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Polynomial

Polynomial. Done by: Abdulla Abbas AT1000093 Ibrahim Ali AT1001673 Mohammed Juma AT1001684 Abdulrahman Ali AT1000195 GRADE : 11-13. Ms. Lakshmi Krishnan. Task 1. Part A: 10= A - 6 = A + B(χ - χ 0 ) -17= A+ B(χ - χ 0 ) + C( χ - χ 0 )(χ - χ 1 )

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Polynomial

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  1. Polynomial Done by: Abdulla Abbas AT1000093 Ibrahim Ali AT1001673 Mohammed JumaAT1001684 Abdulrahman Ali AT1000195 GRADE: 11-13 Ms. Lakshmi Krishnan

  2. Task 1 Part A: 10= A -6= A + B(χ-χ0) -17= A+ B(χ-χ0) + C(χ-χ0)(χ-χ1) 82= A+ B(χ-χ0) + C(χ-χ0)(χ-χ1) + D(χ-χ0)(χ-χ1)(χ-χ2) SystemOfEquations

  3. TASK 1 Part B: 10=A -6=A + B(χ-χ0) -6=10 + B[1-(-1)] B=-8 -17=A + B(χ-χ0) + C(χ-χ0)(χ-χ1) -17=10 + (-8)[2-(-1)] + C[2-(-1)](2-1) C=-1 Solving the system

  4. TASK 1 Part B: 82=A + B(χ-χ0) + C(χ-χ0)(χ-χ1) + D(χ-χ0)(χ-χ1)(χ-χ2) 82=A + B[5-(-1)] + (-1)[5-(-1)](5-1) +D[5-(-1)](5-1)(5-2) D=2

  5. task 1 Finding the polynomial Part C: A= 10 B= -8 C= -1 D= 2 P(χ)= (10) + (-8)[x-(-1)] + (-1)[x-(-1)](x-1) + (2)[x-(-1)](x-1)(x-2) = 10 - 8x - 8 + [-x-(-1)][x-(-1)] + (2x + 2)[x-(-1)](x - 2) = 2 - 8x - x2 + (x - x) + 1(2x3 - 2x + 2x -2)(x - 2) = 1 + 2 - 8x - x2 + (2x2 - 2)(x - 2) = -8x - x2 + 3 + (2x3 - 4x2 - 2x + 4) = 2x3 - 5x2 - 10x + 7

  6. task 1 Part D: P(χ)= A + B(χ - χ0) + C(χ - χ0)(χ - χ1) + D(χ- χ0)(χ - χ1)(χ - χ2) + E(χ - χ0)(χ - χ1)(χ - χ2)(χ - χ3) General form of the polynomial of degree 4

  7. P(χ)= 2x3 - 5x2 - 10x + 7 Part a: Firstzero lies between -2 & -1 P(-2) = 2(-2)3 - 5(-2)2 - 10(-2) + 7 P(-2) =-9 P(-1) = 2(-1)3 - 5(-1)2 - 10(-1) + 7 P(-1) = 10 There are more than one way for finding the zeros of a polynomial function, one of them is Synthetic Substitution. TASK 2

  8. TASK 2 Part a: Second zero lies between 0 & 1 Third zero lies between 3 & 4 P(0) = 2(0)3 - 5(0)2 - 10(0) + 7 P(3) = 2(3)3 - 5(3)2 - 10(3) + 7 P(0) = 7 = -14 P(1) = 2(1)3 - 5(1)2 - 10(1) + 7 P(4) = 2(4)3 - 5(4)2-10(4) + 7 P(1) = -6 = 15

  9. TASK 2 Part b: 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 3.5. Since f (3.5) = -3.5, the zero is between 3.5 and 4. The midpoint of this interval is 3.75. Since f(3.75) is about 4.65625, the zero is between 3.5 and 3.75. The midpoint of this interval is 3.625. Since f(3.625) is about 0.3164. The zero is between 3.625 and 3.75. The midpoint of this interval is 1.6875. Since f(3.6875) is about 2.41943, the zero is between 3.6875 and 3.75. Therefore, the zero is 3.7 to the nearest tenth. The bisection method

  10. a. Choose any value for the width of the walkway w that is less than 6 ft. w = 2 ft b. Write an expression for the area of the garden and walk. THE LENGTH OF THE GARDEN AND THE WALKWAY = (2x+2)ft THE WIDTH OF THE GARDEN AND THE WALKWAY = ( x+2)ft ( 2x+2) ( x+2) = 2x2+4x+2x+4 =(2x2+6x+4)ft2 TASK 3

  11. c. Write an expression for the area of the walkway only. The area of the garden and walk 2x2+6x+4 The area of the garden x(2x)=2x2 The area of the walkway only Is equal to The area of the garden and walk -The area of the garden 2x2+6x+4-2x2 =(6x+4)ft2 TASK 3

  12. TASK 3 b. 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 = 6x + 4 1000 – 4 = 6x 984 = 6x 996 6 = x 166 = x Find the area of the garden x = 166 2x(x) [2(166)](166)=55112 ft2 Finally, I should make the garden 55112 ft2

  13. P(χ)= 2x3 - 5x2 - 10x + 7 Use a graphing program to graph the polynomial found in task 1. TASK 4

  14. Ms. Lakshmi b. Make a PowerPoint to present your project and upload it on a wiki. Wiki Link: http://abdulrahmandailywork.wikispaces.com/ TASK 4 Abdulla Abbas Ibrahim Ali Mohammed Juma Abdulrahman Ali

  15. Thankyoufor listening

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