Unit 1
Polynomials, Radicals & Rationals

 

Algebra II

 

Suggested Unit Length and Description:

 

45 days

 

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)

 

        Reason quantitatively and use units to solve problems.

        Perform arithmetic operations with complex numbers.

        Use complex numbers in polynomial identities and equations.

        Interpret the structure of expressions.

        Perform arithmetic operations on polynomials.

        Understand the relationship between zeros and factors of polynomials.

        Use polynomial identities to solve problems.

        Rewrite rational expressions.

        Understand solving equations as a process of reasoning and explain the reasoning.

        Solve equations and inequalities in one variable.

        Solve systems of equations.

        Analyze functions using different representations.

        Translate between the geometric description and the equation for a conic section.

 

State Standards:

 

N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.

 

N-CN.A.1 Know there is a complex number i such that i2 = −1, and every complex number has the form a + bi with a and b real.

 

N-CN.A.2 Use the relation i2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

 

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 x2 + 4 as (x + 2i)(x 2i).

 

N-CN.C.9(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

 

A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 y4 as (x2)2 (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 y2)(x2 + y2).

 

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).

 

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.

 

A-APR.C.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 y2)2 + (2xy)2 can be used to generate Pythagorean triples.

 

A-APR.D.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

 

A-APR.D.7(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

 

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.

 

A-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

 

A-REI.B.4b Solve quadratic equations in one variable.

b. Solve quadratic equations by inspection (e.g., for x2 = 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.

 

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 x2 + y2 = 3.

 

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.

 

G-GPE.A.2 Derive the equation of a parabola given a focus and directrix.

 

Focus Standards of 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.

 

Instructional Outcomes:

Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:

 

N-Q.A.2

        I can define descriptive modeling

        I can determine appropriate quantities for the purpose of descriptive modeling

 

N-CN.A.1

        I can define i as the square root of −1 or i2 =-1.

        I can define complex numbers.

        I can write complex numbers in the form a+ bi with a and b being real numbers.

N-CN.A.2

        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 i2 = -1 to simplify.

 

N-CN.C.7

        I can solve quadratic equations that have complex solutions.

 

N-CN.C.8(+)

        I can define identity.

        I can give examples of polynomial identifies.

        I can extend polynomial identities to the complex numbers.

 

N-CN.C.9(+)

        I can state the Fundamental Theorem of Algebra.

        I can verify that the Fundamental Theorem of Algebra is true for second degree quadratic polynomials.

 

A-SSE.A.2

        I can identify patterns of factoring.

        I can factor a polynomial or rational expression.

        I can classify expressions by method of factoring.

 

A-APR.B.2

        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

        I can factor polynomials using any method.

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

 

A-APR.C.4

        I can prove polynomial identities.

 

A-APR.D.6

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

 

A-APR.D.7(+)

        I can add, subtract, multiply, and divide rational expressions.

 

A-REI.A.1

        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

        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.

 

A-REI.B.4b

        I can solve quadratic equations by inspection (e.g., for x2 = 49), 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.

 

A-REI.C.6

        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 focusing on pairs of linear equations in two variables.

 

A-REI.C.7

        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-IF.C.7c

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

 

G-GPE.A.2

        I can define a parabola including the relationship of the focus and the equation of the directrix to the parabolic shape (using y= a(x-h)2 + k)

        I can derive the equation of a parabola given the focus and directrix using y= a(x-h)2 + k

 

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?