GeoGebra-5.04/C2/Polynomials/English
| Visual Cue | Narration |
| Slide Number 1
Title slide |
Welcome to the spoken tutorial on Polynomials. |
| Slide Number 2
Learning Objectives |
In this tutorial we learn about,
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| Slide Number 3
System Requirement |
To record this tutorial, I am using:
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| Slide Number 4
Pre-requisites www.spoken-tutorial.org |
To follow this tutorial, learner should be familiar with GeoGebra interface.
For the prerequisite GeoGebra tutorials, please visit this website. |
| Let us first define a polynomial. | |
| Slide Number 5
Polynomial |
An algebraic expression containing one or more terms with non-zero coefficients is a polynomial.
For example x cube plus 3 x squared plus 2 x minus 5 is a polynomial. |
| Cursor on GeoGebra interface. | I have already opened the GeoGebra interface. |
| Point to the input bar. | For this tutorial we will use input bar to solve the polynomials. |
| Let us first start with slope of a polynomial. | |
| Type r(x)=3x-3 >> press Enter.
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In the input bar type,
r within brackets x is equal to 3x minus 3 and press Enter.
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| Type Slope(r) >> press Enter. | Now type Slope within brackets r and press Enter. |
| Point to the slope of line and Algebra view. | Slope of r is shown on the line and in the Algebra view. |
| Now we will define the degree of a polynomial. | |
| Slide Number 6
Degree of polynomial |
The highest power of the variable in a polynomial, is the degree of the polynomial.
For example, p is equal to x raised to the power of 5 minus x raised to the power of 4 plus 3 In this polynomial, degree is '5'. |
| Let’s try some more examples to find the degree of polynomials. | |
| Type Degree(3x^7+4x^6+x+9) | In the input bar type, Degree.
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| Point the value in the Algebra view. | The degree of the polynomial is displayed in the Algebra view as 7. |
| Type Degree(5x^5-4x^2-6)
point to the Algebra view. |
Similarly degree of the polynomial,
5x raised to the power of 5 minus 4x squared minus 6 is 5. |
| Slide Number 7
Assignment Find the degree of the given polynomials 1. x^5-x^4+3 2. 2-y^2-y^3+2y^8 3. 5x^3+4x^2+7x |
Pause the tutorial and do this assignment. |
| Slide number 8
Zeros of Polynomial |
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. |
| Cursor on the interface.
Press Ctrl +A to select all objects >> press Delete key on keyboard. |
Let us delete all the objects.
Press Ctrl + A to select all objects, then press Delete key on the keyboard. |
| In the input bar type,
p=5x^2-3x+7 >> press Enter. |
To find zeros of the polynomial, in the input bar type,
p is equal to 5x squared minus 3x plus 7 and press Enter. |
| Drag Boundary of Algebra view. | I will drag the boundary of the Algebra view to see the polynomial clearly. |
| Click and drag the Graphics view. | Move the Graphics view, if you cannot see the parabola. |
| p(0)=5(0)^2-3(0)+7 = 7
p(1)=5(1)^2-3(1)+7 = 9 p(2)=5(2)^2-3(2)+7 = 21 p(3)=5(3)^2-3(3)+7 = 43 |
Now we will find the values of p of 0, p of 1, p of 2 and p of 3. |
| Type p(0) >> press Enter
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In the input bar type p, then type 0 within brackets and press Enter.
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| Type p(1) >> press Enter.
Type p(2) >> press Enter. Type p(3) >> press Enter. |
Similarly I will type p of 1, p of 2 and p of 3. |
| Point to p(1), p(2), p(3) values in Algebra view. | Values of p of 1, p of 2 and p of 3 are displayed in the Algebra view. |
| Slide Number 9
Assignment Find the values of p of 0, p of 1 and p of 2 for the given polynomials. 1. p=2+t+t^2-t^3 2. p=(x-1)(x+1) 3. p=x^3 |
Pause the tutorial and complete this assignment. |
| Press Ctrl +A >> press Delete key. | I will clear the interface once again. |
| Roots of the polynomial. | Now let us find the roots of the polynomial. |
| In the input bar type, p= x^2-x-2 >> press Enter. | In the input bar type,
p is equal to x squared minus x minus 2 and press Enter. |
| Point to the polynomial in Algebra and Graphics view. | Polynomial p of x is displayed in the Algebra view.
Its graph, a parabola, is displayed in the Graphics view. |
| Drag the Graphics view. | If required, drag the Graphics view to view the parabola clearly. |
| Type Root(p) >> press Enter. | Next type Root within brackets p and press Enter. |
| Point to the roots in Graphics and Algebra views.
A(-1,0) and B(2,0) |
Roots of the polynomial p are displayed
as points A and B in Algebra and Graphics views. |
| Type q=x^2-5x+6 | Let us type one more polynomial.
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| Point to the polynomial in the Algebra view.
Point to the graph in the Graphics view. |
Polynomial q of x is displayed in the Algebra view.
Its graph, a parabola, is displayed in the Graphics view. |
| Type Root(q) >> press Enter. | Type Root within brackets q and press Enter. |
| Point to the roots in Algebra and Graphics views.
C(2,0) and D(3,0). |
Roots of the polynomial q are displayed
as points C and D in the Algebra and Graphics views. |
| Point to coincided B and C.
Click on Move tool >> drag the labels. |
Here we see that the points B and C coincide with each other.
