Factoring Extended to the Complex Realm

Factoring Extended to the Complex Realm

Student Outcomes

• Students solve quadratic equations with real coefficients that have complex solutions. Students extend polynomial identities to the complex numbers.
• Students note the difference between solutions to the equation and the x-intercepts of the graph of said equation.

New York State Common Core Math Algebra II, Module 1, Lesson 39

Worksheets for Algebra 2

Classwork

Opening Exercise
Rewrite each expression as a polynomial in standard form.
a. (𝑥 + 𝑖)(𝑥 − 𝑖)
b. (𝑥 + 5𝑖)(𝑥 − 5𝑖)
c. (𝑥 −(2 +𝑖))(𝑥 − (2 − 𝑖))

Exercises 1–4
Factor the following polynomial expressions into products of linear terms.

1. 𝑥² + 9
2. 𝑥² + 5
3. Consider the polynomial 𝑃(𝑥) = 𝑥⁴ − 3𝑥² − 4.
   a. What are the solutions to 𝑥⁴ − 3𝑥² − 4 = 0?
   b. How many 𝑥-intercepts does the graph of the equation 𝑦 = 𝑥⁴ − 3𝑥² − 4 have? What are the coordinates of the 𝑥-intercepts?
   c. Are solutions to the polynomial equation 𝑃(𝑥) = 0 the same as the 𝑥-intercepts of the graph of 𝑦 = 𝑃(𝑥)?
   Justify your reasoning.
4. Write a polynomial 𝑃 with the lowest possible degree that has the given solutions. Explain how you generated each answer.
   a. −2, 3, −4𝑖, 4𝑖
   b. −1, 3𝑖
   c. 0, 2, 1 + 𝑖, 1 − 𝑖
   d. √2, −√2, 3, 1 + 2𝑖
   e. 2𝑖, 3 − 𝑖
   

Lesson Summary

• Polynomial equations with real coefficients can have real or complex solutions or they can have both.
• If a complex number is a solution to a polynomial equation, then its conjugate is also a solution.
• Real solutions to polynomial equations correspond to the 𝑥-intercepts of the associated graph, but complex solutions do not.

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