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Advanced Algebra Support Unit 2: Polynomial Functions KEY STANDARDS ADDRESSED: Use complex numbers in polynomial identities and equations MCC9‐12.N.CN.8 (+) Extend polynomial identities to the complex numbers. MCC9‐12.N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Interpret the structure of expressions. MCC9‐12.A.SSE.1 Interpret expressions that represent a quantity in terms of its context. MCC9‐12.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. MCC9‐12.A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. MCC9‐12.A.SSE.2 Use the structure of an expression to identify ways to rewrite it. Write expressions in equivalent forms to solve problems MCC9‐12.A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. Perform arithmetic operations on polynomials MCC9‐12.A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Understand the relationship between zeros and factors of polynomials MCC9‐12.A.APR.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). MCC9‐12.A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Use polynomial identities to solve problems MCC9‐12.A.APR.4 Prove polynomial identities and use them to describe numerical relationships. n MCC9‐12.A.APR.5 (+) Know and apply that the Binomial Theorem gives the expansion of (x + y) in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) Solve systems of equations MCC9‐12.A.REI.7 Solve a simple system consisting of a linear equation and a polynomial equation in two variables algebraically and graphically. Represent and solve equations and inequalities graphically MCC9‐12.A.REI.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 approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial functions. Analyze functions using different representations MCC9‐12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. MCC9‐12.F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Wed/T Sept 10H 11 EQ: How can we write a polynomial in standard form? Worksheet 1 EQ: How do we add, subtract, multiply, and divide 6.2 Algebra 2 Book 6.3 Algebra 2 Book polynomials? EQ: In which operations does closure apply? Interpret the structure of expressions. MCC9‐12.A.SSE.1 Interpret expressions that represent a quantity in terms of its context. MCC9‐12.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. MCC9‐12.A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. MCC9‐12.A.SSE.2 Use the structure of an expression to identify ways to rewrite it. Perform arithmetic operations on polynomials MCC9‐12.A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Fri Tue Sept 12 16 EQ: How are polynomials factored? Worksheet 2 EQ: How can we write polynomials in factored form? 6.4 Algebra 2 Book Interpret the structure of expressions. MCC9‐12.A.SSE.1 Interpret expressions that represent a quantity in terms of its context. MCC9‐12.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. MCC9‐12.A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. MCC9‐12.A.SSE.2 Use the structure of an expression to identify ways to rewrite it. Wed/T Sept 17 H 18 EQ: How are polynomials solved by factoring? MCC9‐12.A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. 6.4 Algebra 2 Book Fri Sept 19 QUIZ Mon – Sept 22 Friday – 26 Quiz over writing polynomials in standard form, adding, subtracting, and multiplying polynomials, factoring polynomials, solve polynomials by factoring EQ: What is the Remainder Theorem and what does it tell us? EQ: How are polynomials divided using long division and synthetic division? Worksheet 3 Worksheet 4 Worksheet 5 6.5 Algebra 2 Book Understand the relationship between zeros and factors of polynomials MCC9‐12.A.APR.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). MCC9‐12.A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. MonWed Sept 29 – Oct 1 EQ: What is the Rational Root Theorem and what does it tell us? MCC9‐12.A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Worksheet 6 6.6 Alg 2 Book ThMon EQ: What is the Fundamental Theorem Algebra and what does it tell us? Oct 2-6 EQ: Which sets of numbers can be solutions to polynomial equations? EQ: How can a write a polynomial equation given the zeros of the function? 6.7 Alg 2 Book Worksheet 7 Worksheet 8 Worksheet 9 MCC9‐12.N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Use complex numbers in polynomial identities and equations MCC9‐12.N.CN.8 (+) Extend polynomial identities to the complex numbers. Tue Oct 7 Wed Oct 8 Thur- Oct 9 Fri 10 QUIZ Quiz over dividing polynomials, solving polynomials for all zeros, writing equations of polynomials EQ: How is the end behavior of a polynomial function determined? Graphing Investigation Analyze functions using different representations MCC9‐12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. MCC9‐12.F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 6.8 Lesson Opener EQ: How are polynomials graphed? 6.8 Algebra 2 Book EQ: What are the important characteristics of polynomial graphs? Worksheet 10 \MCC9‐12.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. MCC9‐12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Mon Oct 13 Student Holiday Worksheet 10 Tue – Wed Oct 14 15 EQ: How is a system of equations with a polynomial and linear equation solved algebraically? Worksheet 11 Solve systems of equations MCC9‐12.A.REI.7 Solve a simple system consisting of a linear equation and a polynomial equation in two variables algebraically and graphically. Represent and solve equations and inequalities graphically MCC9‐12.A.REI.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 approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial functions. Thur- Oct 16 Fri 17 EQ: How are polynomial inequalities solved algebraically? Worksheet 12 Represent and solve equations and inequalities graphically MCC9‐12.A.REI.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 approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial functions. Mon Oct 27 Review MonTue Oct 27 28 EQ: How can we apply Pascal’s Triangle to expand? MCC9‐12.A.APR.5 (+) Know and apply that the n Binomial Theorem gives the expansion of (x + y) in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) WedThur Oct 29 – 30 Review for Polynomial Unit Test Essential Questions Due Worksheet 13 Pascal’s Triangle Activity 1 Activity 2 Activity 3 Worksheet 14 12.2 Algebra 2 Book Fri Oct 31 Polynomial Unit Test Notebook Due Essential Questions- Answer essential questions in complete sentences and provide an example of each. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. How can we write a polynomial in standard form? How can we write a polynomial in factored form? How do we add, subtract, multiply, and divide polynomials? In which operations does closure apply? How can we apply Pascal’s Triangle to expand? What is the Remainder Theorem and what does it tell us? What is the Rational Root Theorem and what does it tell us? What is the Fundamental Theorem Algebra and what does it tell us? How can we solve polynomial equations? Which sets of numbers can be solutions to polynomial equations? What is the relationship between zeros and factors? What characteristics of polynomial functions can be seen on their graphs? How can we solve a system of a linear equation with a polynomial equation? Vocabulary – Define and give an example of each Coefficient Degree Power Multiplicity Pascal’s Triangle Remainder Theorem x-intercept zero of a function polynomial function end behavior polynomial long division Synthetic Division Rational Zero Theorem Fundamental Theorem of Algebra Local Maximum Local Minimum Increasing Interval Decreasing Interval Real Numbers Integers Rational Numbers Irrational Numbers Complex Numbers Imaginary Numbers Zero product property Natural Numbers Whole Numbers