More on Polynomials - Arizona State University terri/courses/271resources/... x-intercepts, roots,...

Outlines
More on Polynomials
Terri Miller
February 16, 2009
Terri Miller More on Polynomials

Outlines Part I: Review of Previous Lecture Part II: Todays Lecture
Outline of Part I
1 Summary of the Previous Lecture Vocabulary End Behavior Skills
2 Extrema local finding local

Outlines Part I: Review of Previous Lecture Part II: Todays Lecture
Outline of Part II
3 Extrema Global Extrema
4 Intersections Graphically finding the intersection of two polynomials
5 Multiplicity multiplicity of a root sketch of the graph graphing with calculator
6 Long division of Polynomials

Previous Lecture Extrema
Part I
Review of Previous Lecture

Vocabulary End Behavior Skills
Vocabulary
polynomial
term
coefficient
leading term
leading coefficient
degree of a polynomial
x-intercepts, roots, zeros

End Behavior
leading term determines
degree even, both ends go in same direction
degree odd, the ends go in opposite directions
leading coefficient > 0, right hand side goes up
leading coefficent < 0, right hand side goes down

Skills
Find the x-intercepts and y-intercept of a polynomial function.
Describe the end behaviors of a polynomial function.
Determine the minimal degree of a polynomial given its graph.
Use a graphing utility to find a local maximum or local minimum of a polynomial function.

local finding local
extrema - maxima and minima
the points B, C , and D are local extrema the lowest in an area is called a local minimum, C the highest in an area is called a local maximum, B and D

local finding local
the points B, C , and D are local extrema
the lowest in an area is called a local minimum, C the highest in an area is called a local maximum, B and D

local finding local
the points B, C , and D are local extrema the lowest in an area is called a local minimum, C
the highest in an area is called a local maximum, B and D

local finding local
the points B, C , and D are local extrema
the lowest in an area is called a local minimum, C
the highest in an area is called a local maximum, B and D

local finding local
the points B, C , and D are local extrema the lowest in an area is called a local minimum, C the highest in an area is called a local maximum, B and D

local finding local
local extrema - using TI-83/84
Enter the function in your calculator’s function menu.
y1 = − 25
126 (4x4 − 13x3 − 62x2 + 76x − 126)
graph the function using the window and scale
x : [−4, 6], xscl = 1; y : [−75, 100], yscl = 25
use the “calc” key; the second function on the “trace” key

local finding local
find the coordinates of C by using the minimum option
you should get
find the maximum similarly

Extrema Intersections
Multiplicity Long division of Polynomials
Part II
Todays Lecture

Global Extrema

Global Extrema
the global maximum is the highest point the graph reaches over its entire domain
the global minimum is the lowest point the graph reaches over its entire domain
there may not be a global maximum or minimum
our example has a global maximum but no global minimum

Graphically finding the intersection of two polynomials

Example
Find the intersection of
y1 = − 25
126 (4x4 − 13x3 − 62x2 + 76x − 126),
and y2 = 3x
2 − 2x2 + x − 12.
Enter the function(s) in your calculator’s function menu.
graph the functions using the same window and scale as before
use the “calc” key; the second function on the “trace” key
find the coordinates of a point of intersection by using the intersect option

Example
Find the intersection of
y1 = − 25
126 (4x4 − 13x3 − 62x2 + 76x − 126),
and y2 = 3x
2 − 2x2 + x −

More on Polynomials - Arizona State University terri/courses/271resources/... x-intercepts, roots,...

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