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Lesson 15.2.notebook
May 15, 2012
Lesson 15.2 Graphs and Zeros of Polynomials
Lesson 15.2
When I die, I want to go peacefully like my Grandfather did, in his sleep ­­ not screaming, like the passengers in his car. Lesson 15.2.notebook
May 15, 2012
Polynomial Definitions:
The highest exponent in a set of terms. This
Degree: defines the maximum number of zeros of a polynomial.
Branches:
The "parts" of a polynomial graph. Break the graph up into increasing, and decreasing to get the branches.
Extreme Point (Vertex or Critical Point):
Vertex: A maximum or minimum point on a graph.
Critical Point: Occurs when the horizontal or vertical tangent line = 0.
Lesson 15.2.notebook
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Definition : Zero of a Function
A zero of a function f is an x­value, c, for which f (c) = 0
Rational Root Theorem:
Given a polynomial
p is the integer factor of the constant term
q is the integer factor of the leading coefficient
All possible rational roots are in the form x=±
Find the zeros of g(x) = 6x 3 + 17x 2 ­ 24x ­ 35
Lesson 15.2.notebook
Property : the Remainder Theorem
If p(x) is a polynomial, then p(c) equals the remainder when p(x) is divided by the quantity (x ­ c).
Factor Theorem: The quantity (x ­ c) is a factor of the polynomial p(x) iff p(c) = 0.
Show that x ­ 5 is a factor of f(x) = x 3 ­ 9x 2 ­ x + 105. Write f(x) in factored form
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Lesson 15.2.notebook
May 15, 2012
The Fundamental Theorem of Algebra A polynomial function has at least one zero in the set of complex numbers.
Corollary: An n th degree polynomial function
has exactly n zeros in the set of complex numbers, counting
multiple zeros.
Corollary: If a polynomial has only real
coefficients, then any nonreal complex zeros appear in conjugate
forms.
Let f(x) = x3­ 4x2 ­ 3x+2. Show that ­ 1 is a zero of f(x). Find
the other two zeros, and check by graphing.
Lesson 15.2.notebook
May 15, 2012
Let g(x) = x3­ 4x2 ­ 3x+18. Show that ­ 2 is a zero of g(x). Find
the other two zeros, and check by graphing.
Let h(x) = x3­ 4x2 ­ 3x+54. Show that ­ 3 is a zero of h(x). Find
the other two zeros, and check by graphing.
Lesson 15.2.notebook
May 15, 2012
Use the rational root theorem to find all zeros, real and non real of f(x) = x3­ 4x2 +22x+68. Assignment:
678 ­ 679 QR 1 ­ 10, 1 ­ 17 odd
Lesson 15.2.notebook
Lesson 15.2 Day 2
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Lesson 15.2.notebook
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Lesson 15.2.notebook
Exploration 15.2b
2. Mixed Number Form
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Lesson 15.2.notebook
May 15, 2012
Property : Sums and Products of the zeros of a cubic function If the function p(x) = ax 3 + bx 2 + cx + d has zeros z 1, z 2, z 3 , then
z 1 + z 2 + z 3 = ­ b / a
z 1 z 2 + z 1 z 3 + z 2 z 3 = c / a
z 1 z 2 z 3 = ­ d / a
Find the particular equation of a cubic function w/ integer coefficients if the function's zeros have the given sum, product and sum of pairwise products.
sum = ­5 / 3 sum of pairwise products = ­ 58 / 3
product = 40 / 3
Lesson 15.2.notebook
Assignment: p. 679 – 680 19 – 27 odd, 34, 37
May 15, 2012