Applications-of-GeoGebra/C2/Roots-of-Polynomials/English-timed
From Script | Spoken-Tutorial
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. |