PoojaMoolya at 11:05, 10 October 2019

2019-10-10T11:05:33Z

Nancyvarkey at 08:07, 28 November 2018

2018-11-28T08:07:24Z

Madhurig at 05:45, 27 November 2018

2018-11-27T05:45:32Z

Madhurig: Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- || '''Slide Number 1 ''' '''Title slide ''' || Welcome to the spoken tutorial on '''Polynomials'''. |- || '''Slide Numbe..."

2018-11-26T11:38:01Z

Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- || '''Slide Number 1 ''' '''Title slide ''' || Welcome to the spoken tutorial on '''Polynomials'''. |- || '''Slide Numbe..."

New page

{|border=1
||'''Visual Cue'''
||'''Narration'''
|-

|| '''Slide Number 1 '''

'''Title slide '''
|| Welcome to the spoken tutorial on '''Polynomials'''.

|-
|| '''Slide Number 2'''

'''Learning Objectives'''
|| In this tutorial we learn about,

* Polynomials of one variable
* Slope of a linear polynomial
* Degree of the polynomials
* Zeros of the polynomials
* Roots of the polynomials
* Remainder theorem
* Factorization of polynomials
|-
|| '''Slide Number 3'''

'''System Requirement'''
|| To record this tutorial, I am using;

'''Ubuntu Linux''' OS version 16.04

'''GeoGebra''' version 5.0438.0-d

|-
|| '''Slide Number 4'''

'''Pre-requisites'''

'''www.spoken-tutorial.org'''
|| To follow this tutorial, learner should be familiar with GeoGebra interface.

For the prerequisite '''GeoGebra '''tutorials, please visit.
|-
||
|| Let us first define a polynomial.

|-
|| '''Slide Number 5'''

'''Polynomial'''
|| An algebraic expression containing one or more terms with non-zero coefficients is a polynomial.

For example

'''x cube plus 3 x squared plus 2 x minus 5''' is a polynomial.

|-
|| Cursor on '''GeoGebra''' interface.
|| I have already opened the '''GeoGebra''' interface.

|-
|| Point to the input bar.
|| For this tutorial we will use '''input bar''' to solve the polynomials.

|-
||
|| Let us first start with slope of a polynomial.

|-
|| Type '''r(x)=3x-3''' >> press '''Enter'''.

Point to '''Algebra''' and '''Graphics''' views.
|| In the '''input bar''' type,

'''r''' within brackets ''' x is equal to 3x minus 3 '''

and press '''Enter'''.

The linear polynomial is displayed in the '''Algebra''' and '''Graphics views'''.

|-
|| Type '''Slope(r)''' >> press '''Enter'''.
|| Now type '''Slope ''' within brackets '''r''' and press '''Enter'''.

|-
|| Point to the slope of line and '''Algebra''' view.
|| '''Slope '''of r''' '''is shown on the line and in the '''Algebra''' '''view'''.

|-
||
|| Now we will define the degree of a polynomial.

|-
|| '''Slide Number 6'''

'''Degree of polynomial'''
|| The highest power of the variable in a polynomial, is the degree of the polynomial.

For example,

'''p is equal to x raised to the power of 5 minus x raised to the power of 4 plus 3'''

In this polynomial, degree is '5' .

|-
||
|| Let’s try some more examples to find the degree of polynomials.

|-
|| Type '''Degree(3x^7+4x^6+x+9)'''
|| In the '''input bar''' type, '''Degree'''.

In place of '''polynomial''' type,

'''3x raised to the power of 7 plus 4x raised to the power of 6 plus x plus 9'''

Press '''Enter'''.

|-
|| Point the value in the Algebra view.
|| The degree of the polynomial is displayed in the '''Algebra view''' as 7.

|-
|| Type '''Degree(5x^5-4x^2-6)'''

point to the '''Algebra view'''.
|| Similarly degree of the polynomial

'''5x raised to the power of 5 minus 4x squared minus 6 ''' is 5.

|-
|| '''Slide Number 7'''

'''Assignment '''

Find the degree of the given polynomials

1. x^5-x^4+3

2. 2-y^2-y^3+2y^8

3. 5x^3+4x^2+7x

|| Pause the tutorial and do this assignment.

|-
|| '''Slide number 8'''

'''Zeros of Polynomial '''
|| Now I will explain about zeros of the polynomial.

Zero of a polynomial '''p of x''' is a number ''''r'''' such that '''p of r''' is equal to zero.

|-
|| Cursor on the interface.