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| Slide Number 10
Assignment Find the roots of the following polynomials. 1. f= x^2-2x+1 2. g=2x+1 3. h=x^2-1 |
Pause the tutorial and do this assignment. |
| Next we will use Remainder theorem to divide polynomials. | |
| Slide Number 11
Remainder theorem |
Let p of x be any polynomial of degree greater or equal to 1.
And 'a' be any real number. If p of x is divided by a linear polynomial x minus a, then the remainder is p of a. Dividend is equal to Divisor multiplied by Quotient plus remainder. |
| Click on File >> New Window. | Let us open a new Geogebra window.
Click on File and New Window. |
| Illustrations for polynomial division
Type, p1=3x^2+x-1 press Enter. |
In the input bar type,
p1 is equal to 3x squared plus x minus 1 and press Enter. |
| Type p2=x+1 press Enter. | Then type p2 is equal to x plus 1
and press Enter. |
| Point p1 and p2. | Now we will divide the polynomial p1 with p2. |
| In the input bar type, Division.
Point to the options. Select Division(<Dividend Polynomial>, <Divisor Polynomial>) |
In the input bar type, Division.
Two options appear. Select the second option that contains polynomials. |
| Type p1.
Type p2. Press Enter. |
In place of Dividend Polynomial type p1.
In place of Divisor Polynomial type p2. Then press Enter. |
| Point to lines. | Two lines intersecting each other appear in the Graphics view.
These lines represent division of the polynomials p1 and p2. |
| q =3x - 2
r = 1. Point to the quotient and remainder. |
Quotient and remainder of the division are shown as a list.
L1 is equal to within curly braces 3x minus 2 comma 1. Here quotient is 3x-2 and remainder is 1. |
| Go to View menu >> Select Graphics 2 check box. | To show the second set of polynomials, I will open the Graphics 2 view. |
| Drag boundary to see the Graphics 2 view. | I will drag boundary to see the Graphics 2 view clearly. |
| Point to the input bar. | Then I will type polynomials q1 and q2 in the input bar. |
| Type
q1=4x^3-3x^2-x +1 >> press Enter. |
q1 is equal to 4x cube minus 3x squared minus x plus 1
Press Enter. |
| Type q2=x+1 >> press Enter. | q2 is equal to x plus 1 and press Enter. |
| Type,
Division(q1, q2) |
Type Division, followed by polynomials q1 comma q2 within brackets and press Enter. |
| L2={4x2 -7x +6, -5}
q = 4x2 -7x +6 r = -5 Point to the quotient and remainder. |
Quotient and remainder of the division are shown as a list.
L2 is equal to within curly braces 4x squared minus 7x plus 6 comma minus 5 Here quotient is 4x squared minus 7x plus 6 and remainder is -5. |
| Slide Number 12
Assignment Solve the exercises based on remainder theorem. 1. p1=x^4+x^3-2x^2+x , p2=x-1 2. p1=x^3+3x^2+3x+1, p2=2x+5 3. p1=3x^3+7x, p2=3x+7 |
Pause the video and solve the exercises based on remainder theorem. |
| Factorization of polynomials | Let us now factorize the polynomials. |
| Click on File and New Window. | Let us open a new GeoGebra window.
Click on File and New Window. |
| type, p=x^2-5x+6 >> press Enter. | In the input bar type,
p is equal to x squared minus 5x plus 6 and press Enter. |
| Type Factors >> select Factors(<Polynomial>) option.
Type p(x) >> press Enter. |
Type Factors and select Factors Polynomial option.
In the place of the polynomial type p within brackets x and press Enter. |
| Drag boundary of Algebra view. | Drag boundary to see the Algebra view clearly. |
| M1= {{x-3,1}, {x-2, 1}}
Point to (x-3)(x-2). |
M1 is displayed in the Algebra view.
Here (x minus 3) and (x minus 2) are factors of the polynomial p of x. |
| Let us try another example. | |
| Type Factors( x^3-2x^2-x+2 ) | Type Factors then type within brackets x cube minus 2x squared minus x plus 2
and press Enter. |
| M2={{x-2,1},{x-1, 1},{x+1,1}}
Point M2 in the Algebra view. |
M2 is displayed in the Algebra view.
x minus 2 x minus 1 x plus 1 are the factors of the polynomial. |
| Slide Number 13
Assignment Solve the exercises based on factorization 1. p=x^3-2x^2-x+2 2. p=12x^2-7x+1 3. p=2x^2+7x+3 4. p=x^3+13x^2+32x+20 |
Pause the video and solve the exercises based on factorization. |
| Let us summarize what we have learnt. | |
| Slide Number 14
Summary |
In this tutorial we have learnt about,
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| Slide Number 15
About Spoken Tutorial project |
The video at the following link summarizes the Spoken Tutorial project.
Please download and watch it. |
| Slide Number 16
Spoken Tutorial workshops |
The Spoken Tutorial Project team:
For more details, please write to us. |
| Slide Number 17
Forum for specific questions: Do you have questions in THIS Spoken Tutorial?
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Please post your timed queries in this forum. |
| Slide Number 18
Acknowledgement |
Spoken Tutorial Project is funded by NMEICT, MHRD, Government of India.
More information on this mission is available at this link. |
| This is Madhuri Ganapathi from, IIT Bombay signing off.
Thank you for watching. |
