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# The polynomial project - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about ' The polynomial project' - jethro

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### The polynomial project

By: 1- Ali Ahmed Ali Alfalasi

2- Khalid AbdulrahmanMohdAlrum

4- MajidYousif Ahmed Alhaway

• 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 :

• 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)

• 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

Task 2 pairs of numbers. The 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.

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

F(3)= 2(3)³ - 5(3)² - 10(3) +7 pairs of numbers.

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 pairs of numbers. Real 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 pairs of numbers. 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. pairs of numbers.

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

area of garden

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

= (2(30))(30)

= 1800 ft

Task 4 pairs of numbers. Using Technology:

• Use a graphing program to graph the polynomial found in