Math 3 Unit 3a: Polynomial and Rational Representations
Approximate Time Frame: 3-4 weeks
Connections to Previous Learning:
In previous units, students have worked with quadratic functions in various forms to determine their roots and vertices. They have used the factored form to
determine if and where zeros exist. They have examined the equations of quadratic functions to determine the function's end behavior, and used all of this
information to produce a graph of a quadratic.
Focus of this Unit:
Students will build on their knowledge of quadratics to include higher degree polynomial functions. They will use the factored form to determine zeros and the
sign of the leading coefficient to determine end behavior. Multiplicity of factors will be used to help determine the behavior of the graph at its roots. These
notions will all be extended to rational functions, which will initially be viewed as the reciprocal of a polynomial, and ultimately as the ratio of two polynomials.
From the High School Algebra Progression Document, pp. 6-8:
Polynomials form a rich ground for mathematical explorations that reveal relationships in the system of integers. For example, students can explore the
sequence of squares
1, 4, 9, 16, 25, 36…
And notice the difference between them – 3, 5, 7, 9, 11 – are consecutive odd integers. This mystery is explained by the polynomial identity.
(𝑛 + 1)𝑛 − 𝑛2 = 2𝑛 + 1
A more complex identity,
(𝑥 2 + 𝑦 2 )2 = (𝑥 2 − 𝑦 2 )2 + (2𝑥𝑦)2
allows students to generate Pythagorean triples. For example, taking x =1 and y = 2 in this identity yields 52 = 32 + 42
Viewing polynomials as functions leads to explorations of a different nature. Polynomial functions are, on the one hand, very elementary, in that, unlike
trigonometric and exponential functions, they are built up out of the basic operations of arithmetic. On the other hand, they turn out to be amazingly flexible,
and can be used to approximate more advanced functions such as trigonometric and exponential functions. Although students only learn the complete story
here if and when they study calculus, experience with constructing polynomial functions satisfying given conditions is useful preparation not only for calculus,
but for understanding the mathematics behind curve-fitting methods used in applications to statistics and computer graphics.
A simple step in this direction is to construct polynomial functions with specified zeros. This is the first step in a progression which can lead, as an extension
topic, to constructing polynomial functions whose graphs pass through any specified set of points in the plane.
The analogy between polynomials and integers carries over to the idea of division with remainder. Just as in Grade 4 students find quotients and remainders of
integers, in high school they find quotients and remainders of polynomials. The method of polynomial long division is analogous to, and simpler than, the
method of integer long division.
A particularly important application of polynomial division is the case where a polynomial 𝑝(𝑥) is divided by a linear factor of the form 𝑥 − 𝑎 for a real number a.
In this case the remainder is the value 𝑝(𝑎) of the polynomial at 𝑥 = 𝑎. It is a pity to see this topic reduced to “synthetic division,” which reduced the method to
a matter of carrying numbers between registers, something easily done by a computer, while obscuring the reasoning that makes the result evident. It is
important to regard the Remainder Theorem as a theorem, not a technique.
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A consequence of the Remainder Theorem is to establish the equivalence between linear factors and zeros that is the basis of much work with polynomials in
high school: the fact that 𝑝(𝑎) = 0 if and only if 𝑥 − 𝑎 is a factor of 𝑝(𝑥) . It is easy to see if 𝑥 − 𝑎 is a factor then 𝑝(𝑎) = 0. But the Remainder Theorem tells us
that we can write
𝑝(𝑥) = (𝑥 − 𝑎)𝑞(𝑥) + 𝑝(𝑎) for some polynomial 𝑞(𝑥).
In particular, if 𝑝(𝑎) = 0 then) (𝑥) = (𝑥 − 𝑎)𝑞(𝑥) , so 𝑥 − 𝑎 is a factor of 𝑝(𝑥) .
Creating Equations
Students have been writing equations, mostly linear equations, since middle grades. At first glance it might seem that the progression from middle grades to
high schools is fairly straightforward: the repertoire of functions that is acquired during high school allows students to create more complex equations, including
equations arising from linear and quadratic functions, and simple rational and exponential functions; students are no longer limited largely to linear equations in
modeling relationships between quantities with equations in two variables.
Connections to Subsequent Learning:
The next unit will call on students' understanding of modeling and various functions, including linear, exponential, piecewise and absolute value, polynomial and
rational to compare and choose appropriate models. Use of technology to facilitate the inspection of data and potential models will again become essential.
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Math 3 Unit 3a: Polynomial and Rational Representations
Desired Outcomes
Standard(s):
Interpret the structure of expressions
 A.SSE.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).
Create equations that describe numbers or relationships.
 A.CED.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.
 A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Understand solving equations as a process of reasoning and explain the reasoning.
 A.REI.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.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Interpret functions that arise in applications in terms of the context
 F.IF.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 function in increasing, decreasing, positive, or negative, relative maximums and minimums,
symmetries, end behavior, and periodicity.
 F.IF.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.
Analyze functions using different representations
 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.
 F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
 F.IF.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.
Understand the relationship between zeros and factors of polynomials.
 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)
 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.
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Math 3 Unit 3a: Polynomial and Rational Representations
Represent and solve equations and inequalities graphically.
 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 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.
Use polynomial identities to solve problems.
 A.APR.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.
Rewrite rational expressions
 A.APR.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.
Build new functions from existing functions.
 F.BF.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.
 F.BF.4 Find inverse functions.
 F.BF.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) = 2 x3 or
f(x) = (x+1)/ (x-1) for x ≠ 1.
