Download CCSS.Math.Content.HSA.APRE.A.1

The Complex Number System
Name: ____TEACHER COPY ______ _______
CCSS.Math.Content.HSN.CN.C.9
Date: ______________________ Period: _____
Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Lesson Plan:
1. Hook:
Radical Sign: I love you! So why can't we be together?
Complex Number: It's complex.
2. Introduction and Vocabulary:
In order to understand and simplify complex numbers, we need to first discuss some vocabulary used with them: real
numbers, coefficients, radical sign, square root, commutative, associative, distributive, binomial, trinomial, factor,
polynomial, imaginary numbers and complex numbers.
A wonderful introductory lecture on the fundamental theorem of algebra can be found at
https://www.khanacademy.org/math/algebra2/polynomial_and_rational/fundamental-theorem-ofalgebra/v/fundamental-theorem-of-algebra-intro
3. Guided Practice / Review:
Factor the following expressions using real or complex factors.
1. x2 + 9
2. x2 + 4x + 3
3. x2 + 36
4. 4x2 - 3x + 2
Check student work and review any discrepancies.
4. Independent Practice :
Students will review Fundamental Theorem of Algebra and the quadratic formula and apply them in problems
that involve complex number solutions. After reviewing as a group, have the students work independently on the
specific examples.
Check student work and review any discrepancies.
5. Exit Slip:
Find the real and complex roots of the function p(x) = 4x2 - 3x + 2.
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The Complex Number System
Name: ____________________ ______ _______
CCSS.Math.Content.HSN.CN.C.9
Date: ______________________ Period: _____
Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra states that any polynomial of degree n has n roots. These roots (or
solutions) may be real solutions or complex solutions.
𝑝(𝑥) = 𝑎𝑥 𝑛 + 𝑏𝑥 𝑛−1 + ⋯ + 𝑘
Let's focus on using this theorem with quadratics (n = 2).
Quadratic Formula
For an equation in the form:
Use the discriminant as a quick way to tell how many
and what type of solutions you will have:
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
Use the values of a, b, and c to solve for x:
𝑥=
−𝑏 ±
√𝑏 2
Discriminant
− 4𝑎𝑐
2𝑎
Example:
The discriminant (D) is: D = b2 – 4ac
 If D > 0, then there are 2 real solutions
 If D = 0, then there is 1 real solutions
 If D < 0, then there are 2 complex solutions
Use the discriminant to test how many solutions the
equation to the left has:
3x2 + 5x - 2 = 0
D = b2 – 4ac =
a=
b=
c=
Solve each equation. Include complex solutions.
x2 + 32 = 0
2. 6x2 + 24 = 0
3. x2 + 36 = 0
4. x2 + 100 = 0
5. 3x2 + 108 = 0
6. x2 + 12 = 0
7. x2 - 3x + 3 = 0
8. x2 + 5x + 4 = 0
9. 4x2 - 3x + 2 = 0
1.
Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
All Rights Reserved by NewMathTeacher.Net