Module 1: Students connect polynomial arithmetic to computations with whole numbers and integers. Students learn that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. This unit helps students see connections between solutions to polynomial equations, zeros of polynomials, and graphs of polynomial functions. Polynomial equations are solved over the set of complex numbers, leading to a beginning understanding of the fundamental theorem of algebra. Application and modeling problems connect multiple representations and include both real world and purely mathematical situations.
Unit 1: Polynomials from Base 10 to Base X (A.SSE.2, A.APR.4)
Students begin this module with Topic A, Constructions. Major constructions include an equilateral triangle, an angle bisector, and a perpendicular bisector. Students synthesize their knowledge of geometric terms with the use of new tools and simultaneously practice precise use of language and efficient communication when they write the steps that accompany each construction.
Unit 2: Factoring: It's Uses and ObstaclesPolynomials from Base 10 to Base X (N.Q.2, A.SSE.2, A.APR.2)
This topic focuses on factoring polynomials and the advantages of factored form of a polynomial to both solve equations and sketch graphs of polynomial functions. Students solve problems involving real world situations and develop fluency with creating equations and functions given a verbal description, visual representation or graph. This topic concludes with a discussion of polynomial division with remainder, further strengthening the connection between the remainder, the factors and zeros of a polynomial equation, and graphs of polynomial functions.
Unit 3: Solving and Applying Equations (Polynomial, Rational and Radical) (A.APR.6, A.REI.1, 2)
Students solve polynomial, rational, and radical equations, and apply these types of equations to real-world situations. They examine the conditions under which an extraneous solution is introduced. They rewrite rational expressions in different forms and work with radical expressions as part of this process. Students work with systems of equations that include quadratic and linear equations and apply their work to understanding the definition of a parabola.
Unit 4: Complex Numbers Overcome All Obstacle (N.CN.1, 2, 7)
Students extend their facility with solving polynomial equations to working with complex zeros. Complex numbers are introduced via their relationship with geometric transformations. The topic concludes with students realizing that every polynomial function can be written as a product of linear factors, which is not possible without complex numbers.