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
 Explicit vocabulary instruction with regard to the features and types of equations, functions, and graphs.
 Guided discussion regarding the connections between graphs, tables, equations, expressions and the contexts they represent.
Understandings: Students will understand …
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A polynomial function, in the variable x, may be written as p ( x)  an x n  an 1 x n 1  an  2 x n  2   a2 x 2  a1 x1  a0 x 0 , with an  0 ,
where n is a non-negative integer (called the degree of the polynomial) and each coefficient, ai is a real number for i = 0, 1, ..., n. Otherwise, the
polynomial may be equal to 0, and in this case, we say the degree is undefined.
The degree of a polynomial helps to determine the end behavior of its graph
The zeros of each factor of a polynomial determine the x-intercepts of its graph
Graphs of rational functions are often discontinuous, due to values that are not in the domain of the function
The Remainder Theorem can be used to determine roots of polynomials
The long division algorithm for polynomials can be used to determine horizontal or oblique asymptotes of rational functions.
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Math 3 Unit 3a: Polynomial and Rational Representations
Essential Questions:
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What is a polynomial?
How does the degree of a polynomial affect its graph?
How do the factors of a polynomial affect its graph?
How is the domain of a rational function related to its graph?
How can rewriting the equation of a rational function (using long division of polynomials) give further information about its graph?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
1. Make sense of problems and persevere in solving them.
*2. Reason abstractly and quantitatively. Use factors to determine zeros of polynomials, leading coefficients to determine direction of ends of the graph of a
polynomial. Also, use zeros of the polynomial in the denominator of a rational function to determine location of vertical asymptotes, and compare degrees
of numerator and denominator to determine horizontal asymptotes.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
*5. Use appropriate tools strategically. Use a computer algebra system to factor polynomials; use graphing technology to explore graphs of various
polynomials and rational functions; use technology (graphs, tables) to solve for a point of intersection of two functions.
*6. Attend to precision. Graph a polynomial or rational function while carefully indicating zeros, asymptotes, y-intercept, and other significant features.
*7. Look for and make use of structure. Recognize that a quadratic function with a repeated factor does not intersect the x-axis in two unique points, and
extend this notion to write equations of polynomials or rational functions which have graphs that do not pass through the x-axis at the root of a factor
having even multiplicity.
8. Look for express regularity in repeated reasoning.
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Math 3 Unit 3a: Polynomial and Rational Representations
Prerequisite Skills/Concepts:
Advanced Skills/Concepts:
Students should already be able to:
 Factor a quadratic expression over the integers.
 Determine zeros from a factored form of a quadratic.
 Manipulate a quadratic function between various forms to
determine key features, including zeros, the vertex,
maximum/minimum value, and end behavior.
 Determine domain and range of functions involving simple
rational expressions.
Some students may be ready to:
 Find the conjugate of a complex number and use them to find quotients of
complex numbers or solve for complex roots of a polynomial.
 Determine the magnitude of a complex number.
 Graph a complex number.
 Calculate roots of complex numbers.
 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.
 Determine whether a rational function has a horizontal or oblique asymptote.
 Calculate the equation of a horizontal or oblique asymptote of a rational function.
 Determine if and where a rational function crosses its horizontal (or oblique)
asymptote.
 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.
d. (+) Graph rational functions, identifying zeros and asymptotes when suitable
factorizations are available, and showing end
behavior.
 Write equations of rational functions, given the graph.
 Roughly graph the reciprocal function of a given graph (given polynomial,
reciprocal of polynomial, or ratio of two polynomials).
 A.APR.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.
 F.BF.4 Find Inverse functions.
b. (+) Verify by composition that one function is the inverse of
another.
c. (+) Read values of an inverse function from a graph or a table,
given that the function has an inverse.
d. (+) Produce an invertible function from a non-invertible function
by restricting the domain.
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Knowledge: Students will know…
Skills: Students will be able to …
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The Remainder Theorem and how it applies to determining
whether a number is a root of the polynomial.
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Prove and use polynomial identities
Graph a polynomial given in factored form, indicating all intercepts and directions
of end behaviors.
Use the structure of an expression to identify ways to rewrite it.
Create equations in one, two, or more variables and use them to solve problems.
Construct viable arguments to justify a solution method.
Calculate and interpret the average rate of change of a function.
Identify the effect on the graph of replacing 𝑓(𝑥)by 𝑓(𝑥) + 𝑘, 𝑘𝑓(𝑥), 𝑓(𝑘𝑥), and
𝑓(𝑥 + 𝑘).
Write the equation of a polynomial function given its graph or defining
characteristics of its graph.
Solve radical and rational equations in one variable, checking for extraneous
solutions.
Perform the long division algorithm for polynomials in order to rewrite simple
rational expressions in different forms; use computer algebra systems to perform
the same on complicated examples.
Use technology (graphs, tables) to solve the equation f(x) = g(x), where f(x) and/or
g(x) are polynomial or rational functions.
Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions).
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).
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.
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.
Solve an equation of the form f(x) = c for a simple function f that has an inverse
and write an expression for the inverse. For example, f(x) = 2 x3 or f(x) = (x+1)/ (x-1)
for x ≠ 1.
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Math 3 Unit 3a: Polynomial and Rational Representations
Academic Vocabulary:
Critical Terms:
Supplemental Terms:
Polynomial
Degree of term
Degree of polynomial
Factor of polynomial
Root/zero/x-intercept
Multiplicity of factor
Rational function
Vertical asymptote
Horizontal asymptote
Oblique asymptote
integer
term
coefficient
domain
range
discontinuity
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