PoojaMoolya at 10:07, 10 October 2019

2019-10-10T10:07:45Z

PoojaMoolya: Created page with "{|border=1 ||'''Time''' ||'''Narration''' |- |00:01 | Welcome to the spoken tutorial on '''Polynomials'''. |- |00:05 | In this tutorial we learn about, Polynomials of one..."

2019-10-10T09:43:13Z

Created page with "{|border=1 ||'''Time''' ||'''Narration''' |- |00:01 | Welcome to the spoken tutorial on '''Polynomials'''. |- |00:05 | In this tutorial we learn about, Polynomials of one..."

New page

{|border=1
||'''Time'''
||'''Narration'''

|-
|00:01
| Welcome to the spoken tutorial on '''Polynomials'''.

|-
|00:05
| In this tutorial we learn about,

Polynomials of one variable

Slope of a linear polynomial

|-
|00:14
|Degree of the polynomials

Zeros of the polynomials

Roots of the polynomials

|-
|00:23
| Remainder theorem

Factorization of polynomials

|-
|00:28
| To record this tutorial, I am using:

'''Ubuntu Linux''' OS version 16.04

'''GeoGebra''' version 5.0438.0-d

|-
|00:41
| To follow this tutorial, learner should be familiar with '''GeoGebra interface'''.

For the prerequisite '''GeoGebra '''tutorials, please visit this website.

|-
| 00:52
| Let us first define a polynomial.

|-
|00:55
| An algebraic expression containing one or more terms with non-zero coefficients is a polynomial.

|-
|01:03
| For example

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

|-
|01:08
| I have already opened the '''GeoGebra''' interface.

|-
|01:13
| For this tutorial we will use '''input bar''' to solve the polynomials.

|-
|01:18
| Let us first start with slope of a polynomial.

|-
|01:22
| In the '''input bar''' type, '''r''' within brackets ''' x is equal to 3x minus 3 '''
and press '''Enter'''.

|-
|01:31
| The linear polynomial is displayed in the '''Algebra''' and '''Graphics views'''.

|-
|01:36
| Now type '''Slope''' within brackets '''r''' and press '''Enter'''.

|-
|01:42
| '''Slope''' of '''r''' is shown on the line and in the '''Algebra view'''.

|-
|01:47
| Now we will define the degree of a polynomial.

|-
|01:51
| The highest power of the variable in a polynomial, is the degree of the polynomial.

|-
|01:57
| For example,

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

|-
|02:04
| In this polynomial, degree is '5'.

|-
| 02:07
| Let’s try some more examples to find the degree of polynomials.

|-
|02:13
| In the '''input bar''' type, '''Degree'''.

|-
|02:15
| In place of '''polynomial''' type, '''3x raised to the power of 7 plus 4x raised to the power of 6 plus x plus 9'''

|-
|02:25
| Press '''Enter'''.

|-
|02:27
| The degree of the polynomial is displayed in the '''Algebra view''' as 7.

|-
|02:32
| Similarly degree of the polynomial, '''5x raised to the power of 5 minus 4x squared minus 6''' is 5.

|-
|02:42
| Pause the tutorial and do this assignment.

|-
|02:47
| 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.

|-
|02:59
| Let us delete all the '''objects'''.

|-
|03:02
| Press '''Ctrl + A''' to select all '''objects''', then press '''Delete''' key on the keyboard.

|-
|03:09
| To find zeros of the polynomial, in the '''input bar''' type,

'''p is equal to 5x squared minus 3x plus 7''' and press '''Enter'''.

|-
|03:20
| I will drag the boundary of the '''Algebra view ''' to see the polynomial clearly.

|-
|03:25
| Move the '''Graphics view''', if you cannot see the parabola.

|-
|03:29
| Now we will find the values of '''p of 0, p of 1''', '''p of 2''' and '''p of 3'''.

|-
|03:36
| In the '''input bar''' type '''p''', then type 0 within brackets and press '''Enter'''.

|-
|03:44
| The value of '''p of 0''' is displayed in the '''Algebra view'''.

|-
|03:48
| Similarly I will type '''p of 1''', '''p of 2''' and '''p of 3'''.

|-
|03:57
|Values of '''p of 1''', '''p of 2''' and '''p of 3''' are displayed in the '''Algebra view'''.

|-
|04:06
| Pause the tutorial and complete this assignment.

|-
|04:11
| I will clear the interface once again.

|-
|04:14
| Now let us find the roots of the polynomial.

|-
|04:18
| In the '''input bar''' type, '''p is equal to x squared minus x minus 2''' and press '''Enter'''.

|-
|04:27
| Polynomial p of x is displayed in the Algebra view.

|-
|04:31
| Its graph, a parabola, is displayed in the '''Graphics view'''.

|-
|04:36
| If required, drag the '''Graphics view''' to view the parabola clearly.

|-
|04:41
| Next type '''Root''' within brackets '''p''' and press '''Enter'''.

|-
|04:48
| Roots of the polynomial p are displayed as points '''A''' and '''B''' in '''Algebra''' and '''Graphics views'''.

|-
|04:56
| Let us type one more polynomial.

