 Project Done 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, x 0, x 1, and x 2 in px=A+B(x-x 0 )+C(x-x 0 )(x-x 1 )+D(x-x 0 )(x-x 1 )(x-x 2 ) And simplify: px=10+(-8)(x- -1)-1(x- -1)(x-1)+2(x- -1)(x-1)(x-2) px=2x 3 -5x 2 -10x+7

D. Write the general form of the polynomial of degree 4 for 5 pairs of numbers. px=A+B(x-x 0 )+C(x-x 0 )(x-x 1 )+D(x-x 0 )(x-x 1 ) (x-x 2 )+E(x-x 0 )(x-x 1 )(x-x 2 )(x-x 3 )

The Bisection Method for Approximating Real Zeros The 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 1.5 Since f (1.5) = , the zero is between 1.5 and 2. The midpoint of this interval is Since f (1.75) is about 0.172, the zero is between 1.5 and The midpoint of this interval is Since f (1.625) is about The zero is between and The midpoint of this interval is Since f (1.6875) is about -0.41, the zero is between and Therefore, the zero is 1.7 to the nearest tenth. 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=2x 3 -5x 2 -10x+7 First 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 Since f (3.75) is about 4.65, the zero is between 3.50 and The midpoint of this interval is Since f (3.625) is about The zero is between 3.50 and The midpoint of this interval is Since f (3.56) is about -1.73, the zero is between 3.56 and The midpoint of this interval is Since f (3.59) is about The zero is between 3.59and The midpoint of this interval is Since f (3.605) is about Therefore, the zero is 3.6 to the nearest tenth.

Task 3 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.

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 ) = 4w 2 +2wx+4wx+2x 2 = 4w 2 +2x 2 +6wx

C. Write an expression for the area of the walkway only. 2w+2x× 2w× x – (2x×x)

You have enough gravel to cover 1000ft 2 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=4w 2 +2x 2 +6wx– (2x 2 ) 1000=4w 2 +6wx 1000=64+24x 936=24x x=39

Area of gadenan = 2 x 2 =2(39) 2 = 3042 So 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