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Term 2 Grade 11 core project Due date : 26.Feb.2012 The polynomial project

Term 2 Grade 11 core project Due date : 26.Feb.2012 The polynomial project . Student name : Shaikha Saeed , Meead Asker , Mouza Darwish Section : 11.52 . Introduction .

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Term 2 Grade 11 core project Due date : 26.Feb.2012 The polynomial project

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  1. Term 2 Grade 11 core project Due date : 26.Feb.2012The polynomial project Student name : ShaikhaSaeed , Meead Asker , MouzaDarwish Section : 11.52

  2. Introduction Polynomial is important and related our modern life and used in many fields includes : architecture , agriculture , engineering filed such as electrical and civil engineering , physics and various other sciences . Without the science of polynomial the building and surveys will not be able

  3. Approximation by mean of polynomial Many scientific experiments produce pairs of numbers [x,p(x)] that can be related by a formula . If the pairs form a function, you can fit a polynomial to the pairs in exactly one way .

  4. Task one Find the polynomial that gives the following values :- 𝑝(𝑥)=𝐴+𝐵(𝑥−𝑥0)+𝐶(𝑥−𝑥0)(𝑥−𝑥1)+𝐷(𝑥−𝑥0)(𝑥−𝑥1)(𝑥−𝑥2)

  5. Write the system of equations in A,B,C and D that you can use to find the desired polynomial 10 = A -6 = A + BC( 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

  6. *Solve the system obtained from part a :-

  7. c. Find the polynomial that represents the four ordered pairs. P(x)=10-8(x+1)-1(x+1)(x-1)+2(x+1)(x-1)(x+2) P(x)=2x3-5x2-10x+7 P(-1)=A+2B P(2)=A+3B+3C p(5)=A+6B+24c+72D -10-6=B -17=10-24+3C 82+62=D 2 -3=c 72 -8=B 3 2=D -1=c

  8. 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)+D(x-x0) (x-x1) (x-x2)+E(x-x0) (x-x1) (x-x2) (x-x3) P(x)=A+B(x+1)+c(x+1)(x-1)+D(X+1)(x-1)(x-2)+E(x+1)(x-1)(x-2)(x-5)

  9. The bisection method for approximating real zeros • The bisection method can be used to approximate zeros of polynomial functions like 𝑓(𝑥)=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

  10. Task two 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. P(x)=2x3-5x2-10x+7 P(-2) = -9 P(-2) = 10 MID POINT = -1,5 P(3) = -14 P(4) = 15 MID POINT = 3,5 P(0) = 7 P(1) = -6 MID POINT = 0,5 Task 2: Find the zeros of the polynomial found in task 1.

  11. b. Find to the nearest tenth the third zero using the Bisection Method for Approximating Real Zeros.

  12. 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 = 3

  13. b. Write an expression for the area of the garden and walk. A garden + walkway=(2x+6)(x+6) c. Write an expression for the area of the walkway only. A walkway = (2x+6)(x+16)-2x.x= (2x+6)(x+6)-2x2

  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? (2x+6)(x+6)-2x2<1,000

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