PoojaMoolya: Created page with "{|border=1 ||'''Time''' ||'''Narration''' |- || 00:01 || Welcome to this tutorial on '''Roots of Polynomials'''. |- || 00:06 || In this tutorial, we will learn: To plot gra..."

2020-10-21T07:21:27Z

Created page with "{|border=1 ||'''Time''' ||'''Narration''' |- || 00:01 || Welcome to this tutorial on '''Roots of Polynomials'''. |- || 00:06 || In this tutorial, we will learn: To plot gra..."

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

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|| 00:01
|| Welcome to this tutorial on '''Roots of Polynomials'''.

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|| 00:06
|| In this tutorial, we will learn: To plot graphs of '''polynomial''' equations

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|| 00:13
|| About '''complex numbers''', '''real''' and '''imaginary roots'''

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|| 00:18
|| To find '''extrema''' and '''inflection points'''

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|| 00:22
|| To follow this tutorial, you should be familiar with

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|| 00:25
|| '''GeoGebra''' interface

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|| 00:28
|| Basics of '''coordinate system'''

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|| 00:31
|| '''Polynomials'''

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|| 00:33
|| If not, for relevant tutorials, please visit our website.

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||00:38
|| Here I am using:

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|| 00:41
|| '''Ubuntu Linux''' operating system version 14.04

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|| 00:46
|| '''GeoGebra 5.0.388.0 hyphen d'''
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|| 00:53
|| Let us begin with the '''binomial theorem'''.

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||00:57
|| '''''a''''' and '''''b''''' are '''real numbers'''.

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|| 01:01
||'''index''' '''''n''''' is a positive integer.

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|| 01:05
||'''''r''''' lies between 0 and '''''n'''''.

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|| 01:09
||'''Binomial theorem''' states that '''''a''''' plus '''''b''''' raised to '''''n''''' can be expanded as shown.

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|| 01:18
|| '''Quadratic Equations and Roots'''

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|| 01:21
||A '''second degree polynomial''', '''y equals''' '''''a''''' '''x squared plus''' '''''b''''' '''x plus''' '''''c''''' has '''roots''' given by values of '''''x'''''.

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|| 01:31
||'''''x''''' is equal to '''ratio''' of minus '''''b''''' plus or minus '''squareroot''' of '''''b''''' '''squared''' minus 4 '''''a c''''' to 2 '''''a'''''.

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|| 01:41
||Where '''discriminant''' '''Delta''' is equal to '''''b''''' '''squared''' minus 4 '''''a c'''''

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|| 01:49
||When '''Delta''' is less than 0, '''roots''' are '''complex'''

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|| 01:54
||When '''Delta''' is equal to 0, '''roots''' are '''real''' and equal

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|| 01:59
||When '''Delta''' is greater than 0, '''roots''' are '''real''' and unequal
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|| 02:05
||When '''roots''' are '''real''', '''''ax''''' '''squared plus''' '''''b x''''' '''plus''' '''''c''''' equals 0 has '''extremum''' '''''xv''''' '''comma''' '''''yv'''''

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|| 02:16
||'''''xv''''' equals '''minus''' '''''b''''' '''divided by 2''' '''''a''''' and '''''yv''''' '''equals''' '''''axv''''' '''squared plus''' '''''bxv''''' '''plus''' '''''c'''''
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|| 02:28
|| I have already opened the '''GeoGebra''' interface.
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|| 02:33
|| Click on '''View''' tool and select '''CAS''' to open the '''CAS''' view.
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|| 02:40
|| In line 1 in '''CAS''' view, type the following line.

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|| 02:45
||'''f x''' in parentheses '''colon equals x caret 2 minus 2 space x minus 3'''.

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||02:47
||To type '''caret''' symbol, hold '''Shift''' key down and press 6.

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|| 03:03
||The space indicates multiplication.

Press '''Enter'''.
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|| 03:10
|| Drag boundary to see '''Algebra''' view properly.
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|| 03:15
|| Observe the '''equation f of x''' in '''Algebra''' view.

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|| 03:20
||The '''degree''' of this '''quadratic polynomial f of x''' is 2.
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|| 03:26
|| Drag boundary to see '''Graphics''' view properly.
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||03:31
|| Click in '''Graphics''' view to see '''Graphics View''' toolbar.
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||03:37
|| Under '''Move Graphics View''', click on '''Zoom Out''' tool.

