Polynomials

POLYNOMIALSMaths in English

An experiment at school…

ΠΡΟΤΥΠΟ ΠΕΙΡΑΜΑΤΙΚΟ ΛΥΚΕΙΟ ΒΑΡΒΑΚΕΙΟ Υ ΣΧΟΛΗΣ Σχ. Έτος 2011-2012

Μέσα στην τάξη του Β2 μία ομάδα μαθητών/τριών αποφασίσαμε να κάνουμε ένα πείραμα με τίτλο «Μαθηματικά στα Αγγλικά». Επιλέξαμε το κεφάλαιο των Μαθηματικών «Πολυώνυμα», το οποίο διδασκόμαστε εκείνο το διάστημα, το μεταφράσαμε στα Αγγλικά και το παρουσιάσαμε στην τάξη στα Αγγλικά. Βοηθούς στο πείραμά μας είχαμε τον καθηγητή των Μαθηματικών Δρ Γ. Κόσσυβα και την καθηγήτρια των Αγγλικών Δρ Λ. Νιτσοπούλου. Βοηθητικό υλικό είχαμε ένα βιβλίο Μαθηματικών στα Αγγλικά, που διδάσκεται σε Αμερικανικά κολλέγια μέσης εκπαίδευσης και ένα γλωσσάρι μαθηματικών όρων που έφτιαξε η καθηγήτρια κ. Νιτσοπούλου. Στη συνέχεια, μαζί με ομάδα μαθητών/τριών από το Β4 προχωρήσαμε σε σύγκριση των δύο σχολικών βιβλίων (Ελληνικό και Αμερικανικό) και παραθέσαμε τα αποτελέσματα της σύγκρισης.

ΤΟ ΠΕΙΡΑΜΑ ΜΑΣ

ΣΚΟΠΟΙΗ ανάπτυξη και η ενθάρρυνση της

ομαδοσυνεργατικής προσέγγιση της μαθησιακής διδασκαλίας.

Η μελέτη της διαθεματικότητας (Μαθηματικά- Αγγλικά ).

Η βιωματική μάθηση με την λύση ασκήσεων στα αγγλικά.

Η ενθάρρυνση της δημιουργικότητας των μαθητών στην ψηφιακή τεχνολογία.

Μια πρώτη προσέγγιση της ερευνητικής εργασίας(project) στο πλαίσιο του αναλυτικού προγράμματος.

Η διερευνητική προσέγγιση της μάθησης.Η διαπολιτισμική σκέψη και προσέγγιση των

μαθητών/τριών με την σύγκριση Αμερικανικού-Ελληνικού σχολικού βιβλίου Μαθηματικών .

ΣΤΟΧΟΙΗ ανάπτυξη της διαμεσολαβητικής

ικανότητας των μαθητών /τριώνΗ διαπολιτισμική προσέγγιση του μαθήματος

μέσα από τη σύγκριση Ελληνικού – Αμερικανικού βιβλίου μαθηματικών.

USABILITY OF POLYNOMIALS

Polynomials can be used to model many aspects of the physical world.Polynomials are used to model the height of thrown objects.

Polynomials are also good for approximating other more complicated functions, for that we use Taylor series.

Polynomials are also used to model the trajectory of a cannonball

TAYLOR SERIES In mathematics, a Taylor series is a representation of a function as an

infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

The concept of a Taylor series was formally introduced by the English mathematician Brook Taylor in 1715. If the Taylor series is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.

It is common practice to approximate a function by using a finite number of terms of its Taylor series. Taylor's theorem gives quantitative estimates on the error in this approximation. Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial. The Taylor series of a function is the limit of that function's Taylor polynomials, provided that the limit exists. A function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function that is equal to its Taylor series in an open interval (or a disc in the complex plane) is known as an analytic function.

Degrees of Polynomials3 2 3 2 3 3 2 2( 3 1) ( 3 1) 3 3 1 1 0x x x x x x x x (zero Polynomial)

3 2 3 3 3 2(2 1) ( 2 2 3) 2 2 1 2 2 3 2 2x x x x x x x x x x

(Polynomial of degree 2)

3 2 3 2 3 2 3 2

3 2 3 2

( 2 5 7) (4 5 3) 2 5 7 4 5 3

(1 4) (2 5) 5 (7 3) 5 3 5 10

x x x x x x x x x x

x x x x x x

3 2 3 2 3 2 3 2

3 2 3 2

( 2 5 7) (4 5 3) 2 5 7 4 5 3

(1 4) (2 5) 5 (7 3) 5 3 5 10

x x x x x x x x x x

x x x x x x

(Polynomial of degree 3)

2 2 2 2(5 2 1) ( 5 3 2) 5 2 1 5 3 2 5 3x x x x x x x x x

(Polynomial of degree 1)

DEFINITIONS

Definition of a monomial

A monomial x is an algebraic expression in the form of nax , where a is any real number and n is a non-negative integer

Definition of a polynomial in x (polynomial) A polynomial in x is an algebraic expression of the form

1 2 11 2 1 0...n n

n na x a x a x a x a , where 0na

EQUATION BETWEEN 2 POLYNOMIALSTwo polynomials

1 2 11 2 1 0...n n

n na x a x a x a x a

1 2 11 2 1 0...m m

m mb x b x b x b x a

m n

are equal whenever

0 0a b , 1 1, ..., n na b a b και 1 2 ... 0n n ma a a

NUMERICAL VALUE OF A POLYNOMIAL

In order to find the value of the polynomial , we replace the X with a certain real number r

