chapter 5: complex numbers · 4 day 1 - complex numbers swbat: simplify negative radicals using...
TRANSCRIPT
Algebra 2 and Trigonometry
Chapter 5: Complex Numbers
Name:______________________________
Teacher:____________________________
Pd: _______
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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: x
2 – 9 = 0 2) Solve for x: x
2 + 9 = 0
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
√
√ √ √
√
Imaginary Numbers
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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.
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 ½, 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 .
i
1 -1
-i
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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.
Add 3 + 3i and -4 + 2i graphically. Subtract 3 + 4i from -2 + 2i
4
2
-2
-4
-10 -5 5 10
(1+4i)+(5+i)=6+5i
1+4i
5+i
6+5i
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Student Try It!
1. Represent the complex number 2 + 3i graphically.
2. Add graphically: 3. Graphically Subtract
(-2+4i) and (4+i) (-1+i) from (3+2i)
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Challenge
Summary/Closure:
Exit Ticket:
10
Day 2 - Operations with Complex Numbers
SWBAT: add, subtract, multiply and divide complex numbers.
Warm - Up: Express each expression in terms of i and simplify.
1) √ 2) √ √ 3) i49 4) i246
All operations on complex numbers are exactly the same as you
would do with variables… just make sure there is no power of in
your final answer. Your answer should be in a + bi form. Example: Multiplying binomials ( )( )
( )
Concept 1: Adding and Subtracting Complex Numbers Example 1: (4 + 3i) + (2 + 5i) =
Example 2: (5 + 3i) – (2 + 8i) =
* When multiplying Binomials don’t forget to F.O.I.L!
*
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Concept 2: Multiplying/Dividing Complex Numbers
1. (2 + 6i)2 2. (9 + 3i)(9 – 3i)
3.(2 + 5i)(3 – 3i) 4. √
√
√
√ 6.
√
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Concept 3: Rationalizing Denominators containing Just as you are not allowed to leave a radical in a denominator of a fraction, you are not allowed to leave an in the denominator. This is because is a radical!!
If the denominator of the fraction is a monomial, multiply the fraction by a
Fancy Form of 1 of
. Simplify as normally required.
Example:
( )
If the denominator of the fraction is a binomial, multiply the fraction by a
Fancy Form of 1 of the CONJUGATE of the denominator. Simplify as normally required.
Example:
( )
( )( )
( )
Remember! The conjugate of a binomial is the result of reversing the sign between the two terms. Example: ( ) and ( ) are conjugates.
Conjugate: change the sign in the middle of a binomial.
If the binomial is (a + b), the conjugate is _______
Find each complex conjugate.
a) i – 3 ______________________
b) 3i + 4 ______________________
c) -1 – 6i ______________________
d) 11i ______________________
Complex Conjugates have the same ________ parts but __________ imaginary parts
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Concept 3 (Continued) : Dividing Complex Numbers
Example 2:
Example 3:
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Additive Inverse and Multiplicative Inverse The additive inverse of a number is its opposite. The multiplicative inverse is its reciprocal.
Example: Find the additive inverse and multiplicative inverse of .
Additive inverse: ( )
Multiplicative inverse:
then rationalize the denominator to get
.
Additive inverse: the “opposite.” The additive inverse is the number, when added to a number, gives you 0. 0 is called the “additive identity.”
If the binomial is (a + b), the additive inverse is ________
If the monomial is a, the additive inverse is ______
Multiplicative inverse: the “reciprocal.” The multiplicative inverse is the number, when multiplied by a number, gives you 1. 1 is called the “multiplicative identity.”
If the binomial is (a + b), the multiplicative inverse is ______
If the monomial is a, the multiplicative inverse is ______
Example 4: Write the multiplicative inverse of 2 + 4i.
Try these:
1. Write the multiplicative inverse of 3 – 2i.
2. Write the multiplicative inverse of -2 + 3i.
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Challenge
Summary/Closure:
Exit Ticket:
16
Day 3 – Review Complex Numbers
3. gfgdg 4.
2.
17
5. 6.
7. fdfdf 8.
9. 10.
11. 12.
13. 14.
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15. 16.
17. 18.
19. 20.
21. 22.
19
23. 24.
25. 26.
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Day 4 - Complex Roots of Quadratic Equations
SWBAT: Solve quadratic equations with complex roots
Warm - Up:
1. gbfg 2.
Everything we did with quadratics earlier, this time we with solve quadratic equations with complex numbers
Determine the complex roots of a quadratic function by:
o the method of completing the square.
o the quadratic formula.
NOTE! Often, the regents will “warn” you that you need the quadratic formula by saying that you need to “find the roots in a + bi form.” This is code that the roots will be imaginary, and you cannot solve by factoring.
Example:
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Example 1: Determine the roots of by completing the square.
Example 2: Solve the following using the quadratic formula.
Solve for x:
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Classwork
Solve each of the following. Make sure you get each problem in standard form FIRST before you solve.
1. Solve for x and express the roots in terms of i:
2. Solve for x and express your answer in simplest form:
3. Express the roots of the equation in form.
4. x(x-1) = 56 5. (x + 8)2 = 25
6. 4
6
5
3
x
x
Answers:
1. { } { }
{
}
4. { } 5. { } 6. { }
1.
2.
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3.
4.
5.
6.
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Challenge
SUMMARY
Exit Ticket
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Day 5 - Nature of the Roots
SWBAT: use the discriminant to describe the roots of a quadratic function and the graph of the
function
Warm - Up
Real numbers: Any number that is not the square root of a negative number Imaginary numbers: Numbers that involve square roots of negative numbers (numbers
that contain the imaginary unit . Rational numbers: Numbers that can be written as the ratio of two numbers (as long as the denominator is not 0.)
