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ُ The polynomial project

ُ 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)

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ُ The polynomial project

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  1. ُThe polynomial project

  2. Task1: find the polynomial that gives the following value P(X)=A+B(X-X0)+C(X-X0)(X-X1)+D(X-X0)(X-X1)(X-X2)

  3. 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)

  4. b. Solve the system obtained from part a. • A=10 • -6=A+2B • -17=A+3B+3C • 82=A+6B+24C+72D • A=10, B=-8, C=-1, D=2

  5. c. Find the polynomial that represent the four ordered pairs. • P(X)=A+B(X-X0)+C(X-X0)(X-X1)+D(X-X0)(X-X1)(X-X2) • 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

  6. d. Write the general form of the polynomial of degree 4 for 5 pairs of numbers. • P(X)=A+B(X-X0)+C(X-X0)(X-X1)(X-X2)+E(X-X0)(X-X1)(X-X2)(X-X3)

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

  8. 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+7 • 2(-2)3-5(-2)2-10(-2)+7= -9 • 2(-1)3-5(-1)2-10(-1)+7= 10 • 2(0)3-5(0)2-10(0)+7= 7 • 2(1)3-5(1)2-10(1)+7= -6 • 2(3)3-5(3)2-10(3)+7= -14 • 2(4)3-5(4)2-10(4)+7= 15

  9. 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 = 3.5 • Since f(3.5)=-3.5 the zero between 3.5 and 4the mid point = 3.75 • Since f(3.75)=4.656 the zero between 3.5 and 3.75the mid point = 3.625 • Since f(3.625)=0.316 the zero between 3.625 and 3.75the mid point=3.6875 • Since 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

  10. 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.

  11. a. Choose any value for the width of the walk way W that is less than 6ft. W=5ft

  12. 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

  13. 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

  14. 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 X • 1000=30X+1001000-100=30X900/30=XX=30 • Find Area • =(2X)(X)=[2(30)](30)=1800ft2

  15. Task4: Using technology

  16. a. Use a graphing program to graph the polynomial found in task1. P(X)=2X3-5X2-10X-7

  17. Group members… • 1. MayedHassan • 2. AmerAbdulsalam • 3. SalimAdel • 4. IshaqAbdulla

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