Unit 4 - Polynomials 4 weeks Overview: By introducing the basic concept of complex number, the form of complex numbers and basic operations with complex numbers, Unit 4 starts to lay the foundation for working with complex numbers which will be further extended in this unit. There are multiple connections among standards in this unit that are linked to prior grades, but also the connection to several mathematical practice standards are found in Unit 4. Students should have multiple opportunities to engage in mathematical practice standards 3,5 and 7. (Schwols and Dempsey, 2012. Common Core Standards for High School Mathematics: A Quick-Start Guide) Essential Question(s): In general, how many data points do you need to find the degree of a polynomial? What s the difference between zeros, roots, solutions, and x-intercepts? Why do we bother with different forms of a function? 8 Mathematical Practices (MP) 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Content Standards MP Student Skills Reason quantitatively and use units to solve problems.* N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. Perform arithmetic operations with complex numbers. N-CN.A.1 Know there is a complex number i such that i 2 = -1, and 7 every complex number has the form a + bi with a and b real. N-CN.A.2 Use the relation i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply 6,7 R e v i s e d 1 0. 1 5. 1 5 Page 1
complex numbers. N-CN.A.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Represent complex numbers and their operations on the complex plane. N-CN.B.4 (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. N-CN.B.5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, ( 1 3 i ) 3 8 because ( 1 3 i) has modulus 2 and argument 120 N-CN.B.6 (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. Use complex numbers in polynomial identities and equations N-CN.C.7 Solve quadratic equations with real coefficients that have complex solutions. N-CN.C.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x 2 + 4 as (x + 2i)(x 2i) N-CN.C.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Interpret the structure of expressions A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 4 2 2 2 2 y as ( x ) ( y ), thus recognizing it as a difference of squares that can be 2 2 2 2 factored as ( x y )( x y ) 5 7 R e v i s e d 1 0. 1 5. 1 5 Page 2
Write expressions in equivalent forms to solve problems A-SSE.B.3* Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 1,,7 Understand the relationship between zeros and factors of polynomials A-APR.B.2 Know and apply the Remainder Theorem: For a polynomial 3 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). A-APR.B.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. 3 Use polynomial identities to solve problems A-APR.C.4 Prove polynomial identities and use them to describe 3 numerical relationships. For example, the polynomial identity (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. A-APR.C.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y) n 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. Understand solving equations as a process of reasoning and explain the reasoning A-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 3,6 R e v i s e d 1 0. 1 5. 1 5 Page 3
Solve equations and inequalities in one variable A-REI.B.4 Solve quadratic equations in one variable. b. Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a +/- bi for real numbers a and b. Solve systems of equations A-REI.C. 6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. A-REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x 2 + y 2 = 3. Represent and solve equations and inequalities graphically A-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 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, rational, absolute value, exponential, and logarithmic functions.* Interpret functions that arise in applications in terms of the context F-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is 1,2,3,6,7 1,3,5 4,6 R e v i s e d 1 0. 1 5. 1 5 Page 4
F-IF.B.6* increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.* Analyze functions using different representations F-IF.C.7* F-IF.C.8 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as t t 12t 10 y (1.02), y (0.97), y (1.01), y (1.2), and classify them as representing exponential growth or decay. F-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Build a function that models a relationship between two quantities F-BF.A.1* Write a function that describes a relationship between two quantities.* t 1,4,5,7 1,5,6 3,7 1,3,5,6,8 R e v i s e d 1 0. 1 5. 1 5 Page 5
a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Build new functions from existing functions F-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, 3,5,7 kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F-BF.B.4a Find the inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression 3 for the inverse. For example, f ( x) 2x or ( x 1) f ( x) for x 1 ( x 1) 4 Translate between the geometric description and the equation for a conic section G-GPE.A.2 Derive the equation of a parabola given a focus and directrix. 3 Planning for Instruction Uses mathematical properties and structure of polynomial expressions to create equivalent expressions that aid in solving mathematical and contextual Resources Tool Box Unit Readiness Check Summative Assessment Questions R e v i s e d 1 0. 1 5. 1 5 Page 6
problems with two steps required. Uses mathematical properties and relationships to reveal key features of polynomial functions, using them to sketch graphs and identify characteristics of the relationship between two quantities, and applying the remainder theorem where appropriate. Calculates and interprets the average rate of change of polynomial functions (presented symbolically or as a table) over a specified interval, and estimates the rate of change from a graph. Builds functions that model mathematical and contextual situations, and uses the models to solve, interpret and generalize about problems. Given multiple functions in different forms (algebraically, graphically, numerically and by verbal description), writes multiple equivalent versions of the functions, and identifies and compares key features. Graphs polynomial functions, showing key features. Uses commutative, associative and distributive properties to perform operations with complex numbers. Identifies the effects of multiple transformations on graphs of polynomial functions, and determines if the resulting function is even or odd. Solves multi-step contextual word problems involving quadratic (with real or complex solutions). Clearly constructs and communicates a complete response based on the graph of an equation in two variables, the principle that a graph is a solution set or the relationship between zeros and factors of polynomials. Vocabulary: R e v i s e d 1 0. 1 5. 1 5 Page 7