Polynomial or Not? Mastering the Definition of Polynomial Functions

Algebra 2 Grades High School 3:30 Video
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Lesson Description

This lesson will teach you how to identify polynomial functions by understanding their definition and examining various examples.

Video Resource

Determining if a Function is a Polynomial

Mario's Math Tutoring

Duration: 3:30
Watch on YouTube

Key Concepts

  • Definition of a polynomial
  • Monomials as building blocks of polynomials
  • Whole number exponents in polynomials
  • Real number coefficients in polynomials

Learning Objectives

  • Students will be able to define a polynomial and monomial.
  • Students will be able to identify whether a given function is a polynomial or not based on its exponents and coefficients.
  • Students will be able to explain why a function is or is not a polynomial using the definition.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the general definition of a function. Then, introduce the concept of a polynomial function as a special type of function. Briefly explain the importance of understanding polynomials in algebra and calculus.
  • Video Viewing and Note-Taking (10 mins)
    Play the 'Determining if a Function is a Polynomial' video by Mario's Math Tutoring. Instruct students to take notes on the definition of a polynomial, the criteria for identifying polynomials (whole number exponents, real number coefficients), and the examples provided in the video.
  • Guided Practice (15 mins)
    Work through additional examples similar to those in the video. Have students identify whether each function is a polynomial and explain their reasoning. Provide immediate feedback and address any misconceptions.
  • Independent Practice (15 mins)
    Provide a worksheet or online quiz with a variety of functions. Students should independently determine whether each function is a polynomial and justify their answers.
  • Wrap-up and Discussion (5 mins)
    Review the key concepts of the lesson. Answer any remaining questions and preview upcoming topics related to polynomials, such as graphing and solving polynomial equations.

Interactive Exercises

  • Polynomial Sorting Activity
    Create a set of cards with various functions written on them. Have students sort the cards into two categories: polynomials and non-polynomials. Students should then justify their sorting decisions.
  • Error Analysis
    Present students with incorrect classifications of functions as polynomials or non-polynomials. Have them identify the error in reasoning and explain the correct classification.

Discussion Questions

  • What are the key characteristics that define a polynomial function?
  • How do you distinguish between the coefficient and the exponent of a term in a polynomial?
  • Why is it important for the exponents in a polynomial to be whole numbers?
  • Can a polynomial have irrational coefficients? Explain why or why not.
  • What is the difference between a monomial and a polynomial?

Skills Developed

  • Critical thinking
  • Mathematical reasoning
  • Attention to detail
  • Application of definitions

Multiple Choice Questions

Question 1:

Which of the following is NOT a requirement for a function to be considered a polynomial?

Correct Answer: Integer coefficients

Question 2:

Which of the following functions is a polynomial?

Correct Answer: f(x) = 2x^3 + x - 7

Question 3:

What type of numbers are allowed as exponents in a polynomial function?

Correct Answer: Only whole numbers

Question 4:

Which of the following is a monomial?

Correct Answer: 3x^2

Question 5:

Why is f(x) = √x not a polynomial?

Correct Answer: The exponent is not a whole number.

Question 6:

Which statement is true about polynomial coefficients?

Correct Answer: They must be real numbers.

Question 7:

Is f(x) = x^(2/3) a polynomial?

Correct Answer: No, because 2/3 is not a whole number.

Question 8:

Which of the following is a polynomial expression?

Correct Answer: x^3 + 3x - 1

Question 9:

Which of the following is NOT a polynomial?

Correct Answer: x^(-1) + 3

Question 10:

What makes 5/x - 8 not a polynomial?

Correct Answer: It has x in the denominator.

Fill in the Blank Questions

Question 1:

A polynomial is a sum of ________.

Correct Answer: monomials

Question 2:

The exponents in a polynomial must be ________ numbers.

Correct Answer: whole

Question 3:

The coefficients in a polynomial must be ________ numbers.

Correct Answer: real

Question 4:

A term with a variable in the denominator, such as 5/x, is _______ a polynomial.

Correct Answer: not

Question 5:

The square root of x, written as √x, can also be expressed as x to the power of ________.

Correct Answer: 1/2

Question 6:

A monomial is a single ____ or ____.

Correct Answer: term/group

Question 7:

A polynomial is typically written in ________ order from the highest degree to the lowest degree.

Correct Answer: descending

Question 8:

If a term contains a variable with a negative exponent, it is _________ a polynomial.

Correct Answer: not

Question 9:

f(x)= sqrt(7)x^4 + 2x is a ________ because sqrt(7) is allowed as a coefficient.

Correct Answer: polynomial

Question 10:

Polynomial functions have coefficients that are ____ numbers.

Correct Answer: real

Educational Standards

A2.A.2.2:Add, subtract, multiply, divide, and simplify polynomial expressions. A2.F.1.5:Analyze the graph of a polynomial function by identifying the domain, range, intercepts, zeros, relative maxima, relative minima, and intervals of increase and decrease.

Teaching Materials

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