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Hart Interactive – Honors Algebra 1 M4+ Lesson 27 HONORS ALGEBRA 1
Lesson 27: Operations with Complex Numbers
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.
Lesson 27: Operations with Complex Numbers
S.19
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Hart Interactive – Honors Algebra 1 M4+ Lesson 27 HONORS ALGEBRA 1
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)�
Lesson 27: Operations with Complex Numbers
S.20
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Hart Interactive – Honors Algebra 1 M4+ Lesson 27 HONORS ALGEBRA 1
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
Lesson 27: Operations with Complex Numbers
S.21
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Hart Interactive – Honors Algebra 1 M4+ Lesson 27 HONORS ALGEBRA 1
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.
Lesson 27: Operations with Complex Numbers
S.22
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Hart Interactive – Honors Algebra 1 M4+ Lesson 27 HONORS ALGEBRA 1
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
Lesson 27: Operations with Complex Numbers
S.23
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Hart Interactive – Honors Algebra 1 M4+ Lesson 27 HONORS ALGEBRA 1
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 27: Operations with Complex Numbers
S.24
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.