Difference between revisions of "GeoGebra-5.04/C2/Polynomials/English-timed"
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'''Ubuntu Linux''' OS version 16.04 | '''Ubuntu Linux''' OS version 16.04 | ||
| − | '''GeoGebra''' version 5. | + | '''GeoGebra''' version 5.0.438.0-d |
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Latest revision as of 15:37, 10 October 2019
| 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 5.0.438.0-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. |
