Students analyze features of polynomials using their equations and graphs, perform arithmetic operations on polynomials, and use polynomial identities to solve problems.
In Unit 3, Polynomials, students will apply skills from the first two units to develop an understanding of the features of polynomial functions. Analysis of polynomial functions for degree, end behavior, number, and type of solutions builds on the work done in Unit 2; advanced topics that will be applied to future function types. Students will write polynomial functions to reveal features of the functions, find solutions to systems, and apply transformations, building from Units 1 and 2. Students will be introduced to the idea of an “identity” in this unit as well as operate with polynomials. Division of polynomials is introduced in this unit and will be explored through the concepts of remainder theorem as well as a prerequisite to rational functions.
Unit 3 begins with Topic A, Polynomial Features and Graphs, where students dive into the features of polynomial functions—focusing on end behavior, real and complex roots, where a function is positive or negative, and review of transformations. In this part of the unit, students will focus on looking for structure in equations indicating roots, degree, leading coefficient, etc., and apply this knowledge to graphs, and vice versa. Students will be introduced to new tools, such as sign charts, to aid in the sketch of a polynomial graph.
In Topic B, Operations with Polynomials, students focus on precision of calculations as well as expressing regularity in repeated reasoning with the binomial theorem, the remainder theorem, and various factoring patterns. Students will make connections between Topic A and Topic B by identifying linear factors, number and kind of roots, and end behavior.
In Topic C, Polynomial Extensions, students will use strategies learned throughout the unit to find the solution to systems of polynomial functions, write polynomial functions, and identify Pythagorean triples. Students will need to identify appropriate “tools” (procedures) that will lead them to the solution of various mathematical problems.
The features of polynomial functions, such as end behavior and function behavior, and the operations with polynomials, such as factoring and division, will be used in the next unit, Rational Functions. The conceptual knowledge gained in this unit will be essential to fully understanding rational functions.
|leading coefficient||Fundamental Theorem of Algebra||difference of two squares|
|pythagorean triples||conjugate||polynomial long division|
|successive differences||end behavior||complex root/real root|
|factors||remainder theorem||difference/sum of two cubes|
|polynomial identities||rate of change|
Internalization of Standards via the Unit Assessment
Internalization of Trajectory of Unit
This assessment accompanies Unit 3 and should be given on the suggested assessment day or after completing the unit.
Classify polynomials through identification of degree and leading coefficient. Graph a polynomial function from a table of values; prove degree using successive differences.
Identify features of polynomial functions including end behavior, intervals where the function is positive or negative, and domain and range of function.
Match and compare equations and graphs of polynomials, and identify transformations.
Sketch polynomial functions using sign charts and analysis of the factored form of the polynomial function.
Multiply polynomials, identifying factors, degrees, and number of real roots. Identify features of a polynomial in standard form.
Add and subtract polynomials. Identify degree, leading coefficient, and end behavior of result.
Divide polynomials by binomials to determine linear factors.
Determine if a binomial is a factor of a polynomial using the remainder theorem.
Identify and factor with difference of two squares in quadratic and quartic polynomials. Describe identity of difference of two squares. Describe the zeros that represent the resultant factors.
Factor difference and sums of cubes. Explain sums and differences of cubes as polynomial identities.
Factor polynomials by grouping in quartic, cubic, and quadratic functions.
Key: Major Cluster Supporting Cluster Additional Cluster