Unit 1
Polynomials, Radicals & Rationals

 

Algebra II

 

Unit Description:

 

Building on their work with linear, quadratic, and exponential functions in Algebra I, students extend their repertoire of functions to include polynomial, rational, and radical functions. In this Algebra II course, rational functions are limited to those whose numerators are of degree at most 1 and denominators of degree at most 2. Radical functions are limited to square roots or cube roots of at most quadratic polynomials. Certain standards in this course require students to revisit the topics of linear, quadratic and/or exponential functions to build conceptual understanding.

Unit One develops the structural similarities between the system of polynomials and the system of integers. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. The unit culminates with the fundamental theorem of algebra. Rational numbers extend the arithmetic of integers by allowing division by all numbers except 0. Similarly, rational expressions extend the arithmetic of polynomials by allowing division by all polynomials except the zero polynomial. A central theme of this unit is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. Students revisit the topic of systems of linear and quadratic equations to continue to build conceptual understanding. (Mathematics Appendix A, p.36-38, with adjustments)

Standards for Mathematical Practice

MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

Louisiana Student Standards for Mathematics (LSSM)

A-SSE: Algebra-Seeing Structure in Expressions

A. Interpret the structure of expressions

A-SSE.A.2

 

 

Use the structure of an expression to identify ways to rewrite it. For example, see as , thus recognizing it as a difference of squares that can be factored as .

*I can identify patterns of factoring.

*I can factor a polynomial or rational expression.

*I can classify expressions by method of factoring.

 

A-APR: Arithmetic with Polynomials and Rational Expressions

B. Understand the relationship between zeros and factors of polynomials

A-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).

*I can define the remainder theorem.

*I can use the remainder theorem to show the relationship between a factor and a zero.

 

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.

*I can factor polynomials using any method.

*I can sketch graphs of polynomials using zeroes and a sign chart.

 

C. Use polynomial identities to solve problems

A-APR.C.4

 

 

Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity can be used to generate Pythagorean triples.

*I can prove polynomial identities.

 

D. Rewrite rational expressions

A-APR.D.6

 

 

Rewrite simple rational expressions in different forms; write in the form , where , , , and are polynomials with the degree of less than the degree of , using inspection, long division, or, for the more complicated examples, a computer algebra system.

*I can rewrite rational expressions using inspection or by long or synthetic division.

A-REI: Reasoning with Equations and Inequalities

A. 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.

*I can demonstrate that solving an equation means that the equation remains balanced during each step.

*I can recall the properties of equality.

*I can explain why, when solving equations, it is assumed that the original equation is equal.

*I can determine if an equation has a solution.

*I can choose an appropriate method for solving the equation.

*I can justify solution(s) to equations by explaining each step in solving a simple equation using the properties of equality, beginning with the assumption that the original equation is equal.

*I can construct a mathematically viable argument justifying a given, or self-generated, solution method.

A-REI.A.2

 

 

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

*I can determine the domain of a rational function.

*I can determine the domain of a radical function.

*I can solve radical equations in one variable.

*I can solve rational equations in one variable.

*I can give examples showing how extraneous solutions may arise when solving rational and radical equations.

B. Solve equations and inequalities in one variable

A-REI.B.4b

 

 

Solve quadratic equations in one variable.

b. Solve quadratic equations by inspection (e.g., for ), 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 for real numbers and .

*I can solve quadratic equations by inspection (e.g., for ), taking square roots, completing the square, the quadratic formula and factoring.

*I can determine appropriate strategies (see first knowledge target listed) to solve problems involving quadratic equations, as appropriate to the initial form of the equation.

*I can recognize when the quadratic formula gives complex solutions.

C. Solve systems of equations

A-REI.C.6

 

 

Solve systems of linear equations exactly and approximately (e.g., with graphs), limited to systems of at most three equations and three variables. With graphic solutions, systems are limited to two variables.

*I can solve systems of linear equations by any method.

*I can justify the method used to solve systems of linear equations exactly and approximately.

*I can graph systems of two linear equations.

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 and the circle .

*I can transform a simple system consisting of a linear equation and quadratic equation in 2 variables so that a solution can be found algebraically and graphically.

*I can explain the correspondence between the algebraic and graphical solutions to a simple system consisting of a linear equation and a quadratic equation in 2 variables.

