Unlocking Polynomial Functions: From Zeros to Equations

Algebra 2 Grades High School 8:06 Video
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Lesson Description

Learn how to construct polynomial functions with rational coefficients from given zeros, including the application of the Irrational Conjugates Theorem. Master the relationship between zeros and factors, and efficiently multiply factors to obtain the polynomial function.

Video Resource

How to Write a Polynomial Function with Rational Coefficients and the Given Zeros

Mario's Math Tutoring

Duration: 8:06
Watch on YouTube

Key Concepts

  • Irrational Conjugates Theorem
  • Relationship between zeros and factors of a polynomial
  • Constructing polynomial functions from given zeros

Learning Objectives

  • Students will be able to identify conjugate pairs of irrational zeros.
  • Students will be able to write factors of a polynomial given its zeros.
  • Students will be able to construct a polynomial function with rational coefficients from a given set of zeros, including irrational zeros.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of rational and irrational numbers. Introduce the Irrational Conjugates Theorem and explain its importance in constructing polynomial functions. Briefly discuss the connection between zeros and x-intercepts.
  • Video Presentation (10 mins)
    Play the YouTube video 'How to Write a Polynomial Function with Rational Coefficients and the Given Zeros' by Mario's Math Tutoring. Instruct students to take notes on the key steps and examples provided.
  • Guided Practice (15 mins)
    Work through the examples from the video step-by-step on the board, pausing to answer questions and clarify any confusion. Emphasize the importance of distributing the negative sign when writing factors and the technique of grouping terms to simplify multiplication.
  • Independent Practice (15 mins)
    Provide students with a set of zeros (including irrational zeros) and have them construct the corresponding polynomial functions individually. Circulate the room to provide assistance as needed.
  • Wrap-up and Discussion (5 mins)
    Review the key concepts and address any remaining questions. Discuss the importance of verifying the solution by checking the zeros of the constructed function using a graphing calculator or online tool.

Interactive Exercises

  • Zero to Function Challenge
    Provide students with a list of zeros and challenge them to be the first to correctly construct the corresponding polynomial function. Offer a small reward to the winner.

Discussion Questions

  • Why is the Irrational Conjugates Theorem important when constructing polynomial functions?
  • How does the factor theorem relate to finding the zeros of a polynomial function?
  • What are some strategies for efficiently multiplying polynomial factors?

Skills Developed

  • Applying the Irrational Conjugates Theorem
  • Polynomial Multiplication
  • Problem-solving

Multiple Choice Questions

Question 1:

If a polynomial has rational coefficients and 2 - √3 is a zero, what is another zero that must exist?

Correct Answer: 2 + √3

Question 2:

Which of the following is the factor corresponding to the zero x = -5?

Correct Answer: x + 5

Question 3:

What does the Irrational Conjugates Theorem state?

Correct Answer: If a polynomial has rational coefficients and an irrational number is a zero, its conjugate is also a zero.

Question 4:

When writing a polynomial function from its zeros, what operation is performed on the factors corresponding to the zeros?

Correct Answer: Multiplication

Question 5:

Given the zeros 1, -1, and 2, which of the following is a factor of the polynomial?

Correct Answer: x + 1

Question 6:

What is the conjugate of -1 + √5?

Correct Answer: -1 - √5

Question 7:

Which of the following statements is true about the coefficients of the polynomial function when applying the Irrational Conjugates Theorem?

Correct Answer: They must be rational.

Question 8:

What is the relationship between the zeros of a polynomial and the x-intercepts of its graph?

Correct Answer: They are the same thing.

Question 9:

What should you do after finding all the factors of a polynomial based on its zeros?

Correct Answer: Multiply the factors.

Question 10:

If a polynomial function has a zero of 4 + √7, what other number MUST also be a zero of the polynomial?

Correct Answer: 4 - √7

Fill in the Blank Questions

Question 1:

The Irrational Conjugates Theorem applies when the polynomial has __________ coefficients.

Correct Answer: rational

Question 2:

If x = 3 is a zero of a polynomial, then (x - 3) is a __________ of the polynomial.

Correct Answer: factor

Question 3:

The conjugate of 5 - √2 is __________.

Correct Answer: 5 + √2

Question 4:

To find a polynomial function from its zeros, you must __________ the factors together.

Correct Answer: multiply

Question 5:

Zeros of a polynomial function correspond to the __________ of the graph.

Correct Answer: x-intercepts

Question 6:

When distributing a negative sign across a binomial, you must change the __________ of each term inside the parentheses.

Correct Answer: sign

Question 7:

If 2 + √3 is a zero, then 2 - √3 is called its __________.

Correct Answer: conjugate

Question 8:

After multiplying the factors, you should __________ like terms to simplify the polynomial.

Correct Answer: combine

Question 9:

A polynomial function with rational coefficients will not have any __________ coefficients in its final form.

Correct Answer: irrational

Question 10:

The x-intercepts of a polynomial function occur when the y-value is equal to __________.

Correct Answer: zero

Educational Standards

A2.A.1.4:Solve polynomial equations with real roots using various methods (e.g., polynomial division, synthetic division, using graphing calculators or other appropriate technology). 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.

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