Outlines

More on Polynomials

Terri Miller

February 16, 2009

Terri Miller More on Polynomials

OutlinesPart I: Review of Previous LecturePart II: Todays Lecture

Outline of Part I

1 Summary of the Previous LectureVocabularyEnd BehaviorSkills

2 Extremalocalfinding local

Terri Miller More on Polynomials

OutlinesPart I: Review of Previous LecturePart II: Todays Lecture

Outline of Part II

3 ExtremaGlobal Extrema

4 IntersectionsGraphically finding the intersection of two polynomials

5 Multiplicitymultiplicity of a rootsketch of the graphgraphing with calculator

6 Long division of Polynomials

Terri Miller More on Polynomials

Previous LectureExtrema

Part I

Review of Previous Lecture

Terri Miller More on Polynomials

Previous LectureExtrema

VocabularyEnd BehaviorSkills

Vocabulary

polynomial

term

coefficient

leading term

leading coefficient

degree of a polynomial

x-intercepts, roots, zeros

Terri Miller More on Polynomials

Previous LectureExtrema

VocabularyEnd BehaviorSkills

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

Terri Miller More on Polynomials

Previous LectureExtrema

VocabularyEnd BehaviorSkills

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 localminimum of a polynomial function.

Terri Miller More on Polynomials

Previous LectureExtrema

localfinding local

extrema - maxima and minima

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

Terri Miller More on Polynomials

Previous LectureExtrema

localfinding local

extrema - maxima and minima

the points B, C , and D are local extrema

the lowest in an area is called a local minimum, Cthe highest in an area is called a local maximum, B and D

Terri Miller More on Polynomials

Previous LectureExtrema

localfinding local

extrema - maxima and minima

the points B, C , and D are local extremathe 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 LectureExtrema

localfinding 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 LectureExtrema

localfinding local

extrema - maxima and minima

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

Terri Miller More on Polynomials

Previous LectureExtrema

localfinding 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

Terri Miller More on Polynomials

Previous LectureExtrema

localfinding 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

Terri Miller More on Polynomials

Previous LectureExtrema

localfinding 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

Terri Miller More on Polynomials

Previous LectureExtrema

localfinding local

find the coordinates of C by using the minimum option

you should get

find the maximum similarly

Terri Miller More on Polynomials

Previous LectureExtrema

localfinding local

find the coordinates of C by using the minimum option

you should get

find the maximum similarly

Terri Miller More on Polynomials

Previous LectureExtrema

localfinding local

find the coordinates of C by using the minimum option

you should get

find the maximum similarly

Terri Miller More on Polynomials

Previous LectureExtrema

localfinding local

find the coordinates of C by using the minimum option

you should get

find the maximum similarly

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

Part II

Todays Lecture

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

Global Extrema

3 ExtremaGlobal Extrema

4 IntersectionsGraphically finding the intersection of two polynomials

5 Multiplicitymultiplicity of a rootsketch of the graphgraphing with calculator

6 Long division of Polynomials

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

Global Extrema

the global maximum is the highest point the graph reachesover its entire domain

the global minimum is the lowest point the graph reaches overits 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

ExtremaIntersections

MultiplicityLong division of Polynomials

Global Extrema

the global maximum is the highest point the graph reachesover its entire domain

the global minimum is the lowest point the graph reaches overits 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

ExtremaIntersections

MultiplicityLong division of Polynomials

Global Extrema

the global maximum is the highest point the graph reachesover its entire domain

the global minimum is the lowest point the graph reaches overits 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

ExtremaIntersections

MultiplicityLong division of Polynomials

Global Extrema

the global maximum is the highest point the graph reachesover its entire domain

the global minimum is the lowest point the graph reaches overits 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

ExtremaIntersections

MultiplicityLong division of Polynomials

Graphically finding the intersection of two polynomials

3 ExtremaGlobal Extrema

4 IntersectionsGraphically finding the intersection of two polynomials

5 Multiplicitymultiplicity of a rootsketch of the graphgraphing with calculator

6 Long division of Polynomials

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

Graphically finding the intersection of two polynomials

Example

Find the intersection of

y1 = − 25

126(4x4 − 13x3 − 62x2 + 76x − 126),

andy2 = 3x2 − 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 theintersect option

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

Graphically finding the intersection of two polynomials

Example

Find the intersection of

y1 = − 25

126(4x4 − 13x3 − 62x2 + 76x − 126),

andy2 = 3x2 − 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 theintersect option

