4.2 Polynomials v1 20131118 (period 2).notebook

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Task 4.2 - Polynomial Functions

Learning Targets:

4A - I can describe a polynomial function in standard form and classify special cases of polynomials.

4B - I can apply the Fundamental Theorem of Algebra and connect to finding solutions to polynomial functions.

4C - I can explain how the Division Algorithm, Remainder Theorem, and Factor Theorem are related, and their purpose in finding solutions.

4.2 Polynomials v1 20131118 (period 2).notebook

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I) Polynomials in Standard FormA) Vocabulary

1) Polynomial - A monomial or sum of monomials

2) Degree - Highest exponent

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3) Standard Form - List terms in descending order by degree.

4) Multiplicity - The number of times a root is a root

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II) Fundamental Theorem of AlgebraA) Every polynomial having complex coefficients

and degree ≥ 1, has at least 1 complex root

*Every non-zero polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity

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B) Examples - How many roots? Apply FTA1)

2)

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III) Roots of polynomialsA) Previously Learned

1)_____, ______, _______, _______ mean the same thing.

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2) Factoring Methodsa) b)c)d)e)f)

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3) EX-Find the zeros of the polynomial functions, and their multiplicity. How many are real and how many are imaginary?

a)

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b)

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Practice - Work on "The Fundamental Theorem of Algebra" #1-8

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B) Division Algorithm1) Given a numerator (dividend) and denominator

(divisor), an algorithm that computes their quotient and/or remainder

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B) Division Algorithm

P(x) = D(x) Q(x) + R(x)

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2) Prior- Divide and write in the form:dividend = divisor * quotient + remainder

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Your Turn...Work on #9-13 in packet

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3) New - Divide using Long Division and write in the form:P(x) = D(x) Q(x) + R(x)

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Now your turn...work on #14-16

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3) New - Divide using Sythetic Division & write in form:P(x) = D(x) Q(x) + R(x)

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Now your turn...work on #17-19

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C) The Remainder Theorem1) If P(x) is a polynomial, and "r" is a real number,

then if P(x) is divided by (x-r), then the remainder "R" is P(r).

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2) Example: P(x) = x3+ 2x2+3x- 1

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3) Think! Why would P(r)=0 be important?

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3) Think! Why would P(r)=0 be important?

It means that x=r is a root/solution, and (x-r) is a factor.

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D) Factor Theorem1) If P(x) is a polynomial function, then (x-r) is a

factor of P(x) if and only if R=0.

3) Three methods to determine Remainder R:a) Long Divisionb) Synthetic Division (linear only)c) Remainder Theorem

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2) Example: P(x)= 2x3- x2+ 2x - 3a) Is (x-1) a factor of P(x)?

b) Is (x+2) a factor of P(x)?