Download Unit 5: Polynomial Functions Algebra II Essential Questions

Unit 5: Polynomial Functions
Algebra II
5 Weeks
Essential Questions
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What does the degree of a polynomial tell you about its related polynomial function?
For a polynomial function, how are factors, zeros and x-intercepts related?
For a polynomial function, how are factors and roots related?
Enduring Understandings
1. A polynomial function has distinguishing “behaviors”. You can look at its algebraic form
and know something about its graph. You can look at its graph and know something
about its algebraic form.
2. Knowing the zeros of a polynomial functions can help you understand the behavior of its
graph.
3. If (x-a) is a factor of a polynomials, then the polynomial has value 0 when x=a. If a is a
real number, then the graph of the polynomial has (a,0) as an x-intercept.
4. You can divide polynomials using steps that are similar to the long division steps that you
use to divide whole numbers.
5. The degree of a polynomial equation tells you how many roots the equation has.
6. You can use a pattern of coefficients to write the expansion of (a+b)n.
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Unit 5: Polynomial Functions
Algebra II
5 Weeks
Content
Students will know…
Topics (Pearson):
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(5-1) Polynomial Functions
(5-2) Polynomials, Linear Factors and Zeros
(5-3) Solving Polynomial Equations
(5-4) Dividing Polynomials
(5-6) Fundamental Theorem of Algebra
(5-7) Binomial Theorem
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Polynomial equations
The mathematical significance of zeros
Long and synthetic division
Remainder, Factor and Rational Root
Theorem
Fundamental Theorem of Algebra
Binomial Theorem (Pascal’s Triangle)
21st Learning Expectations
Students will be able to…
 Employ mathematical problem solving skills effectively.
 Make decisions and solve problems in independent and collaborative settings.
21st Century Learning Skills
Students will be able to…
 ML #1 – Make sense of problems and persevere in solving them.
 ML #2 – Reason abstractly and quantitatively.
 ML #4 – Model with mathematics.
 ML #5 – Use appropriate tools strategically.
 ML #7 – Look for and make use of structure.
 ML #8 – Look for and express regularity in repeated reasoning.
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Unit 5: Polynomial Functions
Algebra II
5 Weeks
Connecticut State Standards
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CC.9-12.N.CN.8 (+) Extend polynomial identities to the complex numbers. For example,
rewrite x2 + 4 as (x + 2i)(x - 2i).
CC.9-12.N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for
quadratic polynomials
CC.9-12.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.*
CC.9-12.F.IF.7c Graph polynomial functions, identifying zeros when suitable
factorizations are available, and showing end behavior
CC.9-12.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).
CC.9-12.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.
CC.9-12.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).
CC.9-12.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.
CC.9-12.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
CC.9-12.A.APR.5 (+) Know and apply that the Binomial Theorem gives the expansion of
(x + y)n 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.)
CC.9-12.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 and minimums; symmetries; end behavior; and periodicity.*
CC.9-12.F.LE.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.
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Unit 5: Polynomial Functions
Algebra II
5 Weeks
Objectives
Students will be able to…
 Write a polynomial function given a polynomial equation.
 Identify the degree of a polynomial equation and state the significance.
 Write a polynomial given its factors or zeros.
 Identify the zeros of a polynomial function by finding the x-intercepts of its graph.
 Factor a polynomial equation.
 Apply the zero-product property.
 Use Pascal’s Triangle to find the coefficients of a binomial to the nth degree.
 Apply the Fundamental Theorem of Algebra.
Assessments
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Quiz EU1 – Graphs of Polynomials
Quiz EU2 – Solving Polynomial Equations
Quiz EU3 – Dividing Polynomials
Quiz EU4 – Fundamental Theorem of Algebra
Quiz EU5 – Binomial Theorem
Unit Test – Polynomials
Resources
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Charles, R., Hall, B., Kennedy, D., Bass, L., Johnson, A., Murphy, S., et al. (2012).
Algebra 2: Common Core. Boston: Pearson.
Graphing Calculator
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