GeoGebra-5.04/C2/Polynomials/English-timed

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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.

Contributors and Content Editors

PoojaMoolya