Polynomials

Students analyze features of polynomials using their equations and graphs, perform arithmetic operations on polynomials, and use polynomial identities to solve problems.

Math

Unit 3

11th Grade

Unit Summary

In Unit 3, students will apply skills from the first two units to develop an understanding of the features of polynomial functions. Analysis of polynomial functions for degree, end behavior, number, and type of solutions builds on the work done in Unit 2; advanced topics that will be applied to future function types. Students will write polynomial functions to reveal features of the functions, find solutions to systems, and apply transformations, building from Unit 1 and Unit 2. Students will be introduced to the idea of an “identity” in this unit as well as operate with polynomials. Division of polynomials is introduced in this unit and will be explored through the concepts of remainder theorem as well as a prerequisite to rational functions.

Unit 3 begins with Topic A, Polynomial Features and Graphs, where students dive into the features of polynomial functions—focusing on end behavior, real and complex roots, where a function is positive or negative, and review of transformations. In this part of the unit, students will focus on looking for structure in equations indicating roots, degree, leading coefficient, etc., and apply this knowledge to graphs, and vice versa. Students will be introduced to new tools, such as sign charts, to aid in the sketch of a polynomial graph.

In Topic B, Operations with Polynomials, students focus on precision of calculations as well as expressing regularity in repeated reasoning with the binomial theorem, the remainder theorem, and various factoring patterns. Students will make connections between Topic A and Topic B by identifying linear factors, number and kind of roots, and end behavior.

In Topic C, Polynomial Extensions, students will use strategies learned throughout the unit to find the solution to systems of polynomial functions, write polynomial functions, and identify Pythagorean triples. Students will need to identify appropriate “tools” (procedures) that will lead them to the solution of various mathematical problems.

The features of polynomial functions, such as end behavior and function behavior, and the operations with polynomials, such as factoring and division, will be used in the next unit, Rational Functions. The conceptual knowledge gained in this unit will be essential to fully understanding rational functions.

Pacing: 17 instructional days (14 lessons, 1 flex day, 1 assessment day)

Assessment

The following assessments accompany Unit 3.

Post-Unit

Use the resources below to assess student understanding of the unit content and action plan for future units.

Post-Unit Assessment

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

Internalization of Standards via the Unit Assessment

Take unit assessment. Annotate for:
- Standards that each question aligns to
- Purpose of each question: spiral, foundational, mastery, developing
- Strategies and representations used in daily lessons
- Relationship to Essential Understandings of unit
- Lesson(s) that assessment points to

Internalization of Trajectory of Unit

Read and annotate “Unit Summary."
Notice the progression of concepts through the unit using “Unit at a Glance.”
Do all target tasks. Annotate the target tasks for:
- Essential understandings
- Connection to assessment questions

Essential Understandings

The central mathematical concepts that students will come to understand in this unit

Polynomials are described by their degree and leading coefficient. By knowing this information, you can describe the end behavior and number of roots, and make predictions about the shape of the graph.
Solutions of polynomial functions represent the intersection of two polynomial functions or the intersection of the function across the x-axis. The method to finding these intersections include systems of equations, factoring and factoring patterns, quadratic formula, and long division. Solutions to polynomial functions may be either real or complex.
By analyzing the structure of a polynomial function represented as a table, equation, or graph, you can determine efficient methods and procedures to finding solutions, features, and alternative representations of polynomial functions.

Vocabulary

Terms and notation that students learn or use in the unit

polynomials	roots/solutions	degree
leading coefficient	Fundamental Theorem of Algebra	difference of two squares
pythagorean triples	conjugate	polynomial long division
successive differences	end behavior	complex root/real root
factors	remainder theorem	difference/sum of two cubes
polynomial identities	rate of change

Lesson Map

Topic A: Polynomial Features and Graphs

Lesson

Classify polynomials through identification of degree and leading coefficient. Graph a polynomial function from a table of values; prove degree using successive differences.

