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Advanced Algebra Support
Unit 2: Polynomial Functions
KEY STANDARDS ADDRESSED:
Use complex numbers in polynomial identities and equations
MCC9‐12.N.CN.8 (+) Extend polynomial identities to the complex numbers.
MCC9‐12.N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Interpret the structure of expressions.
MCC9‐12.A.SSE.1 Interpret expressions that represent a quantity in terms of its context.
MCC9‐12.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients.
MCC9‐12.A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity.
MCC9‐12.A.SSE.2 Use the structure of an expression to identify ways to rewrite it.
Write expressions in equivalent forms to solve problems
MCC9‐12.A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1),
and use the formula to solve problems.
Perform arithmetic operations on polynomials
MCC9‐12.A.APR.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.
Understand the relationship between zeros and factors of polynomials
MCC9‐12.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).
MCC9‐12.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.
Use polynomial identities to solve problems
MCC9‐12.A.APR.4 Prove polynomial identities and use them to describe numerical relationships.
n
MCC9‐12.A.APR.5 (+) Know and apply that the Binomial Theorem gives the expansion of (x + y) 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. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)
Solve systems of equations
MCC9‐12.A.REI.7 Solve a simple system consisting of a linear equation and a polynomial equation in two
variables algebraically and graphically.
Represent and solve equations and inequalities graphically
MCC9‐12.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 successive approximations. Include cases where f(x) and/or g(x)
are linear, polynomial functions.
Analyze functions using different representations
MCC9‐12.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.
MCC9‐12.F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and
showing end behavior.
Wed/T Sept 10H
11
EQ: How can we write a polynomial in standard form?
Worksheet 1
EQ: How do we add, subtract, multiply, and divide
6.2 Algebra 2 Book
6.3 Algebra 2 Book
polynomials?
EQ: In which operations does closure apply?
Interpret the structure of expressions.
MCC9‐12.A.SSE.1 Interpret expressions that represent
a quantity in terms of its context.
MCC9‐12.A.SSE.1a Interpret parts of an expression,
such as terms, factors, and coefficients.
MCC9‐12.A.SSE.1b Interpret complicated expressions
by viewing one or more of their parts as a single entity.
MCC9‐12.A.SSE.2 Use the structure of an expression to
identify ways to rewrite it.
Perform arithmetic operations on polynomials
MCC9‐12.A.APR.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.
Fri Tue
Sept 12 16
EQ: How are polynomials factored?
Worksheet 2
EQ: How can we write polynomials in factored form?
6.4 Algebra 2 Book
Interpret the structure of expressions.
MCC9‐12.A.SSE.1 Interpret expressions that represent
a quantity in terms of its context.
MCC9‐12.A.SSE.1a Interpret parts of an expression,
such as terms, factors, and coefficients.
MCC9‐12.A.SSE.1b Interpret complicated expressions
by viewing one or more of their parts as a single entity.
MCC9‐12.A.SSE.2 Use the structure of an expression to
identify ways to rewrite it.
Wed/T Sept 17 H
18
EQ: How are polynomials solved by factoring?
MCC9‐12.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.
6.4 Algebra 2 Book
Fri
Sept 19 QUIZ
Mon – Sept 22
Friday – 26
Quiz over writing polynomials in standard form,
adding, subtracting, and multiplying polynomials,
factoring polynomials, solve polynomials by factoring
EQ: What is the Remainder Theorem and what does it tell
us?
EQ: How are polynomials divided using long division and
synthetic division?
Worksheet 3
Worksheet 4
Worksheet 5
6.5 Algebra 2 Book
Understand the relationship between zeros and
factors of polynomials
MCC9‐12.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).
MCC9‐12.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.
MonWed
Sept 29
– Oct 1
EQ: What is the Rational Root Theorem and what does it tell
us?
MCC9‐12.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.
Worksheet 6
6.6 Alg 2 Book
ThMon
EQ: What is the Fundamental Theorem Algebra and what
does it tell us?
