Unit 5: Polynomial Functions Algebra II Honors 9 Class Meetings Essential Questions   How do polynomial functions model real-world problems and their solutions? Why are complex numbers necessary for solving polynomial equations? Enduring Understandings with Unit Goals  EU #1: A polynomial function has distinguishing “behaviors”. Its algebraic form gives information about its graph and its graph gives information about its algebraic form. o Analyze a polynomial function for end behavior, turning points, increasing/decreasing intervals and relative maximum/minimum. o Connect standard and factored polynomial equations to linear factors, zeros, and roots. o Solve polynomial equations using a variety of methods.  EU #2: You can divide polynomials using steps that are similar to the long division steps that you use to divide whole numbers. o Divide polynomials.  EU #3: The degree of a polynomial equation tells you how many roots there are in an equation. o Find real and complex solutions to polynomial equations. Standards Common Core State Standards/College and Career Readiness Anchor Standards :       CCSS.Math.Content.HSF.IF.C.7.C – Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. CCSS.Math.Content.HSA.REI.D.11 – Explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions, use technology to graph the functions. CCSS.Math.Content.HSA.APR.B.2 – Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x-a is p(a), so p(a)=0 if and only if (x-a) is a factor of p(x). CCSS.Math.Content.HSN.CN.C.7 – Solve quadratic equations with real coefficients that have complex solutions. CCSS.Math.Content.HSN.CN.C.8 – Extend polynomial identities to the complex numbers. CCSS.Math.Content.HSN.CN.C.9 – Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. 1 Unit 5: Polynomial Functions Algebra II Honors 9 Class Meetings MSMHS 21st Century Learning Expectations Competency 1: Read and write effectively for a variety of purposes. Competency 3: Employ mathematical problem-solving skills effectively. Competency 7: Make decisions and solve problems in independent and collaborative settings. Competency 8: Take responsibility for his/her own learning and behavior. Unit Content Overview 1. Polynomial Functions (5-1)  Classifying polynomial functions  Describe end behavior  Find turning points  Describe increasing and decreasing intervals 2. Polynomials, Linear Factors, and Zeros (5-2)  Writing polynomials in factored form  Roots, zeros, and x-intercepts (what do they mean and how are they related)  Finding zeros of a polynomial  Writing a polynomial function from its zeros  How multiple zeros affect a graph (finding multiplicity)  Identifying relative minimum/maximum  Maximizing volume 3. Solving Polynomial Equations (5-3)  Solving polynomial equations using factors  Finding real roots by graphing 4. Dividing Polynomials (5-4)  Using polynomial long division  Checking factors  Using synthetic division  The Remainder Theorem  Evaluating a polynomial 5. Rational Root Theorem (5-5)  Finding a rational root 6. The Fundamental Theorem of Algebra  The Fundamental Theorem of Algebra  Finding all the zeros of a polynomial function 2 Unit 5: Polynomial Functions Algebra II Honors 9 Class Meetings Learning Objectives Students will be able to…              Classify a polynomial function. Determine the end behavior of a polynomial function. Identify the turning points of a polynomial function. Determine intervals of decreasing and increasing behavior. Transform a polynomial into factored form. Identify the zeros of a polynomial function by using both a graph and an equation. Create a polynomial function given its factors or zeros. Explain how multiple zeros affect a graph. Identify a relative maximum and minimum. Solve polynomial equations by factoring and graphing. Divide polynomials using long division and synthetic division. Solve equations using the Rational Root Theorem Apply the Fundamental Theorem of Algebra to solve polynomial equations with complex solutions. Assured Learning Experiences Differentiated Instruction/Instructional Strategies  Lecture with notes  Guided notes  Accountable talk  Student-led instruction  Flipped classroom videos o Student notes  Independent problem-solving  Collaborative problem-solving  Performance Task (practical application) o Fish Tank Dimensions Project  Cross-curricular problem solving (independent and collaborative) Interdisciplinary Connections  Language Arts - Word problems  Marine Science – Word problems 3 Unit 5: Polynomial Functions Algebra II Honors 9 Class Meetings Assessments FORMATIVE ASSESSMENTS:           Checking flipped classroom notes (highlight, underline, star important information from the video) Warm-up – classifying polynomials, end behavior Warm-up – sketching a graph using end behavior, degree, roots, and multiplicity Warm-up – solving polynomial equations Warm-up – dividing polynomials ABCD check-up cards Exit Slip – FToA (find all of the zeros of a polynomial function) Student-led instruction Rubric #3 – Mathematics o Performance Task – Fish Tank Dimensions Project Rubric #7 – Problem Solving o Performance Task – Fish Tank Dimensions Project SUMMATIVE ASSESSMENTS:       Polynomial Functions, Linear Factors, and Zeros Quiz 5-1, 5-2 – EU #1 Solving Polynomial Equations Quiz 5-3 – EU #1 Dividing Polynomials Quiz 5-4 – EU #2 Fundamental Theorem of Algebra Quiz 5-5, 5-6 – EU #3 Unit 5 Test Fish Tank Dimensions Project 4