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||03:42
||Click in '''Graphics''' view to see the minimum '''vertex''' of '''parabola f'''.

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|| 03:48
|| Click on '''Move Graphics View''' tool and click in '''Graphics''' background.
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||03:55
||When hand symbol appears, drag '''Graphics''' view, so you can see parabola '''f'''.
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|| 04:03
|| Drag boundaries to see '''CAS''' view properly.
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|| 04:08
|| In line 2 of '''CAS''' view, type '''Root f''' in parentheses.

Press '''Enter'''.
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|| 04:17
|| The '''roots''' appear below, in the same box, in curly brackets.
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|| 04:22
|| Note that these are the '''x-intercepts''' of parabola '''f''' in '''Graphics''' view.
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|| 04:29
|| In line 3 of '''CAS''' view, type '''Extremum f''' in parentheses.

Press '''Enter'''.
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|| 04:38
|| The '''extremum''' appears below, in the same box, in curly brackets.
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||04:44
|| Note that this is the minimum '''vertex''' of parabola '''f''' in '''Graphics''' view.
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|| 04:49
|| 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'''.
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|| 05:07
|| Drag boundary to see '''Algebra''' view properly.

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|| 05:11
||Observe the '''equation g of x''' in '''Algebra''' view.
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|| 05:16
|| Drag boundary to see '''Graphics''' view properly.
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||05:20
|| Uncheck '''f of x''' in '''CAS''' view.

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|| 05:24
||Note that this also unchecks it in '''Algebra''' view and hides parabola '''f''' in '''Graphics''' view.

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|| 05:32
|| Click in and drag '''Graphics''' view so you can see parabola '''g'''.

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|| 05:40
|| Again, drag boundary to see '''CAS''' view properly.

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|| 05:46
|| In line 5 of '''CAS''' view, type '''Root g''' in parentheses. Press '''Enter'''.

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|| 05:56
|| Empty curly brackets appear below.

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|| 05:59
||Parabola '''g''' does not have any '''real roots''' as it does not intersect '''x axis''' at all.

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|| 06:07
||'''Roots''' are said to be '''complex'''.

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|| 06:10
|| In line 6 of '''CAS''' view, type '''Extremum g''' in parentheses.

Press '''Enter'''.
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|| 06:20
|| The '''extremum''' appears below, in the same box, in curly brackets.
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|| 06:26
|| Note that this is the minimum '''vertex''' of parabola '''g''' in '''Graphics''' view.
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|| 06:33
|| While '''Evaluate''' tool is highlighted in '''CAS View''' toolbar, the '''extremum''' appears as '''fractions'''.

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|| 06:42
||'''Minus 5 divided by 2 comma 15 divided by 4'''.

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|| 06:48
|| In line 6, click on the '''extremum''' and click on '''Numeric''' tool.

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|| 06:55
||The '''extremum '''now appears in '''decimal''' form.

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|| 06:59
||'''Minus 2 point 5 comma 3 point 7 5'''.

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|| 07:05
|| Let us look at '''complex numbers'''.

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|| 07:09
|| '''Complex numbers, XY plane'''

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|| 07:13
||A '''complex number''' is expressed as '''''z''''' '''equals''' '''''a''''' '''plus''' '''''bi'''''.

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|| 07:18
||'''''a''''' is the '''real''' part, '''''bi''''' is '''imaginary '''part, '''''a''''' and '''''b''''' are '''constants'''

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|| 07:26
||'''''i''''' is '''imaginary number''' and is equal to '''square root of minus 1'''.

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|| 07:32
||In the '''XY plane''', '''''a''''' '''plus''' '''''bi''''' corresponds to the point '''''a''''' '''comma''' '''''b'''''.

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|| 07:40
||In the '''complex plane''', '''x axis''' is called '''real axis, y axis''' is called '''imaginary axis'''.

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|| 07:48
|| '''Complex numbers, complex plane'''

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|| 07:51
||In '''complex plane''', '''''z''''' is a '''vector'''.

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|| 07:55
||Its '''real axis coordinate''' is '''''a''''' and '''imaginary axis coordinate''' is '''''b'''''.

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|| 08:02
||The length of the '''vector''' '''''z''''' is equal to the '''absolute value''' of '''''z''''' and to '''''r'''''.

