|00:01||Welcome to this tutorial on Roots of Polynomials.|
|00:06||In this tutorial, we will learn: To plot graphs of polynomial equations|
|00:13||About complex numbers, real and imaginary roots|
|00:18||To find extrema and inflection points|
|00:22||To follow this tutorial, you should be familiar with|
|00:28||Basics of coordinate system|
|00:33||If not, for relevant tutorials, please visit our website.|
|00:38||Here I am using:|
|00:41||Ubuntu Linux operating system version 14.04|
|00:46||GeoGebra 5.0.388.0 hyphen d|
|00:53||Let us begin with the binomial theorem.|
|00:57||a and b are real numbers.|
|01:01||index n is a positive integer.|
|01:05||r lies between 0 and n.|
|01:09||Binomial theorem states that a plus b raised to n can be expanded as shown.|
|01:18||Quadratic Equations and Roots|
|01:21||A second degree polynomial, y equals a x squared plus b x plus c has roots given by values of x.|
|01:31||x is equal to ratio of minus b plus or minus squareroot of b squared minus 4 a c to 2 a.|
|01:41||Where discriminant Delta is equal to b squared minus 4 a c|
|01:49||When Delta is less than 0, roots are complex|
|01:54||When Delta is equal to 0, roots are real and equal|
|01:59||When Delta is greater than 0, roots are real and unequal|
|02:05||When roots are real, ax squared plus b x plus c equals 0 has extremum xv comma yv|
|02:16||xv equals minus b divided by 2 a and yv equals axv squared plus bxv plus c|
|02:28||I have already opened the GeoGebra interface.|
|02:33||Click on View tool and select CAS to open the CAS view.|
|02:40||In line 1 in CAS view, type the following line.|
|02:45||f x in parentheses colon equals x caret 2 minus 2 space x minus 3.|
|02:47||To type caret symbol, hold Shift key down and press 6.|
|03:03||The space indicates multiplication.
|03:10||Drag boundary to see Algebra view properly.|
|03:15||Observe the equation f of x in Algebra view.|
|03:20||The degree of this quadratic polynomial f of x is 2.|
|03:26||Drag boundary to see Graphics view properly.|
|03:31||Click in Graphics view to see Graphics View toolbar.|
|03:37||Under Move Graphics View, click on Zoom Out tool.|
|03:42||Click in Graphics view to see the minimum vertex of parabola f.|
|03:48||Click on Move Graphics View tool and click in Graphics background.|
|03:55||When hand symbol appears, drag Graphics view, so you can see parabola f.|
|04:03||Drag boundaries to see CAS view properly.|
|04:08|| In line 2 of CAS view, type Root f in parentheses.
|04:17||The roots appear below, in the same box, in curly brackets.|
|04:22||Note that these are the x-intercepts of parabola f in Graphics view.|
|04:29|| In line 3 of CAS view, type Extremum f in parentheses.
|04:38||The extremum appears below, in the same box, in curly brackets.|
|04:44||Note that this is the minimum vertex of parabola f in Graphics view.|
|04:49|| In line 4 in CAS view, type the following line.
g x in parentheses colon equals x caret 2 plus 5 space x plus 10. Press Enter.
|05:07||Drag boundary to see Algebra view properly.|
|05:11||Observe the equation g of x in Algebra view.|
|05:16||Drag boundary to see Graphics view properly.|
|05:20||Uncheck f of x in CAS view.|
|05:24||Note that this also unchecks it in Algebra view and hides parabola f in Graphics view.|
|05:32||Click in and drag Graphics view so you can see parabola g.|
|05:40||Again, drag boundary to see CAS view properly.|
|05:46||In line 5 of CAS view, type Root g in parentheses. Press Enter.|
|05:56||Empty curly brackets appear below.|
|05:59||Parabola g does not have any real roots as it does not intersect x axis at all.|
|06:07||Roots are said to be complex.|
|06:10|| In line 6 of CAS view, type Extremum g in parentheses.
