3 and -2) also fit the patterns established.

  • Next, we extend to cubic polynomials of the general form p(x) = ax^3 + bx^2 + cx + d and establish relationships:

  • Sum of the Zeroes:

  • α + β + γ = -b/a

  • Sum of Products of Zeroes Taken Two at a Time:

  • αβ + βγ + γα = c/a

  • Product of Zeroes:

  • αβγ = -d/a

  • Example 3 demonstrates these principles in action for a cubic polynomial p(x) = 3x^3 - 5x^2 - 11x - 3 where the zeroes 3, -1, -1/3 are verified, showing these relationships hold true.

  • This section serves as a foundation for understanding polynomials, vital for further algebraic exploration such as polynomial division methods and complex functions.

Memory Aids

🎵 Rhymes Time

  • Zeroes to the left, coefficients right; find their sum, and the product's a delight!

📖 Fascinating Stories

  • Imagine the quadratic fairy who lived in the polynomial forest, her zeroes were found by the magic of coefficients casting their spells.

🧠 Other Memory Gems

  • Remember: S for Sum, P for Product - just link them to b and c!

🎯 Super Acronyms

SPC

  • Sum
  • Product
  • Coefficients - remember these to tackle any polynomial quest.

Examples

  • Example of quadratic polynomial: p(x) = 2x^2 - 8x + 6 results in zeroes 1 and 3.

  • Example of cubic polynomial: p(x) = 2x^3 - 5x^2 - 14x + 8 results in zeroes 3, -2, and -1/2.

Glossary of Terms

  • Term: Zero of a Polynomial

    Definition:

    A value of x such that p(x) = 0 for a polynomial p(x).

  • Term: Quadratic Polynomial

    Definition:

    Polynomial of degree two represented as p(x) = ax^2 + bx + c.

  • Term: Cubic Polynomial

    Definition:

    Polynomial of degree three represented as p(x) = ax^3 + bx^2 + cx + d.

  • Term: Coefficient

    Definition:

    A numerical or constant factor in a term of an algebraic expression.