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|| 08:10
||According to '''Pythagoras’ theorem''', '''''r''''' is equal to '''squareroot of''' '''''a''''' '''squared plus''' '''''b''''' '''squared'''.
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||08:18
|| Let us go back to the '''GeoGebra interface''' we were working on.

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|| 08:24
||We will now use the '''input bar''' instead of '''CAS''' view.
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|| 08:29
|| Click and close '''CAS''' view.
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|| 08:33
|| In '''Algebra''' view, uncheck '''g of x''' to hide it.
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|| 08:38
|| In '''input bar''', type the following line.

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|| 08:42
||'''h x''' in parentheses '''colon equals x caret 3 minus 4 space x caret 2 plus x plus 6'''.

Press '''Enter'''.
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|| 08:58
|| Drag boundaries to see '''Algebra''' and '''Graphics''' views properly.

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|| 09:04
||Observe equation '''h of x''' in '''Algebra''' view.

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|| 09:09
||Function '''h of x''' is graphed in '''Graphics''' view.
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|| 09:13
|| Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view.
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|| 09:22
|| Click on '''Move Graphics View''' and move '''Graphics''' background to see the graph.
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|| 09:30
|| In '''input bar''', type '''Root h''' in parentheses and press '''Enter'''.
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|| 09:38
|| The '''co-ordinates''' of three '''roots''' '''A, B''' and '''C''' appear in '''Algebra''' view.
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|| 09:44
|| The three '''roots''' are also mapped as '''x intercepts''' of the '''curve h of x''' in '''Graphics''' view.
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|| 09:51
|| In '''input bar''', type '''Extremum h''' in parentheses and press '''Enter'''.
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|| 10:01
|| '''Co-ordinates''' of two '''extrema''' '''D''' and '''E''' appear in '''Algebra''' view.
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|| 10:07
|| The two '''extrema''' are also mapped on '''curve h of x''' in '''Graphics''' view.
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|| 10:14
|| '''Point of inflection'''

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|| 10:17
||A '''point of inflection PoI''' on a curve is the point where the '''curve''' changes its direction.

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|| 10:25
||To find the '''co-ordinates''' of '''PoI''' '''''x''''' comma '''''y''''', We equate second '''derivative''' of the given '''function''' to 0.

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|| 10:35
||Solution of this equation gives us '''x''' , '''x co-ordinate''' of '''PoI'''.

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|| 10:41
||Substitute this '''x''' in original '''function''' to get '''y co-ordinate'''.
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|| 10:46
|| Let us find the '''point of inflection''' on '''h of x'''.
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|| 10:51
|| In '''input bar''', type '''Inf''' and scroll down menu to choose '''InflectionPoint Polynomial''' option.
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|| 11:03
|| Instead of highlighted '''Polynomial''', type '''h''' and press '''Enter'''.
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|| 11:09
|| In '''Algebra''' view, '''point of inflection''' appears as point '''F''', below the two '''extrema'''.
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|| 11:16
|| '''F''' is mapped on '''h of x''' in '''Graphics''' view.
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|| 11:21
|| Correlate the '''degree''' of the '''polynomials''' and the number of '''roots''' seen so far.
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|| 11:29
|| Observe that '''functions''' entered in '''CAS''' appear in '''Algebra''' and '''Graphics''' views.
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||11:37
|| '''Functions''' entered in '''input bar''' appear in '''Algebra''' and '''Graphics''' views but not in '''CAS''' view.
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|| 11:46
|| Let us summarize.
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|| 11:48
|| In this tutorial, we have learnt to:

Plot graphs of '''polynomial functions''' using '''CAS''' view and '''input bar'''

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|| 11:57
|| Find '''real roots, extrema''' and '''inflection point(s)'''

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|| 12:02
|| '''Complex roots''' will be covered in another tutorial
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||12:06
|| Assignment:

Plot '''graphs''' and find '''roots''', '''extrema''' and '''inflection points''' for the following '''polynomials'''.
|-
|| 12:17
|| The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
|| 12:25
|| The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
|| 12:34
|| Please post your timed queries on this forum.
|-
|| 12:38
|| '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.

More information on this mission is available at this link.
|-
|| 12:51
|| This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Roots-of-Polynomials/English-timed - Revision history

PoojaMoolya: Created page with "{|border=1 ||'''Time''' ||'''Narration''' |- || 00:01 || Welcome to this tutorial on '''Roots of Polynomials'''. |- || 00:06 || In this tutorial, we will learn: To plot gra..."