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### Transcript of 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

Terri Miller More on Polynomials

• 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

Terri Miller More on Polynomials

• Previous Lecture Extrema

Part I

Review of Previous Lecture

Terri Miller More on Polynomials

• Previous Lecture Extrema

Vocabulary End Behavior Skills

Vocabulary

polynomial

term

coefficient

degree of a polynomial

x-intercepts, roots, zeros

Terri Miller More on Polynomials

• Previous Lecture Extrema

Vocabulary End Behavior Skills

End Behavior

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

Terri Miller More on Polynomials

• Previous Lecture Extrema

Vocabulary End Behavior Skills

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.

Terri Miller More on Polynomials

• Previous Lecture Extrema

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

Terri Miller More on Polynomials

• Previous Lecture Extrema

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

Terri Miller More on Polynomials

• Previous Lecture Extrema

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

Terri Miller More on Polynomials

• Previous Lecture Extrema

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

Terri Miller More on Polynomials

• Previous Lecture Extrema

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

Terri Miller More on Polynomials

• Previous Lecture Extrema

local finding local

local extrema - using TI-83/84

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

Terri Miller More on Polynomials

• Previous Lecture Extrema

local finding local

local extrema - using TI-83/84

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

Terri Miller More on Polynomials

• Previous Lecture Extrema

local finding local

local extrema - using TI-83/84

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

Terri Miller More on Polynomials

• Previous Lecture Extrema

local finding local

find the coordinates of C by using the minimum option

you should get

find the maximum similarly

Terri Miller More on Polynomials

• Previous Lecture Extrema

local finding local

find the coordinates of C by using the minimum option

you should get

find the maximum similarly

Terri Miller More on Polynomials

• Previous Lecture Extrema

local finding local

find the coordinates of C by using the minimum option

you should get

find the maximum similarly

Terri Miller More on Polynomials

• Previous Lecture Extrema

local finding local

find the coordinates of C by using the minimum option

you should get

find the maximum similarly

Terri Miller More on Polynomials

• Extrema Intersections

Multiplicity Long division of Polynomials

Part II

Todays Lecture

Terri Miller More on Polynomials

• Extrema Intersections

Multiplicity Long division of Polynomials

Global Extrema

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

Terri Miller More on Polynomials

• Extrema Intersections

Multiplicity Long division of Polynomials

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

Terri Miller More on Polynomials

• Extrema Intersections

Multiplicity Long division of Polynomials

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

Terri Miller More on Polynomials

• Extrema Intersections

Multiplicity Long division of Polynomials

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

Terri Miller More on Polynomials

• Extrema Intersections

Multiplicity Long division of Polynomials

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

Terri Miller More on Polynomials

• Extrema Intersections

Multiplicity Long division of Polynomials

Graphically finding the intersection of two polynomials

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

Terri Miller More on Polynomials

• Extrema Intersections

Multiplicity Long division of Polynomials

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.

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

Terri Miller More on Polynomials

• Extrema Intersections

Multiplicity Long division of Polynomials

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 −