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Polynomial Project

Polynomial Project. Introduction.

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Polynomial Project

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  1. Polynomial Project

  2. Introduction • The concept of degree of a polynomial is important, because it gives us information about the behavior of the polynomial on the whole. The concept of polynomial functions goes way back to Babylonian times, as a simple need of computing the area of a square is a polynomial, and is needed in buildings and surveys, fundamental to core civilization. Polynomials are used for fields relating to architecture, agriculture, engineering fields such as electrical and civil engineering, physics, and various other science related subjects.

  3. Approximation by Means of Polynomials • 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. For 2 pairs of numbers you can write a polynomial of degree 1. px=A+B(x-x0) For 3 pairs of numbers you can write a polynomial of degree 2. px=A+B(x-x0)+C(x-x0)(x-x1) For 4 pairs of numbers you can write a polynomial of degree 3. px=A+B(x-x0)+C(x-x0)(x-x1)+D(x-x0)(x-x1)(x-x2) And so on.

  4. Task 1 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. a. Write the system of equations in A, B, C, andD that you can use to find the 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)

  6. b. Solve the system obtained from part a. A=10, B=-8, C=-1, and D=2 c. Find the polynomial that represents the four ordered pairs. p(x)=2x3-5x2-10x+7

  7. 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)

  8. Task 2 Find the zeros of the polynomial found in task 1. p(x)=2x3-5x2-10x+7

  9. a. Show that the 3 zeros of the polynomial found in task 1 are: First zero lies between -2 and -1 Since f(-2)=-9 and f(-1)=10 there is at least one zero between -2 and -1 The midpoint of this interval is -1.5 Second zero lies between 0 and 1 Since f(0)=7 and f(1)=-6 there is at least one zero between 0 and 1 The midpoint of this interval is 0.5 Third zero lies between 3 and 4. Since f(3)=-14 and f(4)=15 there is at least one zero between 3 and 4 The midpoint of this interval is 3.5

  10. b. Find to the nearest tenth the third zero using the Bisection Method for Approximating Real Zeros. p(x)=2x3-5x2-10x+7 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 4.7, the zero is between 3.5 and 3.75. The midpoint of this interval is 3.625 Since f(3.625) is about 0.3. The zero is between 3.5 and 3.625 The midpoint of this interval is 3.5625 Since f(3.5625) is about -1.7. The zero is between 3.5625 and 3.625 Therefore, the zero is 3.6 to the nearest tenth.

  11. 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=2

  12. Real World Construction b. Write an expression for the area of the garden and walk. AreaT.W=LTWT AreaT.W=(2x+4)(x+4) c. Write an expression for the area of the walkway only. • Areawalkway= • (2x+4)(x+4)-2x(x)

  13. 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 + 4 ) ( x + 4 ) - 2x2 = 1000 • 2x2 + 8x +4x +16 -2x2 =1000 • 12x + 16 =1000 • 12x = 1000 – 16 • 12x = 984 • X = 83 • Garden = 2x2 • Garden = 2(83)2 • Garden = 13778ft2

  14. Task 4: Using Technology • Use a graphing program to graph the polynomial found in task 1

  15. Done By: Mohammed jassim & Mohammed essa 11.3

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