Unlocking Equations: Finding Equations from Imaginary and Irrational Roots

Algebra 2 Grades High School 13:12 Video
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Lesson Description

Learn how to construct polynomial equations given imaginary and irrational roots by understanding and applying the concept of conjugates. This lesson builds on prior knowledge of finding equations from rational roots and extends it to more complex number systems.

Video Resource

Imaginary Roots Given Find Equation

Kevinmathscience

Duration: 13:12
Watch on YouTube

Key Concepts

  • Imaginary Roots
  • Irrational Roots
  • Conjugates
  • Polynomial Equations

Learning Objectives

  • Identify imaginary and irrational roots of a polynomial equation.
  • Determine the conjugate of a given imaginary or irrational number.
  • Construct a polynomial equation given its imaginary and/or irrational roots.

Educator Instructions

  • Introduction (5 mins)
    Briefly review how to find a polynomial equation from rational roots. Emphasize the difference between rational, irrational, and imaginary numbers. Introduce the concept of imaginary and irrational roots and explain that this lesson will extend the previous method to include these types of roots.
  • Video Presentation (15 mins)
    Play the video 'Imaginary Roots Given Find Equation' by Kevinmathscience. Encourage students to take notes on the key steps: identifying the need for conjugates, finding the conjugates, and constructing the equation.
  • Conjugate Identification (10 mins)
    Practice identifying the conjugate of various imaginary and irrational numbers. Provide examples and have students identify the corresponding conjugates. For example: 2 + 5i, -3 - √2, 7i, √11.
  • Equation Construction (20 mins)
    Work through examples of constructing equations from given imaginary and irrational roots. Start with simpler cases and gradually increase complexity. Emphasize the importance of including the conjugate root. Demonstrate the process of expanding the factored form of the polynomial to obtain the final equation.
  • Independent Practice (15 mins)
    Assign practice problems where students find equations from given imaginary and irrational roots. Circulate to provide assistance and answer questions.

Interactive Exercises

  • Conjugate Matching Game
    Create a matching game where students pair imaginary or irrational numbers with their conjugates.
  • Equation Builder
    Provide students with sets of roots (including imaginary and irrational ones) and have them work in groups to construct the corresponding polynomial equations. Groups can then present their solutions and explain their process.

Discussion Questions

  • Why is it necessary to include the conjugate when dealing with imaginary or irrational roots?
  • How does finding an equation from imaginary/irrational roots differ from finding an equation from rational roots?
  • Can a polynomial equation with real coefficients have only one imaginary root? Explain.

Skills Developed

  • Complex Number Manipulation
  • Polynomial Equation Construction
  • Problem-Solving
  • Attention to detail

Multiple Choice Questions

Question 1:

What is the conjugate of 3 - 2i?

Correct Answer: 3 + 2i

Question 2:

If 2 + √5 is a root of a polynomial equation, what is another root that must also be present?

Correct Answer: 2 - √5

Question 3:

Given roots of i and -i, which of the following is the quadratic equation?

Correct Answer: x² + 1 = 0

Question 4:

What is the first step in finding the equation given the roots 1 + i and 1 - i?

Correct Answer: Write the factors (x - root)

Question 5:

Which number does NOT require the use of a conjugate when determining its related polynomial?

Correct Answer: 6

Question 6:

What is the correct quadratic equation given the roots are 4i and -4i?

Correct Answer: x² + 16 = 0

Question 7:

What is the factored form of a polynomial with roots of 2, i and -i

Correct Answer: (x-2)(x-i)(x+i)

Question 8:

When multiplying (x + 3i)(x - 3i) what is the last term?

Correct Answer: -9i²

Question 9:

What is a root to the polynomial if one of the binomials is (x - 7)

Correct Answer: 7

Question 10:

When combining like terms and simplifying (x+5)(x+5) what is the correct value of c?

Correct Answer: 25

Fill in the Blank Questions

Question 1:

The conjugate of a + bi is a _____ bi.

Correct Answer: minus

Question 2:

If √7 is a root, then _____ is also a root.

Correct Answer: -√7

Question 3:

The factored form of a quadratic equation from roots is (x - root1)(x - _____).

Correct Answer: root2

Question 4:

i² is equal to _____.

Correct Answer: -1

Question 5:

When given irrational and imaginary roots, remember to include their _____.

Correct Answer: conjugates

Question 6:

The opposite of a irrational root is called a _____.

Correct Answer: conjugate

Question 7:

In a polynomial equation, roots are also known as _____.

Correct Answer: zeros

Question 8:

When expanding (x+2)(x+2) the b term value is _____.

Correct Answer: 4x

Question 9:

Another word for finding the product of binomials is _____.

Correct Answer: FOIL

Question 10:

The quadratic formula is used to find _____.

Correct Answer: roots

Educational Standards

A2.A.1.1:Use mathematical models to represent quadratic relationships and solve using factoring, completing the square, the quadratic formula, and various methods (including graphing calculator or other appropriate technology). Find non-real roots when they exist. A2.N.1.2:Simplify, add, subtract, multiply, and divide complex numbers. A2.A.2.2:Add, subtract, multiply, divide, and simplify polynomial expressions.

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