Press '''Ctrl +A''' to select all objects >> press '''Delete''' key on keyboard.
|| Let us delete all the '''objects'''.

Press '''Ctrl +A''' to select all '''objects''', then press '''Delete''' key on the keyboard.

|-
|| In the '''input bar''' type,

'''p=5x^2-3x+7''' >> press '''Enter'''.
|| To find zeros of the polynomial, in the '''input bar''' type,

'''p is equal to 5x squared minus 3x plus 7'''

and press '''Enter'''.

|-
|| Drag Boundary of '''Algebra view'''.
|| I will drag the boundary of the '''Algebra view '''to see the polynomial clearly.

|-
|| Click and drag the '''Graphics view'''.
|| Move the '''Graphics view''' , if you cannot see the parabola.

|-
|| '''p(0)=5(0)^2-3(0)+7 = 7'''

'''p(1)=5(1)^2-3(1)+7 = 9'''

'''p(2)=5(2)^2-3(2)+7 = 21'''

'''p(3)=5(3)^2-3(3)+7 = 43'''
|| Now we will find the values of '''p of 0, p of 1''', '''p of 2''' and '''p of 3'''.

|-
|| Type '''p(0)''' >> press Enter

Point to '''p(0)''' value in the '''Algebra view''' .
|| In the '''input bar''' type '''p''', then type '''0 '''within brackets and press '''Enter'''.

The value of '''p of 0''' is displayed in the '''Algebra view'''.

|-
|| Type '''p(1)''' >> press '''Enter'''.

Type '''p(2)''' >> press '''Enter'''.

Type '''p(3)''' >> press '''Enter'''.
|| Similarly I will type '''p of 1''', '''p of 2''' and '''p of 3'''.

|-
|| Point to '''p(1)''', '''p(2), '''p(3)''' values in '''Algebra view'''.
|| Values of '''p of 1''', '''p of 2''' and '''p of 3''' are displayed in the '''Algebra view.'''

|-
|| '''Slide Number 9'''

'''Assignment'''

Find the values of '''p of 0''', '''p of 1''' and '''p of 2''' for the given polynomials.

1. p=2+t+t^2-t^3

2. p=(x-1)(x+1)

3. p=x^3
|| Pause the tutorial and complete this assignment.

|-
|| Press '''Ctrl +A''' >> press '''Delete''' key.
|| I will clear the interface once again.

|-
|| Roots of the polynomial.
|| Now let us find the roots of the polynomial.

|-
|| In the '''input bar''' type, '''p= x^2-x-2''' >> press '''Enter'''.
|| In the '''input bar''' type,

'''p is equal to x squared minus x minus 2''' and press '''Enter'''.

|-
|| Point to the polynomial in '''Algebra''' and '''Graphics''' view.
|| Polynomial '''p of x''' is displayed in the '''Algebra view'''.

Its graph, a parabola, is displayed in the '''Graphics view'''.

|-
|| Drag the '''Graphics view'''.
|| If required, drag the '''Graphics view''' to view the parabola clearly.

|-
|| Type '''Root(p)''' >> press '''Enter'''.
|| Next type '''Root''' within brackets '''p '''and press '''Enter'''.

|-
|| Point to the roots in '''Graphics''' and '''Algebra''' views.

'''A(-1,0)''' and '''B(2,0)'''
|| Roots of the polynomial p are displayed

as points '''A''' and '''B''' in '''Algebra''' and '''Graphics views. '''

|-
|| Type '''q<nowiki>=x^2-5x+6</nowiki>'''
|| Let us type one more polynomial.

'''q is equal to x squared minus 5x plus 6''' and press '''Enter'''.

|-
|| Point to the polynomial in the '''Algebra view'''.

Point to the graph in the '''Graphics view'''.
|| Polynomial '''q of x''' is displayed in the '''Algebra view.'''

Its graph, a parabola, is displayed in the '''Graphics view.'''

|-
|| Type '''Root(q)''' >> press Enter.
|| Type '''Root''' within brackets '''q''' and press '''Enter'''.

|-
|| Point to the roots in Algebra and Graphics views.

'''C(2,0''') and '''D(3,0)'''.
|| Roots of the polynomial '''q''' are displayed

as points '''C''' and '''D''' in '''Algebra '''and''' Graphics views'''.

|-
|| Point to coincided '''B''' and '''C'''.

