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ُ The polynomial project. Task1: find the polynomial that gives the following value. P(X)=A+B(X-X 0 )+C(X-X 0 )(X-X 1 )+D(X-X 0 )(X-X 1 )(X-X 2 ). 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) - PowerPoint PPT Presentation
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�ُ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