GeoGebra-5.04/C2/Polynomials/English-timed

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 5.0438.0-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.

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