Standards

A.APR.A.1F.IF.C.7.CF.IF.C.9

Lesson

Identify features of polynomial functions including end behavior, intervals where the function is positive or negative, and domain and range of function.

Standards

F.BF.B.3F.IF.B.4F.IF.B.6F.LE.A.3

Lesson

Match and compare equations and graphs of polynomials, and identify transformations.

Standards

F.BF.B.3F.IF.B.4F.IF.C.7.C

Lesson

Sketch polynomial functions using sign charts and analysis of the factored form of the polynomial function.

Standards

A.APR.B.3

Topic B: Operations with Polynomials

Lesson

Multiply polynomials, identifying factors, degrees, and number of real roots. Identify features of a polynomial in standard form.

Standards

A.APR.A.1F.BF.B.3

Lesson

Add and subtract polynomials. Identify degree, leading coefficient, and end behavior of result.

Standards

A.APR.A.1F.BF.A.1.B

Lesson

Divide polynomials by binomials to determine linear factors.

Standards

A.APR.A.1A.APR.B.2A.APR.B.3

Lesson

Determine if a binomial is a factor of a polynomial using the remainder theorem.

Standards

A.APR.B.2

Lesson

Identify and factor with difference of two squares in quadratic and quartic polynomials. Describe identity of difference of two squares. Describe the zeros that represent the resultant factors.

Standards

A.APR.B.3A.SSE.A.2

Lesson

Factor difference and sums of cubes. Explain sums and differences of cubes as polynomial identities.

Standards

A.APR.C.4

Lesson

Factor polynomials by grouping in quartic, cubic, and quadratic functions.

Standards

F.IF.C.8.A

Topic C: Polynomial Extensions

Lesson

Use polynomial identities to determine Pythagorean triples.

Standards

A.APR.C.4

Lesson

Identify the solution(s) to systems of polynomial functions.

Standards

A.REI.D.11

Lesson

2 days

Write polynomial functions from solutions of that polynomial function.

Standards

A.APR.B.3

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

Arithmetic with Polynomials and Rational Expressions

Arithmetic with Polynomials and Rational Expressions

A.APR.A.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.
Arithmetic with Polynomials and Rational Expressions

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).
Arithmetic with Polynomials and Rational Expressions

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.
Arithmetic with Polynomials and Rational Expressions

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.

Building Functions

Building Functions

F.BF.A.1 — Write a function that describes a relationship between two quantities Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Building Functions

F.BF.A.1.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.
Building Functions

F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(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.

Interpreting Functions

Interpreting Functions

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 increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Interpreting Functions

F.IF.B.6 — 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. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Interpreting Functions

F.IF.C.7 — Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Interpreting Functions

F.IF.C.7.C — Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
Interpreting Functions

F.IF.C.8 — Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Interpreting Functions

F.IF.C.8.A — Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Interpreting Functions

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.

Linear, Quadratic, and Exponential Models

Linear, Quadratic, and Exponential Models

F.LE.A.3 — Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

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. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Seeing Structure in Expressions

Seeing Structure in Expressions

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).

Foundational Standards

Standards covered in previous units or grades that are important background for the current unit

Building Functions

Building Functions

F.BF.A.1 — Write a function that describes a relationship between two quantities Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Building Functions

F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(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.

Geometry

Geometry

8.G.B.7 — Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Interpreting Functions

Interpreting Functions

F.IF.A.3 — Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
Interpreting Functions

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 increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Interpreting Functions

F.IF.C.7.A — Graph linear and quadratic functions and show intercepts, maxima, and minima.
Interpreting Functions

F.IF.C.8.A — Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Interpreting Functions

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.

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

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.
Reasoning with Equations and Inequalities

A.REI.B.3 — Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Reasoning with Equations and Inequalities

A.REI.D.10 — Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Seeing Structure in Expressions

Seeing Structure in Expressions

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).

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

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