lesson 3: operations with complex...

Hart Interactive – Honors Algebra 1 M4+ Lesson 3 HONORS ALGEBRA 1

Lesson 3: Operations with Complex Numbers

Module 4+ Complex Numbers

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Opening Exercise

Since complex numbers are built from real numbers, we should be able to add, subtract, multiply, and divide

them. They should also satisfy the commutative, associative, and distributive properties, just as real numbers

do.

Let’s check how some of these operations work for complex numbers.

Addition with Complex Numbers

1. Compute (3 + 4𝑖𝑖) + (7 − 20𝑖𝑖).

Subtraction with Complex Numbers

2. Compute (3 + 4𝑖𝑖) − (7 − 20𝑖𝑖).

3. In general terms, we can say (𝑎𝑎 + 𝑏𝑏𝑖𝑖) + (𝑐𝑐 + 𝑑𝑑𝑖𝑖) = (______ + ______) + (______ + ______)𝑖𝑖

Multiplication with Complex Numbers

4. Compute (1 + 2𝑖𝑖)(1 − 2𝑖𝑖).

5. In general terms, we can say (𝑎𝑎 + 𝑏𝑏𝑖𝑖) ∙ (𝑐𝑐 + 𝑑𝑑𝑖𝑖) = ______ + ______ + ______ + ______

= (______ − ______) + (______ + ______)𝑖𝑖

Addition of variable expressions is a matter of

rearranging terms according to the properties

of operations. Often, we call this combining

like terms. These properties of operations

apply to complex numbers.

Multiplication is similar to polynomial

multiplication, using the addition,

subtraction, and multiplication

operations and the fact that 𝑖𝑖2 = −1.




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Multiplication with Complex Numbers

6. Verify that −1 + 2𝑖𝑖 and −1 − 2𝑖𝑖 are solutions to 𝑥𝑥2 + 2𝑥𝑥 + 5 = 0.

7. Rewrite each expression as a polynomial in standard form.

A. (𝑥𝑥 + 𝑖𝑖)(𝑥𝑥 − 𝑖𝑖)

B. (𝑥𝑥 + 5𝑖𝑖)(𝑥𝑥 − 5𝑖𝑖)

C. �𝑥𝑥 − (2 + 𝑖𝑖)��𝑥𝑥 − (2 − 𝑖𝑖)�




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Factor the following polynomial expressions into products of linear terms.

8. 𝑥𝑥2 + 9

9. 𝑥𝑥2 + 5

Reverse Your Thinking

Can we construct an equation if we know its solutions? When a polynomial equation is written in factored

form 𝑎𝑎(𝑥𝑥 − 𝑟𝑟1)(𝑥𝑥 − 𝑟𝑟2)⋯ (𝑥𝑥 − 𝑟𝑟𝑛𝑛) = 0, the solutions to the equation are 𝑟𝑟1, 𝑟𝑟2, … , 𝑟𝑟𝑛𝑛.

Write a polynomial 𝑃𝑃 with the lowest possible degree that has the given solutions. Explain how you

generated each answer. Write your answer in standard form. Be careful about which factors to multiply first

in Exercise 10!

10. −2, 3, −4𝑖𝑖, 4𝑖𝑖

11. 3 + 𝑖𝑖, 3 − 𝑖𝑖




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Lesson Summary Adding two complex numbers is comparable to combining like terms in a polynomial expression.

Multiplying two complex numbers is like multiplying two binomials, except one can use 𝑖𝑖2 = −1 to

simplify more.

Complex numbers satisfy the associative, commutative, and distributive properties.




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Homework Problem Set

1. Express each of the following in 𝑎𝑎 + 𝑏𝑏𝑖𝑖 form.

A. (2 + 5𝑖𝑖) + (4 + 3𝑖𝑖)

B. (−1 + 2𝑖𝑖) − (4 − 3𝑖𝑖) C. (4 + 𝑖𝑖) + (2 − 𝑖𝑖) − (1 − 𝑖𝑖) D. (5 + 3𝑖𝑖)(5 − 3𝑖𝑖) E. (2 − 𝑖𝑖)(2 + 𝑖𝑖) F. (1 + 𝑖𝑖)(2 − 3𝑖𝑖) + 3𝑖𝑖(1 − 𝑖𝑖) − 𝑖𝑖

2. Express each of the following in 𝑎𝑎 + 𝑏𝑏𝑖𝑖 form.

A. (1 + 𝑖𝑖)2 B. (1 + 𝑖𝑖)4 C. (1 + 𝑖𝑖)6




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3. Evaluate 𝑥𝑥2 − 6𝑥𝑥 when 𝑥𝑥 = 3 − 𝑖𝑖.

4. Evaluate 4𝑥𝑥2 − 12𝑥𝑥 when 𝑥𝑥 = 32 −

𝑖𝑖2.

5. Show by substitution that 5−𝑖𝑖√55 is a solution to 5𝑥𝑥2 − 10𝑥𝑥 + 6 = 0.

6. Use the fact that 𝑥𝑥4 + 64 = (𝑥𝑥2 − 4𝑥𝑥 + 8)(𝑥𝑥2 + 4𝑥𝑥 + 8) to explain how you know that the graph of

𝑦𝑦 = 𝑥𝑥4 + 64 has no 𝑥𝑥-intercepts. You need not find the solutions.

lesson 3: operations with complex...

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