Chapter 5: Complex Numbers - White Plains Public …

[Pages:40]Algebra 2 and Trigonometry

Chapter 5: Complex Numbers

Name:______________________________ Teacher:____________________________ Pd: _______

Table of Contents

Day 1: Chapter 5-1: The Complex Numbers SWBAT: Simplify expressions involving complex numbers

Pgs. #4 - 9 Hw: pg 208 in textbook. #3-51 odd

Day 2: Chapter 4-9: Operations with Complex Numbers SWBAT: (1) Add, Subtract and Multiply Complex Numbers

Pgs. #10 ? 15 HW: pg 215-216 in textbook. #3 ? 55(odd)

Day 3: REVIEW OF COMPLEX NUMBERS SWBAT:

Define the complex unit Simplify powers of Graphing complex numbers on the complex plane Operate on complex numbers Rationalize denominators involving Pgs. #16 ? 19 HW: Finish the pages in this section

Day 4: Chapter 5-2: Complex Roots of Quadratic Equations SWBAT: Solve quadratic equations with imaginary roots

Pgs. #20 - 24 Hw: pg 219 in textbook. #3 ? 13(odd)

Day 5: Chapter 5-2: Nature of Roots SWBAT: use the discriminant to describe the roots of a quadratic function and the graph of the function

Pgs. #25 - 29 Hw: pg 202 in textbook. #3 ? 8, 9-14, 16-26 (even)

Day 6: Chapter 5-2: Sum and Product of Roots SWBAT: Find the roots of higher order polynomials

Pgs. #30-33 HW: pg 223 in textbook. #3-11, 18 - 23

Day 7 ? Solving Higher Order Polynomial Functions SWBAT: find the roots of polynomials functions Pgs. #34 - 37 pg 227 in textbook #5, 6, 8, 10,11,12,16,17

Day 8: REVIEW OF Properties of Quadratic Equations SWBAT:

Determine the nature of the roots of a quadratic Determine the sum and the roots of a quadratic Write the equation of a quadratic given the roots Solve Higher Order Polynomial Functions HW: Packet

HOMEWORK ANSWER KEYS ? STARTS AT PAGE 38-40

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COMPLEX NUMBERS In this part of the unit we will:

Define the complex unit Simplify powers of Graphing complex numbers on the complex plane Operate on complex numbers Rationalize denominators involving Review, from previous packet, this time using Determine the nature of the roots of a quadratic Determine the sum and the roots of a quadratic Write the equation of a quadratic given the roots

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Day 1 - Complex Numbers

SWBAT: simplify negative radicals using imaginary numbers, 2) simplify powers if i, and 3)

graph complex numbers.

Warm - Up:

1) Solve for x: x2 ? 9 = 0

2) Solve for x: x2 + 9 = 0

Imaginary Numbers

Until now, we have never been able to take the square root of a negative number. From

this point on, we define . is called the complex unit, and now all operations on radicals can be performed on negative numbers.

Examples:

Simplifying radicals

=

Adding (or subtracting like radicals)

(

)

In order to simplify negative square roots, do it exactly as you would regular radicals, but have one of the factors be -1. simplifies to

Example:

Positive Square Root

Negative Square Roots

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Concept 1: Simplifying Negative Radicals

Teacher Modeled

Student Try it!

=

=

= =

=

Concept 2: Simplifying Powers of i

You may not leave powers of in your answer.

After , the pattern start repeating,

meaning that

,

... etc.

i

-1

1

-i

To reduce powers of divide the power by 4, and the remainder is your new power of .

Example:

Evaluate

.

A cheat on reducing a power of : Divide the power by 4. You will get some number, and that number will either have no decimal (no remainder), or a .25, .5, or .75.

.25 represents , which means a remainder of ____ when divided by 4.

. 5 represents ?, which means , which means a remainder of ____ when divided by 4.

.75 represents , which means a remainder of ____ when divided by 4.

The equivalent power of is

.

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Teacher Modeled

i17 = i22 = i83 = i100 =

Student Try It!

Evaluate: 1. i14

2. i7

3. 4i14

4. 5i2 + 2i4

5. i39

6. 2i5 +7i7

Concept 3: Graphing Complex Numbers

Due to their unique nature, complex numbers cannot be represented on a normal set of coordinate axes.

In 1806, J. R. Argand developed a method for displaying complex numbers graphically as a point in a coordinate plane. His method, called the Argand diagram, establishes a relationship between the x-axis (real axis) with real numbers and the y-axis (imaginary axis) with imaginary numbers.

In the Argand diagram, a complex number a + bi is the point (a,b) or the vector from the origin to the point (a,b).

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Graph the complex numbers:

1. 3 + 4i

2. 2 - 3i

3. -4 + 2i

4. 3 (which is really means

)

5. 4i (which is really means

)

The Parallelogram Rule for Complex Addition

The parallelogram rule for complex addition says that if you are adding two complex numbers, then the sum of can be represented by the diagonal of the parallelogram that can be drawn using the two original vectors as adjacent sides.

1+4i

4

2

6+5i 5+i

-10

-5

5

10

(1+4i)+(5+i)=6+5i

-2

Add 3 + 3i and -4 + 2i graphically.

-4

Subtract 3 + 4i from -2 + 2i

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Student Try It!

1. Represent the complex number 2 + 3i graphically.

2. Add graphically: (-2+4i) and (4+i)

3. Graphically Subtract (-1+i) from (3+2i)

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