Examples: , ,
Irrational numbers: Numbers that cannot be written as described above.
Examples: √ , √ ,
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For each of the following problems, use the quadratic formula to determine the roots of the equation.
“Nature” of the Roots
Equation ROOTS? Roots are: Real or Imaginary?
Roots are: Rat’l or Irr’l?
Roots are:
= or
1.
2.
3.
4.
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The discriminant is the name given to the expression that
appears under the square root (radical) sign in the quadratic
formula.
Quadratic Formula
Discriminant
The discriminant tells you about the "nature" of the roots of a quadratic equation given
that a, band c are rational numbers. It quickly tells you the number of real roots, or in
other words, the number of x-intercepts, associated with a quadratic equation.
There are four situations:
If b2 – 4ac is: The nature of the roots is: Inequality Number
of Roots
Negative Imaginary b2 – 4ac < 0 2 Zero Real, Rational, Equal b2 – 4ac = 0 1 Positive, Perfect Square
Real, Rational, Unequal b2 – 4ac > 0, perfect square
2
Positive, Non-perfect Square
Real, Irrational, Unequal b2 – 4ac >0, non-perfect square
2
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Regents Questions
1. The roots of the equation are
1) real, rational, and equal 2) real, rational, and unequal 3) real, irrational, and unequal 4) imaginary
2. If a quadratic equation with real coefficients has a discriminant of , then its roots must be
1) equal 2) imaginary 3) real and irrational 4) real and rational
3. Which value of c would make the roots of the equation real, rational, and equal? 1) 9 3) 18 2) 4)
4. Which value of c will make the roots of the equation real and equal?
1) 3) 0 2) 4) 16
5. In the equation , imaginary roots will be generated if
1) 3)
2) 4)
6.
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Challenge
SUMMARY
EXIT TICKET
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Day 6 - Sum and Product of the Roots of a Quadratic Equation
SWBAT: 1) find the sum and product of the roots from a quadratic equation, 2) use the sum and
product of the roots to write a quadratic equation, 3) using the sum and product to find the missing
value of the quadratic equation.
Warm - Up:
1) Find the roots of the quadratic equation by factoring: x2 - 5x - 14 = 0 What is the sum of the roots? What is the product of the roots?
2) Find the roots of the quadratic equation by factoring: x2 – 6x + 8 = 0 What is the sum of the roots? What is the product of the roots? What conclusion can you draw about the sum of the roots of and product of the roots of a quadratic equation in the form ax2+bx + c = 0?
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The sum of the roots of any quadratic
Sum =
The product of the roots of any quadratic
Product =
Concept 1: Calculating the sum and product of roots given a quadratic equation
Example 1: What is the sum and product of the roots of ?
Concept 2: Writing an equation of a quadratic given the roots Example 2: Write a quadratic equation with integer coefficients that has roots of
and -3.
** Always assume a = 1 **
Example 3: Write a quadratic equation that
has √ as a root. Irrational roots come in conjugate pairs, so if
√ is one root, then the other root must be ___________
Concept 3: Writing an equation of a quadratic given the sum and roots
Write a quadratic equation whose roots have the indicated sum and product.
Sum = 4, product = 3
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Concept 4: Finding a root given one of the two roots If one root of x2 – 6x + k = 0 is 4, find the other root.
Try These! Example 1: Write a quadratic equation with the following sum and product:
Sum = 8 Product = 25
Example 2: Given: 3x2 + 6x – 3
a) Find the sum of the roots. b) Find the product of the roots
Example 3: Given the quadratic equation x2 - 8x + k = 0 and r1= 5, find r2. Example 4: Write a quadratic equation with roots 5 and 7. Example 5: Write a quadratic equation with roots 3 + 2i and 3 – 2i.
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Example 6:
Summary/Closure:
Exit Ticket:
34
Day 7 - Chapter 5 Section 8 Solving Higher Degree Polynomial Equations SWBAT: find the roots of polynomials functions Warm - Up: Find the roots of the function graphed below.
When the powers in polynomial equations increase, it becomes more
difficult to find their solutions (roots).
Consider an equation such as
Finding the roots of an equation such as this can prove to be quite a
task. In this course, we will just be touching the surface on techniques
for solving higher degree polynomial equations.
This course will examine
how to solve polynomial
equations of higher degree
using factoring and/or the
quadratic formula, only.
FYI: The equation at the left
factors into
Let's be sure that we understand the vocabulary associated with this type of task.
The following statements are different ways of asking the same thing!! • Solve the polynomial equation P(x) = 0.
• Find the roots of the polynomial equation P(x) = 0.
• Find the zeroes of the polynomial function P(x) (P(x) = 0).
• Factor the polynomial function P(x) = 0 and express the roots.
How many roots should we expect to find? A polynomial of degree n will have n roots, some of
which may be multiple roots (they repeat). For example, is a
polynomial of degree 3 (highest power) and as such will have 3 roots. This equation is really
giving solutions of x = 1 and x = 4 (repeated).
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Concept 1: Solving Solve for the roots of the functions below. Teacher Modeled Student Try it! f(x) =
f(x) =
Concept 2: Solving Teacher Modeled Student Try it! f(x) =
f(x) =
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Concept 3: Solving
Teacher Modeled Student Try it! f(x) =
f(x) =
Concept 4: Solving
Teacher Modeled Student Try it! f(x) =
f(x) =
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Challenge Summary/Closure:
Exit Ticket:
38
HW ANSWERS
39
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