Click on '''Move''' tool >> drag the labels.
|| Here we see that the points B and C coincide with each other.

Using the '''Move''' tool we can move the labels to see them clearly.

|-
|| '''Slide Number 10'''

'''Assignment '''

Find the roots of the following polynomials.

1. f= x^2-2x+1

2. g=2x+1

3. h=x^2-1
|| Pause the tutorial and do this assignment.

|-
||
|| Next we will use '''Remainder theorem''' to divide polynomials.

|-
|| '''Slide Number 11'''

'''Remainder theorem'''
|| Let '''p of x '''be any polynomial of degree greater or equal to 1.

And ''''a' '''be any real number.

If '''p of x''' is divided by a linear polynomial '''x minus a''', then the remainder is '''p of a'''.

Dividend is equal to Divisor multiplied by the Quotient plus remainder.

|-
|| Click on '''File''' >> '''New Window'''.
|| Let us open a new '''Geogebra '''window.

Click on '''File''' and '''New Window'''.

|-
|| Illustrations for polynomial division

Type, '''p1=3x^2+x-1''' press '''Enter'''.

|| In the '''input bar''' type,

'''p1 is equal to 3x squared plus x minus 1 '''

and press '''Enter'''.

|-
|| Type '''p2=x+1''' press '''Enter'''.
|| Then type '''p2 is equal to x plus 1'''

and press '''Enter'''.

|-
|| Point '''p1''' and '''p2'''.
|| Now we will divide the polynomial '''p1''' with '''p2'''.

|-
|| In the input bar type, '''Division'''.

Point to the options.

Select '''Division(<Dividend Polynomial>, <Divisor Polynomial>)'''
|| In the '''input bar''' type, '''Division'''.

Two options appear.

Select the second option that contains polynomials.

|-
|| Type '''p1'''.

Type '''p2'''.

Press '''Enter'''.
|| In place of '''Dividend Polynomial '''type '''p1'''.

In place of '''Divisor Polynomial''' type '''p2'''.

Then press '''Enter'''.

|-
|| Point to lines.
|| Two lines intersecting each other appear in the '''Graphics view'''.

These lines represent division of the polynomials '''p1''' and '''p2'''.

|-
|| '''q <nowiki>=3x - 2 </nowiki>'''

'''r <nowiki>= </nowiki>1. '''

Point to the quotient and remainder.
|| Quotient and remainder of the division are shown as a list.

'''L1 is equal to '''within curly braces''' 3x minus 2 comma 1'''.

Here quotient is '''3x-2''' and remainder is 1.

|-
|| Go to '''View''' menu >> Select '''Graphics 2''' check box.
|| To show the second set of polynomials, I will open the '''Graphics 2 view.'''

|-
|| Drag boundary to see the '''Graphics 2 view'''.
|| I will drag boundary to see the '''Graphics 2 view''' clearly.

|-
|| Point to the '''input bar'''.
|| Then I'll type polynomials '''q1''' and '''q2''' in the '''input bar'''.

|-
|| Type

'''q1=4x^3-3x^2-x +1''' >> press '''Enter'''.
|| '''q1 is equal to 4x cube minus 3x squared minus x plus 1'''

Press '''Enter'''.

|-
|| Type '''q2=x+1''' >> press '''Enter'''.
|| '''q2 is equal to x plus 1 ''' and press '''Enter'''.

|-
|| Type,

'''Division(q1, q2)'''
|| Type '''Division''', followed by polynomials '''q1 comma q2''' within brackets and press '''Enter'''.

|-
|| '''L2={4x2 -7x +6, -5}'''

q <nowiki>= </nowiki>'''4x2 -7x +6'''

r <nowiki>= </nowiki>'''-5'''

Point to the quotient and remainder.
|| Quotient and remainder of the division are shown as a list.

'''L2 is equal to '''within curly braces''' 4x squared minus 7x plus 6 comma minus 5'''

Here quotient is '''4x squared minus 7x plus 6 '''and remainder is '''-5'''.

|-
|| '''Slide Number 12 '''

'''Assignment'''

Solve the exercises based on remainder theorem.

1. p1=x^4+x^3-2x^2+x , p2=x-1

2. p1=x^3+3x^2+3x+1, p2=2x+5

3. p1=3x^3+7x, p2=3x+7
|| Pause the video and solve the exercises based on remainder theorem.

|-
|| Factorization of polynomials
|| Let us now factorize the polynomials.

|-
|| Click on '''File''' and '''New Window.'''
|| Let us open a new '''GeoGebra''' window.

