Welcome to the spoken tutorial on Polynomials.
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
To record this tutorial, I am using: Ubuntu Linux OS version 16.04 GeoGebra version 5.0.438.0-d
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.
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.
I have already opened the GeoGebra interface.
For this tutorial we will use input bar to solve the polynomials.
Let us first start with slope of a polynomial.
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.
Now type Slope within brackets r and press Enter.
Slope of r is shown on the line and in the Algebra view.
Now we will define the degree of a 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.
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.
The degree of the polynomial is displayed in the Algebra view as 7.
Similarly degree of the polynomial, 5x raised to the power of 5 minus 4x squared minus 6 is 5.
Pause the tutorial and do this assignment.
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.
Let us delete all the objects.
Press Ctrl + A to select all objects, then press Delete key on the keyboard.
To find zeros of the polynomial, in the input bar type, p is equal to 5x squared minus 3x plus 7 and press Enter.
I will drag the boundary of the Algebra view to see the polynomial clearly.
Move the Graphics view, if you cannot see the parabola.
Now we will find the values of p of 0, p of 1, p of 2 and p of 3.
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.
Similarly I will type p of 1, p of 2 and p of 3.
Values of p of 1, p of 2 and p of 3 are displayed in the Algebra view.
Pause the tutorial and complete this assignment.
I will clear the interface once again.
Now let us find the roots of the polynomial.
In the input bar type, p is equal to x squared minus x minus 2 and press Enter.
Polynomial p of x is displayed in the Algebra view.
Its graph, a parabola, is displayed in the Graphics view.
If required, drag the Graphics view to view the parabola clearly.
Next type Root within brackets p and press Enter.
Roots of the polynomial p are displayed as points A and B in Algebra and Graphics views.
Let us type one more polynomial. q is equal to x squared minus 5x plus 6 and press Enter.
Polynomial q of x is displayed in the Algebra view.
Its graph, a parabola, is displayed in the Graphics view.
Type Root within brackets q and press Enter.
Roots of the polynomial q are displayed as points C and D in the Algebra and Graphics views.
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.
Pause the tutorial and do this assignment.
Next we will use Remainder theorem to divide polynomials.
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.
Let us open a new Geogebra window. Click on File and New Window.
In the input bar type, p1 is equal to 3x squared plus x minus 1 and press Enter.
Then type p2 is equal to x plus 1 and press Enter.
Now we will divide the polynomial p1 with p2.
In the input bar type, Division. Two options appear. Select the second option that contains polynomials.
In place of Dividend Polynomial type p1. In place of Divisor Polynomial type p2. Then press Enter.
Two lines intersecting each other appear in the Graphics view.
These lines represent division of the polynomials p1 and p2.
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.
To show the second set of polynomials, I will open the Graphics 2 view.
I will drag boundary to see the Graphics 2 view clearly.
Then I will type polynomials q1 and q2 in the input bar.
q1 is equal to 4x cube minus 3x squared minus x plus 1 Press Enter.
q2 is equal to x plus 1 and press Enter.
Type Division, followed by polynomials q1 comma q2 within brackets and press Enter.
Quotient and remainder of the division are shown as a list.
L2 is equal to within curly braces 4xsquared minus7x plus 6 comma minus 5
Here quotient is 4xsquared minus7x plus 6 and remainder is -5.
Pause the video and solve the exercises based on remainder theorem.
Now let us factorize the polynomials.
Let us open a new GeoGebra window.Click on File and New Window.
In the input bar type, p is equal to x squared minus 5x plus 6 and 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 to see the Algebra view clearly.
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 then type within brackets x cube minus 2x squared minus x plus 2 and press Enter.
M2 is displayed in the Algebra view. x minus 2 x minus 1 x plus 1 are the factors of the polynomial.
Pause the video and solve the exercises based on factorization.
Let us summarize what we have learnt.
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
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This is Madhuri Ganapathi from, IIT Bombay signing off.Thank you for watching.