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

Graphically finding the intersection of two polynomials

Example

Find the intersection of

y1 = − 25

126(4x4 − 13x3 − 62x2 + 76x − 126),

andy2 = 3x2 − 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 theintersect option

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

Graphically finding the intersection of two polynomials

Example

Find the intersection of

y1 = − 25

126(4x4 − 13x3 − 62x2 + 76x − 126),

andy2 = 3x2 − 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 theintersect option

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

Graphically finding the intersection of two polynomials

make sure to have the cursor on the function y1 when it asksfor first curve, and y2 when it asks for the second curve

you should get the point

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

Graphically finding the intersection of two polynomials

make sure to have the cursor on the function y1 when it asksfor first curve, and y2 when it asks for the second curve

you should get the point

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

Graphically finding the intersection of two polynomials

make sure to have the cursor on the function y1 when it asksfor first curve, and y2 when it asks for the second curve

you should get the point

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

3 ExtremaGlobal Extrema

4 IntersectionsGraphically finding the intersection of two polynomials

5 Multiplicitymultiplicity of a rootsketch of the graphgraphing with calculator

6 Long division of Polynomials

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

multiplicity

completely factor a polynomial of degree n to get

P(x) = a(x − r1)p1(x − r2)p2(x − r3)p3 · · · (x − rk)pk ,

p1 + p2 + · · · + pk = n

each root, ri , has a factor x − ri with power pi

this power is the multiplicity of the root

the multiplicity gives us more information on the behavior ofthe graph

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

multiplicity

completely factor a polynomial of degree n to get

P(x) = a(x − r1)p1(x − r2)p2(x − r3)p3 · · · (x − rk)pk ,

p1 + p2 + · · · + pk = n

each root, ri , has a factor x − ri with power pi

this power is the multiplicity of the root

the multiplicity gives us more information on the behavior ofthe graph

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

multiplicity

completely factor a polynomial of degree n to get

P(x) = a(x − r1)p1(x − r2)p2(x − r3)p3 · · · (x − rk)pk ,

p1 + p2 + · · · + pk = n

each root, ri , has a factor x − ri with power pi

this power is the multiplicity of the root

the multiplicity gives us more information on the behavior ofthe graph

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

multiplicity

completely factor a polynomial of degree n to get

P(x) = a(x − r1)p1(x − r2)p2(x − r3)p3 · · · (x − rk)pk ,

p1 + p2 + · · · + pk = n

each root, ri , has a factor x − ri with power pi

this power is the multiplicity of the root

the multiplicity gives us more information on the behavior ofthe graph

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

multiplicity and the graph

if the multiplicity of a root is 1, the graph crosses the axis the

same way as y = x + 1 at x = −1

if the multiplicity of a root is even, the graph “”crosses theaxis the same way as y = (x − 1)2 at x = 1, it actually

bounces

if the multiplicity is odd greater than 1, the graph crosses the

axis the same way as y = (x + 2)3 at x = −2

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

multiplicity and the graph

if the multiplicity of a root is 1, the graph crosses the axis the

same way as y = x + 1 at x = −1

if the multiplicity of a root is even, the graph “”crosses theaxis the same way as y = (x − 1)2 at x = 1, it actually

bounces

if the multiplicity is odd greater than 1, the graph crosses the

axis the same way as y = (x + 2)3 at x = −2

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

multiplicity and the graph

if the multiplicity of a root is 1, the graph crosses the axis the

same way as y = x + 1 at x = −1

if the multiplicity of a root is even, the graph “”crosses theaxis the same way as y = (x − 1)2 at x = 1, it actually

bounces

if the multiplicity is odd greater than 1, the graph crosses the

axis the same way as y = (x + 2)3 at x = −2

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

Example

Sketch a graph of the polynomial

P(x) = −7(x − 3)4(x − 1)5(x + 3)

note the leading term−7x10

note the roots and their multiplicity

x = 3, of multiplicity 4; x = 1, of multiplicity 5;

x = −3, of multiplicity 1

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

Example

Sketch a graph of the polynomial

P(x) = −7(x − 3)4(x − 1)5(x + 3)

note the leading term−7x10

note the roots and their multiplicity

x = 3, of multiplicity 4; x = 1, of multiplicity 5;

x = −3, of multiplicity 1

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

Example

Sketch a graph of the polynomial

P(x) = −7(x − 3)4(x − 1)5(x + 3)

note the leading term−7x10

note the roots and their multiplicity

x = 3, of multiplicity 4; x = 1, of multiplicity 5;

x = −3, of multiplicity 1

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

From the leading term:

the degree of the polynomial is even, so both ends do thesame thing

the leading coefficient is negative, so the right side goes downhence both sides go down