The expression that forms is called numerical value of the polynomial for X=r

ARITHMETICAL PRICE OF THE POLYNOMIALLet a polynomial 3 2( 1) ( 1) 2( 1) 4( 1) 1 0P

11 0( ) ......n n

n nP x a x a Ix a x a

If we put in the place of X a real number ρ , then the real number 1

1 0( ) 1 ...n nn nP a this result,is called arithmetic price,or simpler

price of the polynomial for x=ρ

Then the ρ symbol is called root of the polynomial.For example, the price of the polynomial 3 2( ) 2 4 1P x x x x

For x=1 3 2(1) 1 2 1 4 1 1 6P but

For x=-1 3 2( 1) ( 1) 2( 1) 4( 1) 1 0P which means that

-1 is the root of the polynomial P(x)

In order to find X(x) and U(x) we follow a certain procedure1.Make the shape of the division and write the 2

polynomials2.Find the first term of the quotient by

finding the first term of the dividend with the first term X of the division

3. Multiply with x-3 and subtract the product from the dividend. So we find the first partial difference

4.Repeat 2 and 3 with a new dividend we find the second partial difference -4x-1

Adding polynomials

• Simply combine like terms

Same as adding just change same signs

Multiplying

William George HornerWilliam George Horner (1786 – 22 September

1837) was a British mathematician and schoolmaster. The invention of the zoetrope, in 1834 and under a different name (Daedaleum), has been attributed to him.

Horner’s Work Horner published a mode of solving numerical equations of any

degree, now known as Horner's method. According to Augustus De Morgan, he first made it known in a paper read before the Royal Society, 1st July 1819, by Davies Gilbert, headed A New Method of Solving Numerical Equations of all Orders by Continuous Approximation, and published in the Philosophical Transactions for the same year. But this version of the history is comprehensively denied by later historians. De Morgan's advocacy of Horner's priority in discovery led to "Horner's method" being so called in textbooks, but this is a misnomer. Not only did the 1819 paper not contain that method, but it also appeared in an 1820 paper by Theodore Holdred, being published by Horner only in 1830; and the method was by no means novel, having appeared in the work of the Chinese mathematician Zhu Shijie centuries before, and also in the work of Paolo Ruffini.[4]

The method was republished by Horner in the Ladies' Diary for 1838, and a simpler and more extended version appeared in vol. i. of the Mathematician, 1843

CONSTRUCTION OF HORNER’S TABLEFor the construction of the table we follow the next steps:

We fill the first line with the coefficients of the polynomial P x and the first

place of the third line with the first coefficient of P x .

Afterwards the table is completed as following: Every item on the second line results by multiplying the immediate past item

of the third line with p .

Every other item of the third line results by adding the corresponding items from the first and second lines.

The last item of the third line is the difference of the division between P x

and x p , namely the price of the polynomial P x for x p . The other items of

the third line are the coefficients of the quotient of the said division. Let’s work right now on the Horner Configuration to find the quotient and the difference of the division of 5 43 3 6 13P x x x x to 2x .

THE HORNER CONFIGURATIONLet’s say we have 3 23 8 7 2P x x x x . Consequently, we define the division

:P x x p .

The Horner Configuration is a different way of performing the operation of dividing polynomials and can be visually depicted by the following board:

Coefficients of P x

3 -8 7 2 p

3p 3 8p p 3 8 7p p p

3 3 8p 3 8 7p p 3 8 7 2p p p

Coefficients of the quotient Difference

AbilitiesThe degree of the product of the 2 non-zero

polynomials is equal to the total of the degrees of the Z polynomials.

For every pair of polynomial D(x) and δ(x) with δ(x) 0 there are 2unique polynomials X(x) and U(x),so as:

D(x)=δ(x)X(x)+U(x)

Where U(x) is either the zero polynomial or is of less degree than the δ(x)

Just like the division of natural numbers

D(x) •Is called dividend

δ(x) •Is called divisor

X(x) •Is called quotient

COMPARISON AND CONTRAST BETWEEN THE AMERICAN AND THE GREEK MATHS BOOK

Differences

The English book includes more

thorough examples thus making it much easier for the student

to understand .

It includes more exercises and a wider

variety of them.

It also shows the goals and reasons for learning polynomials

and their practical application in real

life.

More differences…It contains useful tips for

study and technology.

It combines both algebra and geometry, revealing all

polynomial’s aspects.

It is easier on the eyes since it contains images that make it more attractive.

It is more interesting making mathematics more approachable towards the

students.

Similarities between the American and the Greek Maths bookThey have the same terminology.The difficulty of the exercises is gradually

increasing.They both provide students with the answers

of the exercises.They also contain examples that show how to

solve the exercises step by step.

Bibliography :, Άλγεβρα β’ λυκείου, Larson’s algebra for college students, www.wikipedia.orgFoundations, Cambridge University Press, 1991Edit&presentation Supervisor:Κωνσταντίνος Ζήκος

Project made by:Βασίλης Zωγόπουλος ,Νεοκλής Κασιμἀτης , Γεωργία Θεοδωρακοπούλου ,Γιάννης Έξαρχος, του Β2 τμήματος του Προτύπου Πειραματικού ΓΕΛ της Βαρβακείου Σχολής

Many thanks to Our teachers Dr Λίλιαν Νιτσοπούλου και Dr Γεώργιο Κόσσυβα who helped us in this project