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

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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.

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