Factoring Extended to the Complex Realm

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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.

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.

𝑥² + 9
𝑥² + 5
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.
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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