�ُThe polynomial project

Task1: find the polynomial that gives the following value

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

P(X)=A+B(X-X0)+C(X-X0)(X-X1)+D(X-X0)(X-X1)(X-X2)

a. Write the system of equations in A,B,C, and D that you can use to find 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)

b. Solve the system obtained from part a.

A=10-6=A+2B-17=A+3B+3C82=A+6B+24C+72DA=10, B=-8, C=-1, D=2

c. Find the polynomial that represent the four ordered pairs.

P(X)=A+B(X-X 0)+C(X-X 0)(X-X 1)+D(X-X 0)(X-X 1)(X-X 2)

P(X)=10+(-8)(X+1)+(-1)(X+1)(X-1)+2(X+1)(X-1)(X-2)P(X)=10+(-8X-8)+(-1)(X2 -1)+2(X2 -1)(X-2)P(X)=(-8X+2)+(-X2 +1)+2(X3 -2X2 -1X+2)P(X)=(-X2 -8X+3)+(2X3 -4X2 -2X+4)P(X)=2X3 -5X2 -10X-7

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

numbers.

P(X)=A+B(X-X 0)+C(X-X 0)(X-X 1)(X-X 2)+E(X-X 0)(X-X 1)(X-X 2)(X-X 3)

Task2: find the zeros of the polynomial found in task 1

a. 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.

2X3 -5X2 -10X+72(-2)3 -5(-2)2 -10(-2)+7= -92(-1)3 -5(-1)2 -10(-1)+7= 102(0)3 -5(0)2 -10(0)+7= 72(1)3 -5(1)2 -10(1)+7= -62(3)3 -5(3)2 -10(3)+7= -142(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

One real zero between 3 and 4 the mid point = 3.5Since f(3.5)=-3.5 the zero between 3.5 and 4the mid point = 3.75Since f(3.75)=4.656 the zero between 3.5 and 3.75the mid point = 3.625Since f(3.625)=0.316 the zero between 3.625 and 3.75the mid point=3.6875Since f(3.6875) is about 2.419 the zero between 3.625 and 3.6875

There fore, the zero is 3.7 to the nearest tenth

Task3: Real world construction

You are planning a rectangular garden. Its length is twice its width. You want a walkway W feet around the garden. Let X be the width of the garden.

a. Choose any value for the width of the walk way W that is less than 6ft.

W=5ft

b. Write expression for the area of the garden and walk

The length =(2X+5+5)The width=(X+5+5)(2X+5+5) (X+5+5)(2X+10) (X+10)=2X2 +20X+10X+100=2X2 +30X+100

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

Area of the walk way=(2X)(X)=(2X+5+5)(X+5+5)-(2X)(X)=2X2 +30X+100-2X2

=30X+100

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

Find X1000=30X+1001000-100=30X900/30=XX=30

Find Area=(2X)(X)=[2(30)](30)=1800ft2

Task4: Using technology

a. Use a graphing program to graph the polynomial found in task1.

P(X)=2X3-5X2-10X-7

Group members…

1. Mayed Hassan2. Amer Abdulsalam3. Salim Adel4. Ishaq Abdulla