Unit 4: Polynomial Functions
Goal: The student will demonstrate the ability to use a problem-solving approach to investigate polynomial functions and equations, both with and without the use of technology.


Objectives – The student will be able to:


a. Determine domain and range, zeros, local maxima and minima, and intervals where the graphs are increasing and decreasing and concavity.

Mathematical Background/Clarifying Examples:
Review properties of linear and quadratic functions which students learned in Algebra 1 and Algebra 2. Review the basic shapes of polynomial function in varying degrees. Students will need to understand the terms above and then identify the key points of polynomial functions of even and odd degrees from the graph. Students can explore the appropriate tools on the calculator which allow them to find zeros, local maxima, and local minima.

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Tutorial: The site provides definitions for the characteristics required to analyze a polynomial function.
http://www.math.com/tables/derivatives/extrema.htm

Example
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Resources:
1. Interactive exploration: This site shows a function of the 6th degree and allows students to change coefficients to discover how the function changes based on it's terms. By using 0's as coefficients of the higher degrees students can look at just quadratics, or cubics, or quintics, etc.
http://www.geogebra.org/en/upload/files/english/Cullen_Stevens/polynomials.html

2. Tutorial: The site provides instructions, with screenshots, on how to find the zeros, local max and min, on the graphing calculator.
http://www.prenhall.com/esm/app/graphing/ti83/Graphing/working_with_graphs/calc/calc.html



b. Use common characteristics of a polynomial function to sketch its graph.

Mathematical Background/Clarifying Examples:
Once students recognize the shape of the curve using end behavior, Leading Coefficient Test, domain and range, and number of turning points by the equation, they can make a general sketch of the curve.

Leading Coefficient Test
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Turning Points
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Resources:
1) Modeling End Behavior: Have all students stand and face the same direction. Their right arm represents the right side of the curve and their left arm represents the left side of the curve. Show a polynomial function and have students demonstrate the end behavior of the function.

2) Matching: Create a set of cards with functions and their graphs of varying degrees and leading coefficients. Have students match work together to match the correct graph and equation. This can be done in varying size groups of students.



c. Analyze a function numerically and graphically to determine if the function is odd, even, or neither.

Mathematical Background/Clarifying Examples:
The words even and odd, when applied to a function describe the symmetry that exists for the graph of the function, f. A function is even, if and only if, whenever the point (x,y) is on the graph of f, then (-x, y) is also on the graph . In other words, a function f is even if, for every number x in its domain, the number -x is also in the domain: f(-x) = f(x). A function f is odd, if and only if, whenever the point (x, y) is on the graph of f then the point (-x, -y) is also on the graph. In other words, a function f is odd if, for every number x in its domain, the number -x is also in the domain and f(-x) = -f(x). Graphically, an even function is symmetric about the y-axis and an odd function is symmetric about the origin.

Determine algebraically whether the function is even, odd, or neither.
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Tutorial: The site gives additional information on even/odd functions, with graphs, and examples on how to determine algebraically.
http://www.mathsisfun.com/algebra/functions-odd-even.html

Resources:
1) Interactive guided tutorial: The site gives an apple with a parent function from each family of functions. It asks students to determine graphically whether they are even, odd, or neither, and them to verify each algebraically. It also gives four additional practice problems.
http://www.analyzemath.com/function/even_odd.html



d. Use the Fundamental Theorem of Algebra to determine the number of zeros of a polynomial.

Mathematical Background/Clarifying Examples:
Students will identify the number of zeros of a polynomial function based on its degree. Students are not finding the zeros, just acknowledging how many zeros, real and imaginary, exist.

Fundamental Theorem of Algebra: Every polynomial function of positive degree with complex coefficients has at least one complex zero. It can also be stated: every polynomial function of positive degree n has exactly n complex zeros if zeros are counted to their full multiplicity.

Resources:
Tutorial: The site describes the Fundamental Theorem of Algebra, rewriting polynomials in its linear factors, and pairs of complex roots.
http://www.mathsisfun.com/algebra/fundamental-theorem-algebra.html



e. Find all rational, irrational, and complex zeros of a polynomial using algebraic methods.

