Lesson 2.1

Identifying Polynomials

Welcome to Unit 2! We are moving beyond parabolas (degree 2) to study "smooth" curves of any degree. But first, we must define exactly what a polynomial *is*.

Introduction

A polynomial is a mathematical expression consisting of variables and coefficients, involving only operations of addition, subtraction, multiplication, and non-negative integer exponents.

Past Knowledge

You know (Degree 1) and (Degree 2).

Today's Goal

Identify the degree, leading coefficient, and whether an expression is a polynomial.

Future Success

This vocabulary is essential for finding "zeros" and graphing high-degree functions later in this unit.

Key Concepts

1. The Rules

For an expression to be a polynomial, the exponents on the variable must be Whole Numbers ().

  • Allowed
  • NOT Allowed (Negative exponent)
    (Fractional exponent)
    (Variable exponent)

2. Standard Form

Write terms in order from highest exponent to lowest.

Degree5(Highest Exponent)
Leading Coeff.3(Number in front)

3. Vocabulary

DegreeNameExample
0Constant
1Linear
2Quadratic
3Cubic
4Quartic
5Quintic

Worked Examples

Example 1: Classifying Polynomials

Basic

Classify .

1

Rewrite in Standard Form

Put the highest exponent term first. Watch the negative sign!

2

Identify Attributes

  • Degree: (Cubic)
  • Leading Coefficient:
  • Constant Term:

Example 2: Is it a Polynomial?

Concept

Determine if the following are polynomials:

A)

NO

. Negative exponents are not allowed.

B)

YES

Coefficients can be fractions () or irrationals (). Only the variable cannot be under a root.

Example 3: Tricky Standard Form

Advanced

Identify the degree and leading coefficient of .

1

Rewrite Carefully

is the same as .

2

Identify

  • Degree: (Quartic)
  • Leading Coefficient: (Positive Fraction)
  • Note: is just a number (coefficient), so it is allowed!

Common Pitfalls

Confusing Degree vs. Terms

is a very simple polynomial, but it has Degree 100. Do not think "simple" means "small degree."

Hidden Exponents

looks nice, but it's . Fractional exponents are NOT allowed in polynomials.

Real-Life Applications

Polynomials are used to model smooth, continuous curves in the real world, such as roller coaster tracks. Engineers use "Cubic Splines" (degree 3 polynomials connected together) to ensure the track doesn't have sharp corners that would hurt passengers.

Practice Quiz

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