GeoGebra-5.04/C2/Polynomials/English

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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,
  • Polynomials of one variable
  • Slope of a linear polynomial
  • Degree of the polynomials
  • Zeros of the polynomials
  • Roots of the polynomials
  • Remainder theorem
  • Factorization of polynomials
Slide Number 3

System Requirement

To record this tutorial, I am using:
  • Ubuntu Linux OS version 16.04
  • GeoGebra version 5.0438.0-d
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.


Point to Algebra and Graphics views.

In the input bar type,

r within brackets x is equal to 3x minus 3

and press Enter.


The linear polynomial is displayed in the Algebra and Graphics views.

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.


In place of polynomial type,


3x raised to the power of 7 plus 4x raised to the power of 6 plus x plus 9


Press Enter.

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


Point to p(0) value in the Algebra view .

In the input bar type p, then type 0 within brackets and press Enter.


The value of p of 0 is displayed in the Algebra view.

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.


q is equal to x squared minus 5x plus 6 and press Enter.

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.


Using the Move tool, we can move the labels to see them clearly.

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,
  • Polynomials of one variable
  • Slope of a linear polynomial
  • Degree of the polynomials
  • Zeros of the polynomials
  • Roots of the polynomials
  • Remainder theorem
  • Factorization of polynomials
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:
  • conducts workshops using spoken tutorials and
  • gives certificates on passing online tests.

For more details, please write to us.

Slide Number 17

Forum for specific questions:

Do you have questions in THIS Spoken Tutorial?

  • Please visit this site
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  • Explain your question briefly
  • Someone from our team will answer them.
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.

Contributors and Content Editors

Madhurig, Nancyvarkey