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

|-
|05:07
| Polynomial '''q of x''' is displayed in the '''Algebra view'''.

|-
|05:12
| Its graph, a parabola, is displayed in the '''Graphics view'''.

|-
|05:17
|Type '''Root''' within brackets '''q''' and press '''Enter'''.

|-
|05:22
|Roots of the polynomial '''q''' are displayed as points '''C''' and '''D''' in the '''Algebra '''and''' Graphics views'''.

|-
|05:30
| Here we see that the points B and C coincide with each other.

|-
|05:35
| Using the '''Move''' tool, we can move the labels to see them clearly.

|-
|05:41
|Pause the tutorial and do this assignment.

|-
|05:46
|Next we will use '''Remainder theorem''' to divide polynomials.

|-
|05:51
| Let '''p of x''' be any polynomial of degree greater or equal to 1.

|-
|05:57
| And ''''a'''' be any real number.

|-
|06:00
| If '''p of x''' is divided by a linear polynomial '''x minus a''', then the remainder is '''p of a'''.

|-
|06:08
| Dividend is equal to Divisor multiplied by Quotient plus remainder.

|-
|06:14
| Let us open a new '''Geogebra''' window. Click on '''File''' and '''New Window'''.

|-
|06:22
| In the '''input bar''' type, '''p1 is equal to 3x squared plus x minus 1''' and press '''Enter'''.

|-
|06:32
| Then type '''p2 is equal to x plus 1''' and press '''Enter'''.

|-
|06:38
|Now we will divide the polynomial '''p1''' with '''p2'''.

|-
|06:43
| In the '''input bar''' type, '''Division'''.

Two options appear.

Select the second option that contains polynomials.

|-
|06:52
| In place of '''Dividend Polynomial''' type '''p1'''.

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

Then press '''Enter'''.

|-
|07:03
| Two lines intersecting each other appear in the '''Graphics view'''.

|-
|07:08
|These lines represent division of the polynomials '''p1''' and '''p2'''.

|-
|07:14
| Quotient and remainder of the division are shown as a list.

|-
|07:19
|'''L1 is equal to within curly braces '''3x minus 2 comma 1'''.

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

|-
|07:30
| To show the second set of polynomials, I will open the '''Graphics 2 view.'''

|-
|07:35
| I will drag boundary to see the '''Graphics 2 view''' clearly.

|-
|07:40
| Then I will type polynomials '''q1''' and '''q2''' in the '''input bar'''.

|-
|07:45
| '''q1 is equal to 4x cube minus 3x squared minus x plus 1''' Press '''Enter'''.

|-
|07:54
| '''q2 is equal to x plus 1 ''' and press '''Enter'''.

|-
|08:01
| Type '''Division''', followed by polynomials '''q1 comma q2''' within brackets and press '''Enter'''.

|-
|08:10
| Quotient and remainder of the division are shown as a list.

|-
|08:15
|'''L2 is equal to within curly braces '''4xsquared minus7x plus 6 comma minus 5'''

|-
|08:24
|Here quotient is '''4xsquared minus7x plus 6''' and remainder is '''-5'''.

|-
|08:31
|Pause the video and solve the exercises based on remainder theorem.

|-
|08:37
| Now let us factorize the polynomials.

|-
|08:40
| Let us open a new '''GeoGebra''' window.Click on '''File''' and '''New Window'''.

|-
|08:47
| In the '''input bar''' type, '''p is equal to x squared minus 5x plus 6''' and press '''Enter'''.

|-
|08:57
| Type '''Factors''' and select '''Factors Polynomial''' option.

|-
|09:02
|In the place of the polynomial type '''p''' within brackets '''x''' and press '''Enter'''.

|-
|09:09
| Drag boundary to see the '''Algebra view''' clearly.

|-
|09:13
| '''M1 '''is displayed in the '''Algebra view'''.

|-
|09:16
|Here '''(x minus 3)''' and '''(x minus 2)''' are factors of the polynomial '''p of x'''.

|-
|09:23
| Let us try another example.

|-
|09:26
|Type '''Factors''' then type within brackets '''x cube minus 2x squared minus x plus 2'''
and press '''Enter'''.

|-
|09:38
| '''M2 '''is displayed in the '''Algebra view'''.

'''x minus 2'''

'''x minus 1'''

'''x plus 1'''

are the factors of the polynomial.

|-
|09:49
| Pause the video and solve the exercises based on factorization.

|-
|09:55
|Let us summarize what we have learnt.

|-
|09:57
| In this tutorial we have learnt about,

Polynomials of one variable

Slope of a linear polynomial

Degree of the polynomials

|-
|10:10
| Zeros of the polynomials

Roots of the polynomials

|-
|10:16
| Remainder theorem

Factorization of polynomials

|-
|10:21
| The video at the following link summarizes the Spoken Tutorial project. Please download and watch it.

|-
|10:28
| The '''Spoken Tutorial Project '''team:

conducts workshops using spoken tutorials and gives certificates on passing online tests.

|-
|10:37
|For more details, please write to us.

|-
|10:40
| Please post your timed queries in this forum.

|-
|10:44
| Spoken Tutorial Project is funded by NMEICT, MHRD, Government of India.
More information on this mission is available at this link.

|-
|10:55
| This is Madhuri Ganapathi from, IIT Bombay signing off.Thank you for watching.

|}

@@ Line 35: / Line 35: @@
 '''Ubuntu Linux''' OS version 16.04
-'''GeoGebra''' version 5.0438.0-d
+'''GeoGebra''' version 5.0.438.0-d
 |-

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

PoojaMoolya at 10:07, 10 October 2019

PoojaMoolya: Created page with "{|border=1 ||'''Time''' ||'''Narration''' |- |00:01 | Welcome to the spoken tutorial on '''Polynomials'''. |- |00:05 | In this tutorial we learn about, Polynomials of one..."