Oct 2-6
EQ: Which sets of numbers can be solutions to polynomial
equations?
EQ: How can a write a polynomial equation given the zeros
of the function?
6.7 Alg 2 Book
Worksheet 7
Worksheet 8
Worksheet 9
MCC9‐12.N.CN.9 (+) Know the Fundamental Theorem
of Algebra; show that it is true for quadratic
polynomials.
Use complex numbers in polynomial identities and
equations
MCC9‐12.N.CN.8 (+) Extend polynomial identities to
the complex numbers.
Tue
Oct 7
Wed
Oct 8
Thur- Oct 9 Fri
10
QUIZ
Quiz over dividing polynomials, solving polynomials for
all zeros, writing equations of polynomials
EQ: How is the end behavior of a polynomial function
determined?
Graphing
Investigation
Analyze functions using different representations
MCC9‐12.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.
MCC9‐12.F.IF.7c Graph polynomial functions,
identifying zeros when suitable factorizations are
available, and showing end behavior.
6.8 Lesson Opener
EQ: How are polynomials graphed?
6.8 Algebra 2 Book
EQ: What are the important characteristics of
polynomial graphs?
Worksheet 10
\MCC9‐12.A.SSE.1a Interpret parts of an expression,
such as terms, factors, and coefficients.
MCC9‐12.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.
Mon
Oct 13
Student Holiday
Worksheet 10
Tue –
Wed
Oct 14 15
EQ: How is a system of equations with a polynomial
and linear equation solved algebraically?
Worksheet 11
Solve systems of equations
MCC9‐12.A.REI.7 Solve a simple system consisting of
a linear equation and a polynomial equation in two
variables algebraically and graphically.
Represent and solve equations and inequalities
graphically
MCC9‐12.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 successive approximations. Include cases where
f(x) and/or g(x) are linear, polynomial functions.
Thur- Oct 16 Fri
17
EQ: How are polynomial inequalities solved
algebraically?
Worksheet 12
Represent and solve equations and inequalities
graphically
MCC9‐12.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 successive approximations. Include cases where
f(x) and/or g(x) are linear, polynomial functions.
Mon
Oct 27
Review
MonTue
Oct 27 28
EQ: How can we apply Pascal’s Triangle to expand?
MCC9‐12.A.APR.5 (+) Know and apply that the
n
Binomial Theorem gives the expansion of (x + y) 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. (The Binomial Theorem
can be proved by mathematical induction or by a
combinatorial argument.)
WedThur
Oct 29 –
30
Review for Polynomial Unit Test
Essential Questions Due
Worksheet 13
Pascal’s Triangle
Activity 1
Activity 2
Activity 3
Worksheet 14
12.2 Algebra 2 Book
Fri
Oct 31
Polynomial Unit Test
Notebook Due
Essential Questions- Answer essential questions in complete sentences and provide an example of each.
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How can we write a polynomial in standard form?
How can we write a polynomial in factored form?
How do we add, subtract, multiply, and divide polynomials?
In which operations does closure apply?
How can we apply Pascal’s Triangle to expand?
What is the Remainder Theorem and what does it tell us?
What is the Rational Root Theorem and what does it tell us?
What is the Fundamental Theorem Algebra and what does it tell us?
How can we solve polynomial equations?
Which sets of numbers can be solutions to polynomial equations?
What is the relationship between zeros and factors?
What characteristics of polynomial functions can be seen on their graphs?
How can we solve a system of a linear equation with a polynomial equation?
Vocabulary – Define and give an example of each
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Coefficient
Degree
Power
Multiplicity
Pascal’s Triangle
Remainder Theorem
x-intercept
zero of a function
polynomial function
end behavior
polynomial long division
Synthetic Division
Rational Zero Theorem
Fundamental Theorem of Algebra
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Local Maximum
Local Minimum
Increasing Interval
Decreasing Interval
Real Numbers
Integers
Rational Numbers
Irrational Numbers
Complex Numbers
Imaginary Numbers
Zero product property
Natural Numbers
Whole Numbers