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Intro to Polynomials

by toni dimella on Apr 10, 2011

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Intro to PolynomialsPresentation Transcript

• Polynomial Functions and Models
Module 12
• Polynomial Functions
A polynomial of degree n is a function of the form
P(x) = anxn + an-1xn-1 + ... + a1x + a0
Where an 0. The numbers a0, a1, a2, . . . , an are
called the coefficients of the polynomial.
The a0is the constant coefficientorconstant term.
The number an, the coefficient of the highest
power, is the leading coefficient, and the term anxn is
• Graphs of Polynomial Functions and Nonpolynomial Functions
• Graphs of Polynomials
Graphs smooth curve
Degree greater than 2
ex. f(x) = x3
These graphs will not have the following:
Break or hole
Corner or cusp
Graphs are lines
Degree 0 or 1
ex. f(x) = 3 or f(x) = x – 5
Graphs are parabolas
Degree 2
ex. f(x) = x2 + 4x + 8
• Even- and Odd-Degree Functions
• Finding Zeros of a Polynomial
Zero- another way of saying solution
Zeros of Polynomials
Solutions
Place where graph crosses the x-axis
(x-intercepts)
Zeros of the function
Place where f(x) = 0
• Using the Graphing Calculator to Determine Zeros
Graph the following polynomial function and determine the zeros.
Before graphing, determine the end behavior and the number
of relative maxima/minima.
In factored form:
P(x) = (x + 2)(x – 1)(x – 3)²
• MultiplicityIf (x-c)k, k 1, is a factor of a polynomial function P(x) and:
K is even
The graph is tangent to the x-axis at (c, 0)
K is odd
The graph crosses the x-axis at (c, 0)
• Multiplicity
y = (x + 2)²(x − 1)³
−2 is a root of multiplicity 2,
and 1 is a root of multiplicity 3.
These are the 5 roots:
−2,  −2,  1,  1,  1.
• Multiplicity
y = x³(x + 2)4(x − 3)5