Difference between revisions of "Applications-of-GeoGebra/C2/Roots-of-Polynomials/English"

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|| The two '''extrema''' are also mapped on '''curve h of x''' in '''Graphics''' view.  
 
|| The two '''extrema''' are also mapped on '''curve h of x''' in '''Graphics''' view.  
 
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'''Point of inflection'''
 
'''Point of inflection'''
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|| Let us summarize.
 
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'''Summary'''
 
'''Summary'''
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* '''Complex roots''' will be covered in another tutorial
 
* '''Complex roots''' will be covered in another tutorial
 
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'''Assignment'''
 
'''Assignment'''
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Plot '''graphs''' and find '''roots''', '''extrema''' and '''inflection points''' for the following '''polynomials'''.
 
Plot '''graphs''' and find '''roots''', '''extrema''' and '''inflection points''' for the following '''polynomials'''.
 
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'''About Spoken Tutorial project'''
 
'''About Spoken Tutorial project'''
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Please download and watch it.
 
Please download and watch it.
 
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'''Spoken Tutorial workshops'''
 
'''Spoken Tutorial workshops'''
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For more details, please write to us.
 
For more details, please write to us.
 
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'''Forum for specific questions:'''
 
'''Forum for specific questions:'''
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|| Please post your timed queries on this forum.
 
|| Please post your timed queries on this forum.
 
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'''Acknowledgement'''
 
'''Acknowledgement'''

Latest revision as of 17:03, 28 June 2018

Visual Cue Narration
Slide Number 1

Title Slide

Welcome to this tutorial on Roots of Polynomials.
Slide Number 2

Learning Objectives

In this tutorial, we will learn:
  • To plot graphs of polynomial equations
  • About complex numbers, real and imaginary roots
  • To find extrema and inflection points
Slide Number 3

Pre-requisites

www.spoken-tutorial.org

To follow this tutorial, you should be familiar with
  • GeoGebra interface
  • Basics of coordinate system
  • Polynomials
  • If not, for relevant tutorials, please visit our website.
Slide Number 4

System Requirement

Here I am using:
  • Ubuntu Linux operating system version 14.04
  • GeoGebra 5.0.388.0-d
Let us begin with the binomial theorem.
Slide Number 5

Binomial Theorem

Binomial theorem states that if a, b Єℝ, index n is a positive integer,

  • 0 ≤ r ≤n, then
  • (a + b)n = nC0 an + nC1 an-1 b1 + nC2 an-2 b2 + … + nCr an-r br + … + nCn bn

Reminder: nC1 = n!/[1! (n-1)!]

a and b are real numbers.

index n is a positive integer.

r lies between 0 and n.

Binomial theorem states that a plus b raised to n can be expanded as shown.

Slide Number 6

Quadratic Equations and Roots

  • A second degree polynomial, y = a x2+ b x+ c has roots
  • x=- b ± sqrt{( b 2-4 ac)/2a }
  • where ▲= b2-4ac
  • When ▲< 0, roots are complex
  • When ▲=0, roots are real and equal
  • When ▲>0, roots are real and unequal
Quadratic Equations and Roots

A 2nd degree polynomial, y equals a x squared plus b x plus c has roots given by values of x.

x is equal to ratio of minus b plus or minus squareroot of b squared minus 4 a c to 2 a.

Where discriminant Delta is equal to b squared minus 4 a c

When Delta is less than 0, roots are complex

When Delta is equal to 0, roots are real and equal

When Delta is greater than 0, roots are real and unequal

Slide Number 7

Quadratic Equations and Roots

  • When roots are real, ax2+b x+ c =0 has extremum (xv, yv)
  • xv = - b/2a and yv= a xv2+ b xv+ c
When roots are real, ax squared plus b x plus c equals 0 has extremum xv comma yv

xv equals minus b divided by 2 a and yv equals axv squared plus bxv plus c

Show the GeoGebra window. I have already opened the GeoGebra interface.
Click on View >> select CAS. Click on View tool and select CAS to open the CAS view.
In line 1 in CAS view, type f(x):=x^2-2x-3 >> press Enter. In line 1 in CAS view, type the following line.

f x in parentheses colon equals x caret 2 minus 2 space x minus 3.

To type caret symbol, hold Shift key down and press 6.

The space indicates multiplication.

Press Enter.

Drag boundary of Algebra view. Drag boundary to see Algebra view properly.
Point to the equation f(x) appearing in Algebra view.

Point to exponent 2 in f(x).

Observe the equation f of x in Algebra view.

The degree of this quadratic polynomial f of x is 2.