F-BF: Building Functions

B. Build new functions from existing functions

F-BF.B.3

 

 

Identify the effect on the graph of replacing by , , , and for specific values of (both positive and negative); find the value of 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.

*I can perform transformation on functions which may involve simple radical, rational, polynomial, exponential, logarithmic. (Simple exponential and logarithmic functions were introduced in Algebra I).

*I can identify the effect a single transformation will have on the function (symbolic or graphic).

*I can use technology to identify effects of single transformations on graphs of functions.

*I can describe the differences and similarities between a parent function and the transformed function.

*I can find the value of , given the graphs of a parent function, f(x), and the transformed function: , , , and .

*I can recognize even and odd functions from their graphs and from their equations.

*I can identify transformations of a function on a graph.

*I can describe the effects of transformations on parent functions.

F-IF: Interpreting Functions

C. Analyze functions using different representations

F-IF.C.7c

 

 

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

*I can graph polynomial, rational, and radical functions accurately.

 

N-Q: Quantities

A. Reason quantitatively and use units to solve problems.

N-Q.A.2

 

 

Define appropriate quantities for the purpose of descriptive modeling.

*I can define descriptive modeling

*I can determine appropriate quantities for the purpose of descriptive modeling

 

N-CN: The Complex Number System

A. Perform arithmetic operations with complex numbers

N-CN.A.1

 

 

Know there is a complex number such that , and every complex number has the form with and real.

*I can define as the square root of −1 or .

*I can define complex numbers.

*I can write complex numbers in the form with and being real numbers.

N-CN.A.2

 

 

Use the relation and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

*I can recognize that the commutative, associative, and distributive properties extend to the set of complex numbers over the operations of addition and multiplication.

*I can use the relation to simplify.

 

C. Use complex numbers in polynomial identities and equations.

N-CN.C.7

 

 

Solve quadratic equations with real coefficients that have complex solutions.

*I can solve quadratic equations that have complex solutions.

 

A-CED: Creating Equations

A. Create equations that describe numbers or relationships

A-CED.A.1

 

 

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

*I can solve equations specified in this standard in one variable.

*I can solve inequalities specified in this standard in one variable.

*I can describe the relationships between the quantities in the problem (for example, how the quantities are changing or growing with respect to each other); express these relationships using mathematical operations to create an appropriate equation or inequality to solve.

*I can create equations and inequalities specified in this standard in one variable and use them to solve problems.

*I can create equations and inequalities specified in this standard in one variable to model real-world situations.

*I can compare and contrast problems that can be solved by the different types of equations specified in this standard.

Enduring Understandings:

         Odd functions begin and end in opposite directions. 

         Even functions begin and end in the same direction. 

         Zeros are the x intercepts of an equation and can be used to find factors and write a polynomial equation.

         Odd and even functions have graphs that are symmetric with respect to the origin or y-axis. 

         Zeros of a polynomial can be found by factoring or by graphing the polynomial.

         Rational functions can be represented as fractional exponents and follow the same rules as regular exponents.

         Solve rational expressions.

         Computational skills applicable to numerical fractions also apply to rational expressions involving variables.

         Radical expressions can be written and simplified using rational exponents.

         Graphs of radical functions look like curves, are not symmetric and can be transformed using predictive indicators. 

         Radicals are the opposite of exponents. 

         Only radicals with a common radicand and index can be added or subtracted. 

         Radical equations can be solved by graphing or inverses.

 

 

Essential Questions:

         How do exponent value, zeros, and factors affect the appearance of a graph?

         How can a polynomial inequality be solved? 

         How can real world data be used to generate a polynomial model? 

         What does it mean to be an odd or even function? 

         How do the elements of a polynomial equation determine its general shape? 

         How can the solutions or zeros of a polynomial be found?  

         How do the zeros of a polynomial relate to its graph?

         What is a rational function and what does its graph look like?

         How are operations extended to rational functions?

         How are rational expressions simplified?

         How are rational equations solved?

         How are rational equations graphed?

         What are radicals and how can they be simplified?

         What do graphs of radical functions look like?

         How can a radical function be identified from its graph?

         How can a radical expression be simplified and combined?

         How can a radical equation be solved?

         How can operations be extended to radical expressions and equations?