Click on '''File''' and '''New Window'''.

|-
|| type, '''p=x^2-5x+6''' >> press '''Enter'''.
|| In the '''input bar''' type,

'''p is equal to x squared minus 5x plus 6''' and press '''Enter'''.

|-
|| Type '''Factors''' >> select '''Factors(<Polynomial>)''' option.

Type '''p(x)''' >> press '''Enter'''.
|| Type '''Factors''' and select '''Factors Polynomial''' option.

In the place of the polynomial type '''p''' within brackets '''x'''

and press '''Enter'''.

|-
|| Drag boundary of '''Algebra view'''.
|| Drag boundary to see the '''Algebra view''' clearly.

|-
|| '''M1= {{x-3,1}, {x-2, 1}} '''

Point to''' (x-3)(x-2)'''.
|| '''M1 '''is displayed in the '''Algebra view'''.

Here '''(x minus 3) '''and''' (x minus 2)''' are factors of the polynomial '''p of x'''.

|-
||
|| Let us try another example.

|-
|| Type '''Factors( x^3-2x^2-x+2 )'''
|| Type '''Factors '''then type within brackets '''x cube minus 2x squared minus x plus 2'''

and press '''Enter'''.

|-
|| '''M2={{x-2,1},{x-1, 1},{x+1,1}}'''

Point '''M2''' in the '''Algebra view'''.
|| '''M2 '''is displayed in the '''Algebra view'''.

'''x minus 2'''

'''x minus 1'''

'''x plus 1'''

are the factors of the polynomial.

|-
|| '''Slide Number 13'''

'''Assignment'''

Solve the exercises based on factorization

1. p=x^3-2x^2-x+2

2. p=12x^2-7x+1

3. p=2x^2+7x+3

4. p=x^3+13x^2+32x+20
|| Pause the video and solve the exercises based on factorization.

|-
||
|| Let us summarize what we have learnt.

|-
|| '''Slide Number 14'''

'''Summary'''
|| In this tutorial we have learnt about,

* Polynomials of one variable
* Slope of a linear polynomial
* Degree of the polynomials
* Zeros of the polynomials
* Roots of the polynomials
* Remainder theorem
* Factorization of polynomials

|-
|| '''Slide Number 15'''

'''About Spoken Tutorial project'''
|| The video at the following link summarizes the Spoken Tutorial project.

Please download and watch it.

|-
|| '''Slide Number 16'''

'''Spoken Tutorial workshops'''
|| The '''Spoken Tutorial Project '''team:

conducts workshops using spoken tutorials and

* gives certificates on passing online tests.

For more details, please write to us.

|-
|| '''Slide Number 17'''

'''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 in this forum.

|-
|| '''Slide Number 18'''

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

More information on this mission is available at this link.

|-
||
|| This is Madhuri Ganapathi from, IIT Bombay signing off.

Thank you for watching.

|}

@@ Line 29: / Line 29: @@
 *'''Ubuntu Linux''' OS version 16.04
-*'''GeoGebra''' version 5.0438.0-d
+*'''GeoGebra''' version 5.0.438.0-d
 |-

@@ Line 26: / Line 26: @@
 '''System Requirement'''
-|| To record this tutorial, I am using;
+|| To record this tutorial, I am using:
-'''Ubuntu Linux''' OS version 16.04
+*'''Ubuntu Linux''' OS version 16.04
+*'''GeoGebra''' version 5.0438.0-d
 '''GeoGebra''' version 5.0438.0-d
 |-
@@ Line 38: / Line 37: @@
 '''www.spoken-tutorial.org'''
-|| To follow this tutorial, learner should be familiar with GeoGebra interface.
+|| To follow this tutorial, learner should be familiar with '''GeoGebra interface'''.
-For the prerequisite '''GeoGebra '''tutorials, please visit.
+For the prerequisite '''GeoGebra '''tutorials, please visit this website.
 |-
 ||

GeoGebra-5.04/C2/Polynomials/English - Revision history

PoojaMoolya at 11:05, 10 October 2019

Nancyvarkey at 08:07, 28 November 2018

Madhurig at 05:45, 27 November 2018

Madhurig: Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- || '''Slide Number 1 ''' '''Title slide ''' || Welcome to the spoken tutorial on '''Polynomials'''. |- || '''Slide Numbe..."