2.6 Summary

Description

Quick Overview

This section outlines key concepts related to polynomials, including definitions, terms, and important theorems.

Standard

In this section, we explore various types of polynomials defined by their terms and degrees, along with their significance as mathematical expressions. The concept of zeros and the Factor Theorem are also explained.

Detailed

Detailed Summary

In this section, we delve into the structure and characteristics of polynomials. A polynomial in one variable is expressed as a sum of terms, each comprising a coefficient and a variable raised to a certain power. The classification of polynomials includes monomials (one term), binomials (two terms), and trinomials (three terms), along with specific types based on their degrees: linear (degree one), quadratic (degree two), and cubic (degree three). The concept of zeros, or roots, of polynomials is crucial, as a real number 'a' is a zero if substituting it into the polynomial results in zero. The Factor Theorem further connects the roots of a polynomial with its factors. This section concludes with specific polynomial identities, illustrating the expansion of binomials and the sum of cubes.

Key Concepts

  • Polynomial: A mathematical expression containing variables of non-negative integer powers.

  • Monomial, Binomial, Trinomial: Different categories of polynomials based on the number of terms.

  • Degree: The highest exponent in a polynomial, indicating its type.

  • Zero of a Polynomial: Values that make the polynomial equal to zero.

  • Factor Theorem: A principle connecting factors with their roots.

Memory Aids

🎡 Rhymes Time

  • Polynomials are fun to see, monomials, binomials, come join me!

πŸ“– Fascinating Stories

  • Imagine climbing a hill (the highest degree). Each step (the terms) counts, but together they show the way.

🧠 Other Memory Gems

  • For degrees: 'L, Q, C' means Linear, Quadratic, Cubic.

🎯 Super Acronyms

PRIME - Polynomials, Roots, Identity, Monomial, Equation.

Examples

  • If p(x) = 2x^3 + 3x^2 - x + 5, then it is a cubic polynomial of degree 3.

  • For p(x) = x^2 - 4, the zero is a = 2 because p(2) = 0.

Glossary of Terms

  • Term: Polynomial

    Definition:

    An algebraic expression in one variable that consists of terms of the form anxn + an–1xn–1 + ... + a2x2 + a1x + a0.

  • Term: Monomial

    Definition:

    A polynomial with only one term.

  • Term: Binomial

    Definition:

    A polynomial with two terms.

  • Term: Trinomial

    Definition:

    A polynomial with three terms.

  • Term: Degree

    Definition:

    The highest exponent of the variable in a polynomial.

  • Term: Zero of a Polynomial

    Definition:

    A value 'a' such that p(a) = 0.

  • Term: Factor Theorem

    Definition:

    States that if x – a is a factor of p(x), then p(a) = 0.