From the roots:

at x = 3 the graph touches the axis and bounces

at x = 1 the graph passses through the axis with a kink

at x = −3 the graph passes through the axis

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

From the leading term:

the degree of the polynomial is even, so both ends do thesame thing

the leading coefficient is negative, so the right side goes downhence both sides go down

From the roots:

at x = 3 the graph touches the axis and bounces

at x = 1 the graph passses through the axis with a kink

at x = −3 the graph passes through the axis

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

From the leading term:

the degree of the polynomial is even, so both ends do thesame thing

the leading coefficient is negative, so the right side goes downhence both sides go down

From the roots:

at x = 3 the graph touches the axis and bounces

at x = 1 the graph passses through the axis with a kink

at x = −3 the graph passes through the axis

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

From the leading term:

the degree of the polynomial is even, so both ends do thesame thing

the leading coefficient is negative, so the right side goes downhence both sides go down

From the roots:

at x = 3 the graph touches the axis and bounces

at x = 1 the graph passses through the axis with a kink

at x = −3 the graph passes through the axis

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

From the leading term:

the degree of the polynomial is even, so both ends do thesame thing

the leading coefficient is negative, so the right side goes downhence both sides go down

From the roots:

at x = 3 the graph touches the axis and bounces

at x = 1 the graph passses through the axis with a kink

at x = −3 the graph passes through the axis

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

the y -intercept is P(0) = −7(−3)4(−1)5(3) = 1701the resulting sketch is:

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

Example

Graph the polynomial using your calculator

P(x) = −7(x − 3)4(x − 1)5(x + 3)

enter the function in your function menu

since all roots are between −3 and 3,set the x window to [−4, 4]leave the y at the default [−10, 10]

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

Example

Graph the polynomial using your calculator

P(x) = −7(x − 3)4(x − 1)5(x + 3)

enter the function in your function menu

since all roots are between −3 and 3,set the x window to [−4, 4]leave the y at the default [−10, 10]

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

Example

Graph the polynomial using your calculator

P(x) = −7(x − 3)4(x − 1)5(x + 3)

enter the function in your function menu

since all roots are between −3 and 3,set the x window to [−4, 4]leave the y at the default [−10, 10]

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

Luck - much behavior is visibleTo see more vertical behavior:

“zoom”

scroll down to zoomfit(this keeps your x range and changes y to best fit it)

this graph is too flat

try another window

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

Luck - much behavior is visibleTo see more vertical behavior:

“zoom”

scroll down to zoomfit(this keeps your x range and changes y to best fit it)

this graph is too flat

try another window

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

Luck - much behavior is visibleTo see more vertical behavior:

“zoom”

scroll down to zoomfit(this keeps your x range and changes y to best fit it)

this graph is too flat

try another window

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

Luck - much behavior is visibleTo see more vertical behavior:

“zoom”

scroll down to zoomfit(this keeps your x range and changes y to best fit it)

this graph is too flat

try another window

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

Luck - much behavior is visibleTo see more vertical behavior:

“zoom”

scroll down to zoomfit(this keeps your x range and changes y to best fit it)

this graph is too flat

try another window

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

Windows

Check these windows for other views.

good locally for behavior at x = 1, 3

x : [−4, 4], xscl = 1; y : [−10, 10], yscl = 1

good locally for behavior at x = −3

x : [−3.1,−2.9], y : zoomfit

good for finding peak around x = −3

x : [−5, 5], xscl = 1, y : [−2000000, 2000000], yscl = 100000

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

multiplicity of a rootsketch of the graphgraphing with calculator

Conclusion

It is helpful to have an idea of the behavior of a graph in order toknow where to look for it on the calculator.

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

3 ExtremaGlobal Extrema

4 IntersectionsGraphically finding the intersection of two polynomials

5 Multiplicitymultiplicity of a rootsketch of the graphgraphing with calculator

6 Long division of Polynomials

Terri Miller More on Polynomials

ExtremaIntersections

MultiplicityLong division of Polynomials

The Problem

divide the polynomial P(x) = 3x4 + 2x3 − x + 2 byD(x) = x2 + 2x − 1

See the file Polynomial Division for this part of the lecture.

Terri Miller More on Polynomials