lesson 27: operations with complex...
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Hart Interactive β 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.
Letβs check how some of these operations work for complex numbers.
Addition with Complex Numbers
1. Compute (3 + 4ππ) + (7 β 20ππ).
Subtraction with Complex Numbers
2. Compute (3 + 4ππ) β (7 β 20ππ).
3. In general terms, we can say (ππ + ππππ) + (ππ + ππππ) = (______ + ______) + (______ + ______)ππ
Multiplication with Complex Numbers
4. Compute (1 + 2ππ)(1 β 2ππ).
5. In general terms, we can say (ππ + ππππ) β (ππ + ππππ) = ______ + ______ + ______ + ______
= (______ β ______) + (______ + ______)ππ
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 ππ2 = β1.
Lesson 27: Operations with Complex Numbers
S.19
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This file derived from ALG II-M1-TE-1.3.0-07.2015
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Hart Interactive β Honors Algebra 1 M4+ Lesson 27 HONORS ALGEBRA 1
Multiplication with Complex Numbers
6. Verify that β1 + 2ππ and β1 β 2ππ are solutions to π₯π₯2 + 2π₯π₯ + 5 = 0.
7. Rewrite each expression as a polynomial in standard form.
a. (π₯π₯ + ππ)(π₯π₯ β ππ)
b. (π₯π₯ + 5ππ)(π₯π₯ β 5ππ)
c. οΏ½π₯π₯ β (2 + ππ)οΏ½οΏ½π₯π₯ β (2 β ππ)οΏ½
Lesson 27: Operations with Complex Numbers
S.20
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©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 β Honors Algebra 1 M4+ Lesson 27 HONORS ALGEBRA 1
Factor the following polynomial expressions into products of linear terms.
8. π₯π₯2 + 9
9. π₯π₯2 + 5
Reverse Your Thinking
Can we construct an equation if we know its solutions? When a polynomial equation is written in factored
form ππ(π₯π₯ β ππ1)(π₯π₯ β ππ2)β― (π₯π₯ β ππππ) = 0, the solutions to the equation are ππ1, ππ2, β¦ , ππππ.
Write a polynomial ππ 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. β2, 3, β4ππ, 4ππ
11. 3 + ππ, 3 β ππ
Lesson 27: Operations with Complex Numbers
S.21
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©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 β 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 ππ2 = β1 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 β’ and licensed by Great Minds. Β©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 β Honors Algebra 1 M4+ Lesson 27 HONORS ALGEBRA 1
Homework Problem Set
1. Express each of the following in ππ + ππππ form.
A. (2 + 5ππ) + (4 + 3ππ)
B. (β1 + 2ππ) β (4 β 3ππ) C. (4 + ππ) + (2 β ππ) β (1 β ππ) D. (5 + 3ππ)(5 β 3ππ) E. (2 β ππ)(2 + ππ) F. (1 + ππ)(2 β 3ππ) + 3ππ(1 β ππ) β ππ
2. Express each of the following in ππ + ππππ form.
A. (1 + ππ)2 B. (1 + ππ)4 C. (1 + ππ)6
Lesson 27: Operations with Complex Numbers
S.23
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©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 β Honors Algebra 1 M4+ Lesson 27 HONORS ALGEBRA 1
3. Evaluate π₯π₯2 β 6π₯π₯ when π₯π₯ = 3 β ππ.
4. Evaluate 4π₯π₯2 β 12π₯π₯ when π₯π₯ = 32 β
ππ2.
5. Show by substitution that 5βππβ55 is a solution to 5π₯π₯2 β 10π₯π₯ + 6 = 0.
6. Use the fact that π₯π₯4 + 64 = (π₯π₯2 β 4π₯π₯ + 8)(π₯π₯2 + 4π₯π₯ + 8) to explain how you know that the graph of
π¦π¦ = π₯π₯4 + 64 has no π₯π₯-intercepts. You need not find the solutions.
Lesson 27: Operations with Complex Numbers
S.24
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©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.