Drag boundary of CAS view. Drag boundary to see Graphics view properly.
Click in Graphics view to see Graphics View toolbar. Click in Graphics view to see Graphics View toolbar.
Move Graphics View>> click on Zoom Out Under Move Graphics View, click on Zoom Out tool.
Click in Graphics view >> minimum vertex of parabola f. Click in Graphics view to see the minimum vertex of parabola f.
Click on Move Graphics View tool >> click in Graphics background. Click on Move Graphics View tool and click in Graphics background.
Hand symbol appears >> drag Graphics view to see parabola f. When hand symbol appears, drag Graphics view, so you can see parabola f.
Drag boundaries of CAS view. Drag boundaries to see CAS view properly.
Type Root(f) in line 2 of CAS view >> press Enter. In line 2 of CAS view, type Root f in parentheses.

Press Enter.

Point to roots in CAS view. The roots appear below, in the same box, in curly brackets.
Point to roots in Graphics view. Note that these are the x-intercepts of parabola f in Graphics view.
Type Extremum(f) in line 3 of CAS view >> press Enter. In line 3 of CAS view, type Extremum f in parentheses.

Press Enter.

Point to extremum in CAS view. The extremum appears below, in the same box, in curly brackets.
Point to minimum vertex of parabola f in Graphics view. Note that this is the minimum vertex of parabola f in Graphics view.
In line 4 in CAS view, type g(x):=x^2+5x+10 >> press Enter. In line 4 in CAS view, type the following line.

g x in parentheses colon equals x caret 2 plus 5 space x plus 10.

Press Enter.

Drag boundary of Algebra View.

Point to the equation g(x) appearing in Algebra view.

Drag boundary to see Algebra view properly.

Observe the equation g of x in Algebra view.

Drag boundary of CAS view. Drag boundary to see Graphics view properly.
Uncheck f of x in CAS view.

Point to Algebra view >> hides parabola f in Graphics view.

Uncheck f of x in CAS view.

Note that this also unchecks it in Algebra view and hides parabola f in Graphics view.

Click >> drag Graphics view to see parabola g. Click in and drag Graphics view so you can see parabola g.
Drag boundary of CAS view. Again, drag boundary to see CAS view properly.
Type Root(g) in line 5 of CAS view >> press Enter. In line 5 of CAS view, type Root g in parentheses.

Press Enter.

Point to empty curly brackets for roots in CAS view. Empty curly brackets appear below.

Parabola g does not have any real roots as it does not intersect x axis at all.

Roots are said to be complex.

Type Extremum(g) in line 6 of CAS view >> press Enter. In line 6 of CAS view, type Extremum g in parentheses.

Press Enter.

Point to extremum in CAS view. The extremum appears below, in the same box, in curly brackets.
Point to the minimum vertex of parabola g in Graphics view. Note that this is the minimum vertex of parabola g in Graphics view.
Point to Evaluate tool.

Point to extremum in form of fractions.

While Evaluate tool is highlighted in CAS View toolbar, the extremum appears as fractions.

Minus 5 divided by 2 comma 15 divided by 4.

Click on the extremum in line 6 and click on Numeric tool.

Point to extremum in form of decimals.

In line 6, click on the extremum and click on Numeric tool.

The extremum now appears in decimal form.

Minus 2 point 5 comma 3 point 7 5.

Let us look at complex numbers.
Slide Number 8

Complex numbers, XY plane


  • A complex number is expressed as z = a + bi: where 'a' = real part, bi = imaginary part, and a and b are constants.
  • Imaginary number, i = sqrt{-1}
  • In the XY plane, a + bi is point (a, b).
  • In the complex plane, x axis = real axis, y axis = imaginary axis.
Complex numbers, XY plane

A complex number is expressed as z equals a plus bi.


a is the real part; bi is imaginary part;a and b are constants


i is imaginary number and is equal to square root of minus 1.


In the XY plane, a plus bi corresponds to the point a comma b.


In the complex plane, x axis is called real axis, y axis is called imaginary axis.

Slide Number 9

Complex numbers, complex plane

  • In complex plane, z is a vector with real axis coordinate a and imaginary axis coordinate b


  • Length of the vector z = |z| = r


  • r = sqrt (a2+b2) (Pythagoras’ theorem)
Complex numbers, complex plane

In complex plane, z is a vector.


Its real axis coordinate is a and imaginary axis coordinate is b.


The length of the vector z is equal to the absolute value of z and to r.


According to Pythagoras’ theorem, r is equal to squareroot of a squared plus b squared.

