Unit 6: Polynomial Functions
Lesson Video
Focus Standards
Learning Focus
Additional Resources
Lesson 1: Scott's March Motivation
Develop Understanding
Introduces polynomial functions and examines how the degree of the polynomial relates to the rate of change. Focuses on a cubic function as the sum of a quadratic function.
NC.M3.A-CED.2
NC.M3.A-SSE.1
NC.M3.A-SSE.1.a
NC.M3.A-SSE.1.b
NC.M3.F-BF.1
NC.M3.F-BF.1.a
NC.M3.F-IF.9
Model patterns of growth with tables, equations, graphs, and diagrams.
Make conjectures about function rates of change.
What patterns do you notice, and how do these patterns connect to our understanding of functions we have studied?
Video (first 4 minutes): 1st, 2nd, 3rd differences
Solidify Understand
Examines properties of the cubic function through graphing and comparing to the quadratic function. Reviews key features of graphs, including domain, range, intercepts, and end behavior. Graphs cubic functions using transformations.
Compare quadratic and cubic functions.
Graph cubic functions.
What are the similarities and differences between quadratic functions and cubic functions?
Lesson 3: Pascal's Pride
Solidify Understanding
Extends multiplication skills to higher-order polynomials and provides various strategies for multiplication (area models and multiplying terms horizontally). Introduces the binomial theorem and how to apply Pascal’s Triangle to expand binomials.
Multiply polynomials.
Raise binomials to powers.
How is multiplying polynomials like multiplying integers?
How can I use strategies that I know for multiplying binomials to multiply polynomials of higher degree?
What are the most efficient and accurate strategies for multiplying polynomials?
Extra Practice Problems
Solidify Understand
Builds on prior knowledge of factoring to introduce long division with polynomials. Connects polynomial division with division of whole numbers. Applies the Polynomial Remainder Theorem to determine if a given expression is a factor.
Divide polynomials.
Write equivalent multiplication statements after dividing.
Know when one polynomial is a factor of another polynomial.
How is dividing polynomials like dividing whole numbers?
How is factoring related to division?
How can the remainder of a polynomial division problem be used?
Video: Dividing Polynomials
Video: Remainder Theorem
Solidify Understand
Deepens understanding of the Fundamental Theorem of Algebra and applies it to cubic functions to find roots. Includes problems with real and complex roots to build on understanding of complex numbers.
Find roots and factors of quadratic and cubic functions.
Write quadratic and cubic equations in factored form.
Identify multiple roots of quadratic and cubic functions.
Do all polynomial functions of degree n have n roots?
Solidify Understand
Builds on work with factoring, polynomial long division, and the quadratic formula to find complex roots of polynomials and write polynomial equations in factored form.
Break down a polynomial function to find the roots.
Write a polynomial function in factored form.
How can we find all the factors and roots of a polynomial?
Lesson 7: What's the End Game?
Develop Understand
Examines the end behavior of polynomials to solidify understanding of how the degree of a polynomial affects the rate of change and end behavior.
Find patterns in the end behavior of polynomial functions.
Describe the end behavior of a function using appropriate notation.
What conclusions can be drawn about the end behavior of polynomial functions?
How does the end behavior of polynomials compare to other functions we know?
Practice Understanding
Builds connection among equations, roots, end behavior, operations, and graphs of polynomials. Uses a couple of features of a polynomial to find the rest, requiring the skills to find roots, know end behavior, create graphs, and write equations.
NC.M3.A-APR.3
NC.M3.A.CED.2
NC.M3.A-SSE.1
NC.M3.A-SSE.2
NC.M3.F-BF.1.a
NC.M3.F-IF.7
NC.M3.N-CN.9
Combine pieces of information about polynomials to write equations and graph them.
Identify features of polynomials from equations and graphs.
How do polynomial roots and end behavior work together to make equations and graphs?