The polynomial project
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The polynomial project. By: 1- Ali Ahmed Ali Alfalasi 2- Khalid Abdulrahman Mohd Alrum 3- Fahad Aabdelqader Mohd Alshaer 4- Majid Yousif Ahmed Alhaway Grade: 11.5. Introduction.

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The polynomial project

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The polynomial project

The polynomial project

By: 1- Ali Ahmed Ali Alfalasi

2- Khalid AbdulrahmanMohdAlrum

3- FahadAabdelqaderMohdAlshaer

4- MajidYousif Ahmed Alhaway

Grade: 11.5


Introduction

Introduction

  • The concept of degree of a polynomial is important, because it gives us information about the behavior of the polynomial on the whole. The concept of polynomial functions goes way back to Babylonian times, as a simple need of computing the area of a square is a polynomial, and is needed in buildings and surveys, fundamental to core civilization. Polynomials are used for fields relating to architecture, agriculture, engineering fields such as electrical and civil engineering, physics, and various other science related subjects.


Task 1 approximation by means of polynomials

Task 1Approximation by Means of Polynomials

  • Find the polynomial that gives the following values :


The polynomial project

  • Write the system of equations in A, B, C, andD that you can use to find the desired polynomial.

    10=A

    -6=A+B(χ-χ0)

    -17=A+B(χ-χ0)+C(χ-χ0)(χ-χ1)

    82=A+B(χ-χ0)+C(χ-χ0)(χ-χ1)+D(χ-χ0)(χ-χ1)(χ-χ2)


The polynomial project

  • Solve the system obtained from part a.

    10=A

    -----------------

    -6=A+B(χ-χ0)

    -6=10+B(1-(-1))

    B=-8

    -----------------

    -17=A+B(χ-χ0)+C(χ-χ1)(χ-χ2)

    -17=10+(-8)(2-(-1))+C (2-(-1))(2-1)

    C=-1

82=A+B(χ-χ0)+C(χ-χ0)(χ-χ1)+D(χ-χ0)(χ-χ1)(χ-χ2)

82=10+(-8)(5-(-1))+(-1)(5-(-1))(5-1)+D(5-(-1)(5-1)(5-1)

D=2


The polynomial project

  • Find the polynomial that represents the four ordered pairs.

    p(x) = A + B( x - x₀ ) + C( x - x₀ )( x - x₁ ) + D ( x - x₀ )( x - x₁ )( x - x₂ )

    = 10 + (-8)( x - (-1) ) + (-1)( x - (-1))( x - 1 ) + 2( x - (-1) )( x - 1 )( x - 2 )

    = 10 - 8x - 8 + (-x - 1)(x - 1) + (2x +2)(x-1)(x-2)

    = 2 - 8x - x² + x - x + 1 + (2x² - 2x + 2x - 2)(x-2)

    = 3 - 8x - x² + (2x² - 2)(x-2)

    = 3 - 8x - x² + 2x³ - 4x² - 2x + 4

    = 2x³ - 5x² - 10x +7


The polynomial project

  • Write the general form of the polynomial of degree 4 for 5 pairs of numbers.

    Pχ=A+B(χ-χ0)+C(χ-χ0)(χ-χ1)+D(χ-χ0)(χ-χ1)(χ-χ2)+E(χ-χ0)(χ-χ1)(χ-χ2)

    (χ-χ3)


Task 2 the bisection method for approximating real zeros

Task 2The Bisection Method for Approximating Real Zeros

  • Find the zeros of the polynomial found in task 1.

  • Find to the nearest tenth the third zero using the Bisection Method for Approximating Real Zeros.


The polynomial project

  • Show that the 3 zeros of the polynomial found in task 1 are:

  • First zero lies between -2 and -1

  • Second zero lies between 0 and 1

  • Third zero lies between 3 and 4.

P(x)=2x³ - 5x² - 10x +7

F(-2) = 2(-2)³ - 5(-2)² - 10(-2) +7

F(-2) = -9

F(-1)= 2(-1)³ - 5(-1)² - 10(-1) +7

F(-1)= 10

F(-2) = 2(-2)³ - 5(-2)² - 10(-2) +7

F(-2) = -9

F(-1)= 2(-1)³ - 5(-1)² - 10(-1) +7

F(-1)= 10

There is one zero between 0 and 1 because the sign changes from positive to negative

There is one zero between -2 and -1 because the sign changes from positive to negative


The polynomial project

F(3)= 2(3)³ - 5(3)² - 10(3) +7

F(3)= -14

F(4)= 2(4)³ - 5(4)² - 10(4) +7

F(4)= 15

There is one zero between 3 and 4 because the sign changes from positive to negative

Since f (3) = -14 and f (4) = 15, there is at least one real zero between 3 and 4.

The midpoint of this interval is 3.5

Since f(3.5) = -3.5, the zero is between 3.5 and 4.

The midpoint of this interval is 3.75.

Since f(3.75) is about 4.65625, the zero is between 3.5 and 3.75.

The midpoint of this interval is 3.625

Since f(3.625) is about 0.3164. The zero is between 3.625 and 3.75.

The midpoint of this interval is 1.6875.

Since f(3.6875) is about 2.41943, the zero is between 3.6875 and 3.75.

Therefore, the zero is 3.7 to the nearest tenth.


Task 3 real world construction

Task 3Real World Construction

  • You are planning a rectangular garden. Its length is twice its width. You want a walkway w feet wide around the garden. Let x be the width of the garden.

  • Let W = 5


The polynomial project

  • Write an expression for the area of the garden and walk.

    The length of the garden and the walkway = 2x + 5 + 5

    The width of the garden and the walkway = x + 5 + 5

    ------------------------------------------------------------------------

    (2x + 5 + 5) (x + 5 + 5)

    = (2x + 10) (x + 10)

    = 2x² + 20x + 10x +100

    = 2x² + 30x + 100


The polynomial project

  • Write an expression for the area of the walkway only.

    The area of the garden without the walkway = (2x)(x)

    ------------------------------------------------------------------------

    Area of walkway = (the whole area) – (the garden area)

    = (2x + 5 + 5) ( x + 5 + 5 ) - (2x)(x)

    = (2x + 10) ( x + 10 ) - 2x²

    = 2x² + 20x + 10x +100 - 2x²

    = 30x + 100


The polynomial project

  • You have enough gravel to cover 1000ft2 and want to use it all on the walk. How big should you make the garden?

    Find X

    1000 = 30x + 100

    1000 - 100 = 30x

    900/30= x

    x = 30

area of garden

(2x)(x) -- x=30 so,

= (2(30))(30)

= 1800 ft


Task 4 using technology

Task 4Using Technology:

  • Use a graphing program to graph the polynomial found in

  • task 1.


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