Show the GeoGebra window. Let us go back to the GeoGebra interface we were working on.

We will now use the input bar instead of CAS view.

Click >> close CAS view. Click and close CAS view.
In Algebra view, uncheck g of x to hide. In Algebra view, uncheck g of x to hide it.
In input bar, type

h(x):=x^3-4x^2+x+6 >> press Enter.

In input bar, type the following line.

h x in parentheses colon equals x caret 3 minus 4 space x caret 2 plus x plus 6.

Press Enter.

Drag boundaries of Algebra >> Graphics view.

Point to the equation h(x) appearing in Algebra view.

Drag boundaries to see Algebra and Graphics views properly.

Observe equation h of x in Algebra view.

Function h of x is graphed in Graphics view.

Under Move Graphics View>> click on Zoom Out tool.

Click in Graphics view.

Under Move Graphics View, click on Zoom Out tool.

Click in Graphics view.

Click on Move Graphics View >> move Graphics background to see graph. Click on Move Graphics View and move Graphics background to see the graph.
In input bar, type Root(h) >> press Enter. In input bar, type Root h in parentheses and press Enter.
Point to co-ordinates of three roots (A, B and C) in Algebra view. The co-ordinates of three roots (A, B and C) appear in Algebra view.
Point to three roots mapped on the curve h of x in Graphics view. The three roots are also mapped as x intercepts of the curve h of x in Graphics view.
In input bar, type Extremum h in parentheses >> press Enter. In input bar, type Extremum h in parentheses and press Enter.
Point to co-ordinates of two extrema (Dand E) in Algebra view. Co-ordinates of two extrema (D and E) appear in Algebra view.
Point to two extrema mapped on the curve h of x in Graphics view. The two extrema are also mapped on curve h of x in Graphics view.
Slide Number 10

Point of inflection

Point of inflection (PoI) on a curve is the point where curve changes direction.

  • To find co-ordinates of PoI (x,y), we equate 2nd derivative of given function to 0
  • Solve to get x (x co-ordinate of PoI)
  • Substitute this x in original function to get y co-ordinate
Point of inflection

A point of inflection PoI on a curve is the point where the curve changes its direction.

To find the co-ordinates of PoI x comma y,

We equate second derivative of the given function to 0.


Solution of this equation gives us x (x co-ordinate of PoI).


Substitute this x in original function to get y co-ordinate.

Let us find the point of inflection on h(x). Let us find the point of inflection on h of x.
In input bar, type Inf >> choose InflectionPoint ( <Polynomial> ) option from menu. In input bar, type Inf and scroll down menu to choose InflectionPoint Polynomial option.
Instead of highlighted Polynomial, type h >> Press Enter. Instead of highlighted Polynomial, type h and press Enter.
Point to the point of inflection in Algebra view. In Algebra view, point of inflection appears as point F, below the two extrema.
Point to F on h(x) in Graphics view. F is mapped on h of x in Graphics view.
point to the function. Correlate the degree of the polynomials and the number of roots seen so far.
Point to CAS>> Algebra >> Graphics views. Observe that functions entered in CAS appear in Algebra and Graphics views.
Point to Algebra >> Graphics views >> CAS view. Functions entered in input bar appear in Algebra and Graphics views but not in CAS view.
Let us summarize.
Slide Number 11

Summary

In this tutorial, we have learnt to:
  • Plot graphs of polynomial functions using CAS view and input bar
  • Find real roots, extrema and inflection point(s)
  • Complex roots will be covered in another tutorial
Slide Number 12

Assignment

  • d(x)=x2-6x+5
  • e(x)=3x3-2x2+0.2x-1
  • f(x)=-2x4-x3+3x2
  • g(x)=x5-7x4+9x3+23x2-50x+24
  • h(x)=(4x+3)/(x-1)
  • i(x)=(3x2-2x-1)/(2x2+3x-2)
Assignment:

Plot graphs and find roots, extrema and inflection points for the following polynomials.

Slide Number 13

About Spoken Tutorial project

The video at the following link summarizes the Spoken Tutorial project.

Please download and watch it.

Slide Number 14

Spoken Tutorial workshops

The Spoken Tutorial Project team conducts workshops and gives certificates.

For more details, please write to us.

Slide Number 15

Forum for specific questions: Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.

Please post your timed queries on this forum.
Slide Number 16

Acknowledgement

Spoken Tutorial Project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.

This is Vidhya Iyer from IIT Bombay, signing off.

Thank you for joining.

Contributors and Content Editors

Madhurig, Nancyvarkey, Vidhya

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