|06:20||The extremum appears below, in the same box, in curly brackets.|
|06:26||Note that this is the minimum vertex of parabola g in Graphics view.|
|06:33||While Evaluate tool is highlighted in CAS View toolbar, the extremum appears as fractions.|
|06:42||Minus 5 divided by 2 comma 15 divided by 4.|
|06:48||In line 6, click on the extremum and click on Numeric tool.|
|06:55||The extremum now appears in decimal form.|
|06:59||Minus 2 point 5 comma 3 point 7 5.|
|07:05||Let us look at complex numbers.|
|07:09||Complex numbers, XY plane|
|07:13||A complex number is expressed as z equals a plus bi.|
|07:18||a is the real part, bi is imaginary part, a and b are constants|
|07:26||i is imaginary number and is equal to square root of minus 1.|
|07:32||In the XY plane, a plus bi corresponds to the point a comma b.|
|07:40||In the complex plane, x axis is called real axis, y axis is called imaginary axis.|
|07:48||Complex numbers, complex plane|
|07:51||In complex plane, z is a vector.|
|07:55||Its real axis coordinate is a and imaginary axis coordinate is b.|
|08:02||The length of the vector z is equal to the absolute value of z and to r.|
|08:10||According to Pythagoras’ theorem, r is equal to squareroot of a squared plus b squared.|
|08:18||Let us go back to the GeoGebra interface we were working on.|
|08:24||We will now use the input bar instead of CAS view.|
|08:29||Click and close CAS view.|
|08:33||In Algebra view, uncheck g of x to hide it.|
|08:38||In input bar, type the following line.|
|08:42||h x in parentheses colon equals x caret 3 minus 4 space x caret 2 plus x plus 6.
|08:58||Drag boundaries to see Algebra and Graphics views properly.|
|09:04||Observe equation h of x in Algebra view.|
|09:09||Function h of x is graphed in Graphics view.|
|09:13|| Under Move Graphics View, click on Zoom Out tool.
Click in Graphics view.
|09:22||Click on Move Graphics View and move Graphics background to see the graph.|
|09:30||In input bar, type Root h in parentheses and press Enter.|
|09:38||The co-ordinates of three roots A, B and C appear in Algebra view.|
|09:44||The three roots are also mapped as x intercepts of the curve h of x in Graphics view.|
|09:51||In input bar, type Extremum h in parentheses and press Enter.|
|10:01||Co-ordinates of two extrema D and E appear in Algebra view.|
|10:07||The two extrema are also mapped on curve h of x in Graphics view.|
|10:14||Point of inflection|
|10:17||A point of inflection PoI on a curve is the point where the curve changes its direction.|
|10:25||To find the co-ordinates of PoI x comma y, We equate second derivative of the given function to 0.|
|10:35||Solution of this equation gives us x , x co-ordinate of PoI.|
|10:41||Substitute this x in original function to get y co-ordinate.|
|10:46||Let us find the point of inflection on h of x.|
|10:51||In input bar, type Inf and scroll down menu to choose InflectionPoint Polynomial option.|
|11:03||Instead of highlighted Polynomial, type h and press Enter.|
|11:09||In Algebra view, point of inflection appears as point F, below the two extrema.|
|11:16||F is mapped on h of x in Graphics view.|
|11:21||Correlate the degree of the polynomials and the number of roots seen so far.|
|11:29||Observe that functions entered in CAS appear in Algebra and Graphics views.|
|11:37||Functions entered in input bar appear in Algebra and Graphics views but not in CAS view.|
|11:46||Let us summarize.|
|11:48|| In this tutorial, we have learnt to:
Plot graphs of polynomial functions using CAS view and input bar
|11:57||Find real roots, extrema and inflection point(s)|
|12:02||Complex roots will be covered in another tutorial|
Plot graphs and find roots, extrema and inflection points for the following polynomials.
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