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ProjectDone by:

KHALFAN ALBALOUSHI

AHMED ALATISHE

Class: 11.08

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

Find the polynomial that gives the following values

A. 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(1- - 1)

-17=A+B(2 - -1)+C(2 - -1)(2-1)

82=A+B(5 - -1)+C(5 - -1)(5-1)+D(5 - -1) (5-1) (5-2)

10=A

-6=A+2B

-17=A+3B+3C

82=A+6B+24C+72D 

Or

B. Solve the system obtained from part a.

We solve the above system by substitution to get:

 A = 10

B = -8

C = -1

D = 2

C. Find the polynomial that represents the four

ordered pairs.  To find the polynomial we substitute the values of A, B, C, D, x0 , x1 , and x2 in

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

And simplify:  

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

 px=2x3-5x2-10x+7 

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

of numbers.   

px=A+B(x-x0)+C(x-x0)(x-x1)+D(x-x0)(x-x1)(x-x2)+E(x-x0)(x-x1)(x-x2)(x-x3) 

 

The Bisection Method for Approximating Real

ZerosThe bisection method can be used to approximate zeros of polynomial functions like fx=x3+x2-3x-3(To the nearest tenth) Since f (1) = -4 and f (2) = 3, there is at least one real zero between 1 and 2.The midpoint of this interval is 1.5Since f(1.5) = -1.875, the zero is between 1.5 and 2. The midpoint of this interval is 1.75. Since f(1.75) is about 0.172, the zero is between 1.5 and 1.75. The midpoint of this interval is 1.625Since f(1.625) is about -0.94. The zero is between 1.625 and 1.75.The midpoint of this interval is 1.6875.Since f(1.6875) is about -0.41, the zero is between 1.6875 and 1.75. Therefore, the zero is 1.7 to the nearest tenth. The diagram below summarizes the results obtained by the bisection method.

The diagram below summarizes the results

obtained by the bisection method.

Task 2 Show that the 3 zeros of the polynomial

found in task 1 are:

 px=2x3-5x2-10x+7First zero lies between -2 and -1

Second zero lies between 0 and 1 

Third zero lies between 3 and 4. 

  

P(-2)=2(-2)3-5(-2)2-10(-2)+7=-9

P(-1)=2(-1)3-5(-1)2-10(-1)+7=10

P(0)=2(0)3-5(0)2-10(0)+7=7

P(1)=2(1)3-5(1)2-10(1)+7=-6

P(3)=2(0)3-5(3)2-10(3)+7=-14

P(4)=2(4)3-5(4)2-10(4)+7=15

B. Find to the nearest tenth the third zero using the Bisection Method for

Approximating Real Zeros. 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 3.50

Since f(3.5) = -3.5, the zero is between 3.50 and 4. The midpoint of this interval is 3.75.

Since f(3.75) is about 4.65 , the zero is between 3.50 and 3.75. The midpoint of this interval is 3.625

Since f(3.625) is about 0.316. The zero is between 3.50 and 3.625.

The midpoint of this interval is 3.56.Since f(3.56) is about -1.73, the zero is between 3.56 and 3.62 .

The midpoint of this interval is 3.59.Since f(3.59) is about -0.80 . The zero is between 3.59and 3.62.

The midpoint of this interval is 3.605.Since f(3.605) is about -0.32.

Therefore, the zero is 3.6 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.

A. Choose any value for the width of the

walkway w that is less than 6 ft.

 W=4

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

Area =length × Width = (2w+2x) × (2w+ x ) = 4w2+2wx+4wx+2x2

= 4w2+2x2+6wx

C . Write an expression for the area of the

walkway only.

2w+2x× 2w× x – (2x×x)

 You have enough gravel to

cover 1000ft2 and want to use it all on the walk. How big

should you make the garden? Area of walk = Length×Width–area of garden

1000=4w2+2x2+6wx– (2x2) 1000=4w2+6wx 1000=64+24x

936=24x x=39

Area of gadenan = 2x2

=2(39)2

= 3042So the area of garden

should equal 3042 ft to cover all the walk.

Task 4: Using Technology:

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