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

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 1APPROXIMATION BY MEANS

OF POLYNOMIALS

Find the polynomial that gives the following values :

X0 X1 X2

X -1 1 2 5P(x) 10 -6 -17 82

Write the system of equations in A, B, C, and D 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)

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

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

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 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.

Show that the 3 zeros of the polynomial found in task 1 are:First zero lies between -2 and -1Second zero lies between 0 and 1Third 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) +7F(-1)= 10

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

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

F(-2) = 2(-2)³ - 5(-2)² - 10(-2) +7 F(-2) = -9 F(-1)= 2(-1)³ - 5(-1)² - 10(-1) +7F(-1)= 10

F(3)= 2(3)³ - 5(3)² - 10(3) +7F(3)= -14 F(4)= 2(4)³ - 5(4)² - 10(4) +7F(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 3.5Since 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 3.625Since 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 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

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

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

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 4USIN G TECHN OL O GY:

Use a graphing program to graph the polynomial found in task 1.