Unit 2 –Modeling with Polynomial Functions
Part 1 – Operations with Polynomials/Quadratics
**Subject to change**
Day
Date
Lesson
Wed
9/10
Polynomial Operations
(Adding, Subtracting, Multiplying)
Thurs
9/11
Fri
9/12
Factoring Polynomials Practice
Mon
9/15
Quiz 1
Complex Number System
Yes, there is a number called i
Tues
9/16
Derive Equation of Parabola
Properties of Parabola & Vertex Form
Wed
9/17
Factoring Polynomials Review
Vertex Form & Completing the Square
Thurs
9/18
Solving Quadratics by Completing the
Square (Review solving using intercepts and
square root method)
Fri
9/19
Proving the Quadratic Formula and Solving
using the Quadratic Formula
Quiz 2
Discriminant & Determining Types of
Solutions
Mon
9/22
Tues
9/23
Review for Test
Wed
9/24
TEST – Unit 2 Part 1
Thurs
9/25
No School – Teacher Workday
Write your homework here!
Math III Unit 2: Modeling with Polynomial Functions Unit Guide
Unit Description
In this unit, students will continue their study of quadratic functions and generalize concepts learned about
quadratic and power functions to polynomial functions. Students will be introduced to complex numbers and
use both the quadratic formula and completing the square to solve quadratic equations. Students will
investigate relationships between degree, factors, and zeroes of a polynomial function.
Essential Questions
By the end of this unit, I will be able to answer the following questions:
 Evaluate which representations of a function are most useful for solving problems in different
mathematical and real world settings. Note: This statement does not have the same meaning in question
format, so we left it the way it is.
 How are the key features identified, described, and interpreted from different representations of
polynomial functions?
 How are the factors, zeroes (both real and complex), and degree of a polynomial related and how do they
determine the shape of the graph of a polynomial function?
 How do the properties of complex numbers compare to the properties of real numbers?
Enduring Understandings
I understand that . . .
 The degree of a polynomial determines the number of solutions or zeros of the corresponding equation or
function.
 There is a complex number 𝑖 such that 𝑖 2 = – 1, and every complex number has the form 𝑎 + 𝑏𝑖.
Unit Skills
I can . . .
Use properties and operate with rational, irrational, and complex numbers
 Explain why the sum or product of two rational numbers is rational. (N-RN.3)
 Explain why the sum of a rational number and an irrational number is irrational. (N-RN.3)
 Explain why the product of a nonzero rational number and an irrational number is irrational. (N-RN.3)
 Add, subtract, and multiply complex numbers. (N-CN.2)
Solve quadratic equations and graph quadratic functions
 Solve quadratic equations with real coefficients that have complex solutions. (N-CN.7)
 Solve quadratic equations by inspection, taking square roots, factoring, completing the square, and using
the quadratic formula. (A-REI.4a,b)
 Determine which method for solving a quadratic equation is most appropriate based on the initial form of
the equation. (A-REI.4b)
 Derive the quadratic formula using the process of completing the square. (A-REI.4a)
 Recognize when the quadratic formula gives complex solutions and write them as 𝑎 ± 𝑏𝑖 for real numbers
𝑎 and 𝑏. (A-REI.4b)
 Show that the Fundamental Theorem of Algebra is true for quadratic polynomials. (N-CN.9)
 Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it
defines. (A-SSE.3b)


Use the process of factoring and completing the square in a quadratic function to show zeros, extreme
values, and symmetry of the graph, and interpret these in terms of a context. (F-IF.8a)
Derive the equation of a parabola given a focus and directrix. (G-GPE.2)
Generalize concepts about quadratic functions to polynomials of higher degree
 Add, subtract, and multiply polynomials. (A-APR.1)
 Solve polynomial equations and systems of polynomial equations approximately by using technology to
graph the functions they define. (A-REI.11)
 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a
rough graph showing key features of the function defined by the polynomial. Key features include
intercepts, relative maxima and minima, and end behavior. (A-APR.3, F-IF.7c)
 Prove polynomial identities and use them to describe numerical relationships. (A-APR.4)
Unit Facts
I know that . . .
𝑚
 A rational number is a real number that can be expressed in the form 𝑛 where 𝑚 and 𝑛 are integers.


An irrational number is a real number that cannot be expressed as the ratio of integers.
A real number is a value that represents a quantity along a continuous number line.


There is a complex number 𝑖 such that 𝑖 2 =– 1.
Every complex number can be written in the form 𝑎 + 𝑏𝑖 for real numbers 𝑎 and 𝑏. For real numbers,
𝑏 = 0.
The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with
complex coefficients has at least one complex root.
The quadratic formula can be used to solve a quadratic equation in standard form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0. The


formula is 𝑥 =








−𝑏±√𝑏 2 −4𝑎𝑐
2𝑎
.
A parabola can be defined geometrically as the locus of points equidistant from a given point, called the
focus, and a given line, called the directrix.
A polynomial of degree 𝑛 is a function of the form 𝑓(𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0 ,
where each coefficient 𝑎𝑘 is a real number, 𝑎0 ≠ 0, and 𝑛 is a non-negative integer.
For each polynomial with complex coefficients, the number of roots is equal to the degree of the
polynomial (taking multiplicity into account).
The degree of a single-variable polynomial is equal to the value of the largest exponent of the variable
terms.
The multiplicity of a root is equal to the power of the factor that corresponds to that root when the
polynomial is written in factored form.
Higher order polynomials often produce relative maxima and minima as opposed to absolute maxima and
minima.
The values of the zeroes, x-intercepts, solutions, and factors of a polynomial are related.
Irrational and imaginary solutions always occur in pairs.