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

Time	Narration
00:01	Welcome to this tutorial on Roots of Polynomials.
00:06	In this tutorial, we will learn: To plot graphs of polynomial equations
00:13	About complex numbers, real and imaginary roots
00:18	To find extrema and inflection points
00:22	To follow this tutorial, you should be familiar with
00:25	GeoGebra interface
00:28	Basics of coordinate system
00:31	Polynomials
00:33	If not, for relevant tutorials, please visit our website.
00:38	Here I am using:
00:41	Ubuntu Linux operating system version 14.04
00:46	GeoGebra 5.0.388.0 hyphen d
00:53	Let us begin with the binomial theorem.
00:57	a and b are real numbers.
01:01	index n is a positive integer.
01:05	r lies between 0 and n.
01:09	Binomial theorem states that a plus b raised to n can be expanded as shown.
01:18	Quadratic Equations and Roots
01:21	A second degree polynomial, y equals a x squared plus b x plus c has roots given by values of x.
01:31	x is equal to ratio of minus b plus or minus squareroot of b squared minus 4 a c to 2 a.
01:41	Where discriminant Delta is equal to b squared minus 4 a c
01:49	When Delta is less than 0, roots are complex
01:54	When Delta is equal to 0, roots are real and equal
01:59	When Delta is greater than 0, roots are real and unequal
02:05	When roots are real, ax squared plus b x plus c equals 0 has extremum xv comma yv
02:16	xv equals minus b divided by 2 a and yv equals axv squared plus bxv plus c
02:28	I have already opened the GeoGebra interface.
02:33	Click on View tool and select CAS to open the CAS view.
02:40	In line 1 in CAS view, type the following line.
02:45	f x in parentheses colon equals x caret 2 minus 2 space x minus 3.
02:47	To type caret symbol, hold Shift key down and press 6.
03:03	The space indicates multiplication. Press Enter.
03:10	Drag boundary to see Algebra view properly.
03:15	Observe the equation f of x in Algebra view.
03:20	The degree of this quadratic polynomial f of x is 2.
03:26	Drag boundary to see Graphics view properly.
03:31	Click in Graphics view to see Graphics View toolbar.
03:37	Under Move Graphics View, click on Zoom Out tool.
03:42	Click in Graphics view to see the minimum vertex of parabola f.
03:48	Click on Move Graphics View tool and click in Graphics background.
03:55	When hand symbol appears, drag Graphics view, so you can see parabola f.
04:03	Drag boundaries to see CAS view properly.
04:08	In line 2 of CAS view, type Root f in parentheses. Press Enter.
04:17	The roots appear below, in the same box, in curly brackets.
04:22	Note that these are the x-intercepts of parabola f in Graphics view.
04:29	In line 3 of CAS view, type Extremum f in parentheses. Press Enter.
04:38	The extremum appears below, in the same box, in curly brackets.
04:44	Note that this is the minimum vertex of parabola f in Graphics view.
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.
05:07	Drag boundary to see Algebra view properly.
05:11	Observe the equation g of x in Algebra view.
05:16	Drag boundary to see Graphics view properly.
05:20	Uncheck f of x in CAS view.
05:24	Note that this also unchecks it in Algebra view and hides parabola f in Graphics view.
05:32	Click in and drag Graphics view so you can see parabola g.
05:40	Again, drag boundary to see CAS view properly.
05:46	In line 5 of CAS view, type Root g in parentheses. Press Enter.
05:56	Empty curly brackets appear below.
05:59	Parabola g does not have any real roots as it does not intersect x axis at all.
06:07	Roots are said to be complex.
06:10	In line 6 of CAS view, type Extremum g in parentheses. Press Enter.
06:20	The extremum appears below, in the same box, in curly brackets.
06:26	Note that this is the minimum vertex of parabola g in Graphics view.
06:33	While Evaluate tool is highlighted in CAS View toolbar, the extremum appears as fractions.
06:42	Minus 5 divided by 2 comma 15 divided by 4.
06:48	In line 6, click on the extremum and click on Numeric tool.
06:55	The extremum now appears in decimal form.
06:59	Minus 2 point 5 comma 3 point 7 5.
07:05	Let us look at complex numbers.
07:09	Complex numbers, XY plane
07:13	A complex number is expressed as z equals a plus bi.
07:18	a is the real part, bi is imaginary part, a and b are constants
07:26	i is imaginary number and is equal to square root of minus 1.
07:32	In the XY plane, a plus bi corresponds to the point a comma b.
07:40	In the complex plane, x axis is called real axis, y axis is called imaginary axis.
07:48	Complex numbers, complex plane
07:51	In complex plane, z is a vector.
07:55	Its real axis coordinate is a and imaginary axis coordinate is b.
08:02	The length of the vector z is equal to the absolute value of z and to r.
08:10	According to Pythagoras’ theorem, r is equal to squareroot of a squared plus b squared.
08:18	Let us go back to the GeoGebra interface we were working on.
08:24	We will now use the input bar instead of CAS view.
08:29	Click and close CAS view.
08:33	In Algebra view, uncheck g of x to hide it.
08:38	In input bar, type the following line.
08:42	h x in parentheses colon equals x caret 3 minus 4 space x caret 2 plus x plus 6. Press Enter.
08:58	Drag boundaries to see Algebra and Graphics views properly.
09:04	Observe equation h of x in Algebra view.
09:09	Function h of x is graphed in Graphics view.
09:13	Under Move Graphics View, click on Zoom Out tool. Click in Graphics view.
09:22	Click on Move Graphics View and move Graphics background to see the graph.
09:30	In input bar, type Root h in parentheses and press Enter.
09:38	The co-ordinates of three roots A, B and C appear in Algebra view.
09:44	The three roots are also mapped as x intercepts of the curve h of x in Graphics view.
09:51	In input bar, type Extremum h in parentheses and press Enter.
10:01	Co-ordinates of two extrema D and E appear in Algebra view.
10:07	The two extrema are also mapped on curve h of x in Graphics view.
10:14	Point of inflection
10:17	A point of inflection PoI on a curve is the point where the curve changes its direction.
10:25	To find the co-ordinates of PoI x comma y, We equate second derivative of the given function to 0.
10:35	Solution of this equation gives us x , x co-ordinate of PoI.
10:41	Substitute this x in original function to get y co-ordinate.
10:46	Let us find the point of inflection on h of x.
10:51	In input bar, type Inf and scroll down menu to choose InflectionPoint Polynomial option.
11:03	Instead of highlighted Polynomial, type h and press Enter.
11:09	In Algebra view, point of inflection appears as point F, below the two extrema.
11:16	F is mapped on h of x in Graphics view.
11:21	Correlate the degree of the polynomials and the number of roots seen so far.
11:29	Observe that functions entered in CAS appear in Algebra and Graphics views.
11:37	Functions entered in input bar appear in Algebra and Graphics views but not in CAS view.
11:46	Let us summarize.
11:48	In this tutorial, we have learnt to: Plot graphs of polynomial functions using CAS view and input bar
11:57	Find real roots, extrema and inflection point(s)
12:02	Complex roots will be covered in another tutorial
12:06	Assignment: Plot graphs and find roots, extrema and inflection points for the following polynomials.
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12:51	This is Vidhya Iyer from IIT Bombay, signing off. Thank you for joining.

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