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Common Core For Algebra

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.

Worksheets for Algebra II, Module 1, Lesson 39

Solutions for Algebra II, Module 1, Lesson 39

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.

- 𝑥
^{2}+ 9 - 𝑥
^{2}+ 5 - Consider the polynomial 𝑃(𝑥) = 𝑥
^{4}− 3𝑥^{2}− 4.

a. What are the solutions to 𝑥^{4}− 3𝑥^{2}− 4 = 0?

b. How many 𝑥-intercepts does the graph of the equation 𝑦 = 𝑥^{4}− 3𝑥^{2}− 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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