Mathematical Background/Clarifying Examples:
Students will utilize their prior knowledge of factoring, the fundamental theorem of algebra, and complex numbers. They will have briefly seen division, but students will need to understand both long division and synthetic division. It would be a good idea to use a pretest to assess each students prior knowledge entering this section. Teach the rational zeros theorem (also called rational roots theorem), Factor Theorem, Remainder Theorem, the conjugate pairs theorem and the effect multiplicity of zeros has on a graph, in order to find all zeros of a polynomial algebraically and then recognize how it affect the curve of each polynomial. After this objective, I recommend graphing higher order polynomials by hand using end behavior, intermediate value theorem, bounds theorem, zeros, multiplicity, and y-intercepts.

Division of Polynomials
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- Long Division: You can begin by asking students to divide integers with long division. Also discuss how to write the quotient when there is a remainder.
- Synthetic Division: Discuss that synthetic division is used when the divisor is linear and how to do the process.

Tutorial and Examples: This site gives step-by step instruction for conducting long and synthetic division. It also discusses the Factor Theorem and Remainder Theorem.
http://www.kkuniyuk.com/M1410203.pdf

Students need to realize that the divisor and quotient in are factors of the dividend. They need to be able to use division and factoring to find all factors of a given polynomial.

Rational Roots Theorem

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Tutorial: The site provides an explanation of the Rational Roots test and a few detailed examples.
http://www.purplemath.com/modules/rtnlroot.htm

Factor Theorem
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Since this is a biconditional statement, the statement and its converse are both true. Be sure to state it both ways for your students.

Remainder Theorem
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Tutorial: Instruction, Examples, and 12 practice questions.
http://www.mathsisfun.com/algebra/polynomials-remainder-factor.html

Tutorial and Practice: This site reviews division of polynomials, factor theorem, and remainder theorem and then gives practice problems with worked out
solutions for each.
http://www.intmath.com/equations-of-higher-degree/2-factor-remainder-theorems.php

Conjugate Pairs Theorem
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Multiplicity
There are two cases when dealing with multiple zeros:
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Bounds Theorem
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Example: This link provides an example of checking the upper and lower bounds. It also has additional information on other theorems from this objective and some practice problems.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut39_zero2.htm

Intermediate Value Theorem
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Video Tutorial: A teacher explains the Intermediate Value Theorem
http://www.calculus-help.com/the-intermediate-value-theorem/

Descartes Rule of Signs
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Writing a polynomial from the zeros
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Example:
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Resources:
1) Tutorial: This tutorial explains long division of polynomials and synthetic division for cases when the divisor is linear.
http://tutorial.math.lamar.edu/Classes/Alg/DividingPolynomials.aspx

2) Tutorial: The site gives step by step explanation of a polynomial long division problem. It also provides four additional examples with worked out solutions.
http://www.purplemath.com/modules/polydiv2.htm

3) Practice: Apply the leading coefficient test and find zeros graphically and algebraically, given the zeros find the polynomial, and apply the conjugate pairs theorem.
http://sites.csn.edu/istewart/mathweb/Accuplacer/accu_coll/polyfun.pdf

4) Lesson Plan using the graphing calculator: Reviews the Fundamental Theorem of Algebra, Complex roots, and multiplicity in a lesson utilizing the graphing calculator.
http://education.ti.com/xchange/US/Math/PrecalculusTrig/11758/Precalc_BackToRoots_TI84.pdf

5) Examples and Practice: The site gives a brief tutorial on finding all zeros of polynomials. It shows how to find the zeros both algebraically and graphically for the same function. The site also explains writing a polynomial with specified zeros. FInally, it gives 27 exercises, answers not included.
http://www.pstcc.edu/facstaff/jahrens/math1130/polyroot.pdf

6) Practice: The site provides practice questions with an option to view the full solution.
http://hotmath.com/help/gt/genericalg2/section_6_9.html



f. Use polynomial functions to model and solve real-world problems.

Mathematical Background/Clarifying Examples:
Look at situations that are modeled by polynomials of varying degrees. In addition to using the skills from previous sections, they can use the graphing calculator to find polynomial regression equations to model data.

Example:
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Resources:
1) Project/Example: This site gives a real-world application where students look at a curve that models the amount of a drug in a patients bloodstream. Students compile knowledge from the unit to graph the function and identify when the drug has left the patients bloodstream.
http://www.regentsprep.org/Regents/math/algtrig/ATE13/PolyResource.htm

2) Modeling with Polynomials: The link provides practice for students to model and solve polynomial functions.
http://www.augustatech.edu/math/molik/PolynomialModels.pdf

3) Quadratic and Cubic Applications: Real world situations that are modeled by quadratic and cubic equations.
http://www.augustatech.edu/math/molik/ExamplesFunPolynomials.pdf