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T h e P o l y n o m i a l P r o j e c t

T h e P o l y n o m i a l P r o j e c t. Student Name: AL-Reem Hamad Reem Mustafa Section: 11.54 Cluster : Applied Engineering (AE -D) Subject : Core – Mathematics . Date & day of submission : 29-feb-2021. Contents:. Introduction.

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T h e P o l y n o m i a l P r o j e c t

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  1. The Polynomial Project Student Name: AL-Reem Hamad Reem Mustafa Section: 11.54 Cluster: Applied Engineering (AE -D) Subject : Core – Mathematics . Date & day of submission : 29-feb-2021

  2. Contents: • Introduction. • More information's. • 3 Tasks.

  3. 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.

  4. More information:- • A polynomial is an expression of finite length constructed from variables. • (also known as indeterminates). • And constants, using only the operations of addition, subtraction, multiplication. • Example of polynomial and it can have: • Constant: “ like: 3,-23, ¼ . • Variable: “ like: X & Y. • Exponents: “ like: the 3 in X3.

  5. polynomial • Each one of the little product-blobs is a “term” • The coefficients: are the numbers in front of the letters: • The one term could: monomial • The two terms could: binomial • The three terms could: trinomial

  6. Polynomial (degree) • Degree: This is a 5th degree polynomial • Degree: This is a 7th degree polynomial • Degree: This is a 1st degree polynomial • Degree: this is a zero degree polynomial This guy has 5 letters
The degree is 5. This guy has 7 letters
The degree is 7. This guy has 1 letter
The degree is 1. Has zero degree

  7. Not polynomial • 2/(x+2) is not, because dividing is not allowed • 1/x is not • 3xy-2 is not, because the exponent is "-2" (exponents can only be 0,1,2,...) • √x is not, because the exponent is "½" But these are allowed: • x/2 is allowed, because it is also (½)x (the coefficient is ½, or 0.5) • also 3x/8 for the same reason (the coefficient is 3/8, or 0.375) • √2 is allowed, because it is a constant (= 1.4142...etc)

  8. The graph of the polynomial:

  9. Task 1: Find the polynomial that gives the following values

  10. a) Write the system of equations in A, B, C, and D that you can use to find the desired polynomial. • 10=A+0+0+0 • -6=A+B(1+1)+C(1+1)(1-1)+0(0-1)(0-1)(0-2) • -6=A+2B • -17=A+B(2+1)+C(2+1)(2-1)+0(0-1)(0-1)(0-2) • -17=A+3B+3C • 82=A+B(5+1)+C(5+1)(4)+D(5+1)(4)(3) • 82=A+6B+24C+72D • A=10 • -6=A2B • -17=A+3B+3C • 82=A+6B+24C+72D

  11. B)Solve the system obtained from part a. A=10 -6=10+2B -6-10=2B -16/2=2B/2 B=-8 A=10 , B=-8 -17=10+3(-8)+3C -17-10+24=3C -3/3 = 3C/3 C=-1 A=10 , B=-8 , C=-1 82=10+6(-8)+24(-1)+72D 82-10+48+24=72D 144/72 = 72D/72 D=2

  12. C)Find the polynomial that represents the four ordered pairs. • P(x)=A+B(X-X0)+C(X-X0)(X-X1)+D(X-X0)(X-X1)(X-X2) • =10+(-8)(X+1)+(-1)(X+1)(X-2)+(2)(X+1)(X-2)(X-5) • =10-8X-8-X-1(X-2)+(2X+2)(X2-5X-2X+10) • =10-8X-8-X2+2X-X+2+2X3-10X2-4X2+20X+2X2-10X-4X+20 • =2X3-13X2-X+24

  13. 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-X2) (X-X2) (X-X3)

  14. Task 2: Find the zeros of the polynomial found in task 1.Show that the 3 zeros of the polynomial found in task 1 are:First zero lies between -2 and -1Second zero lies between 0 and 1Third zero lies between 3 and 4.

  15. First zero lies between -2 and -1 2x3-13x2-x+24 • X=-1 , x=-2 • =2(-1)3-13(-1)2 –(-1)+24 • =10 • 2(-2)3-13(-2)2-(-2)+24 • =-42

  16. Second zero lies between 0 and 1 • 2x3-13x2-x+24 • X=0 , x=1 • 2(0)3-13(0)2 – (0)+24 • =24 • 2(1)3 -13(1)2-(1)+24 • =12

  17. Third zero lies between 3 and 4. • 2x3-13x2-x+24 • X=3 , x=4 • 2(3)3-13(3)2 – (3)+24 • =-42 • 2(4)3-13(4)2 – (4)+24 • =-60

  18. b)Find to the nearest tenth the third zero using the Bisection Method for Approximating Real Zeros. • 1) 2x3-13x2-x+24 • MID POINT= -2+(-1)/2=-1.5 • 2(-1.5)3-13(-1.5)2 – (-1.5)+24 • =-10.5 • 2) 2x3-13x2 –x+24 • MID POINT= -1.5+(-1)/2=-1.25 • 2(-1.25)3-13(-1.25)2 – (-1.25)+24 • =1.03 • 3) 2x3-13x2 –x+24 • MID POINT= -1.25+(-1)/2=-1.125 • 2(-1.125)3-13(-1.125)2 – (-1.125)+24 • =5.82

  19. 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.

  20. A)Choose any value for the width of the walkway w that is less than 6 ft. • W=4FT • B)Write an expression for the area of the garden and walk. 2x+x=2x2 Width (x+8) w=4 w=4 Length (2x+8)

  21. c)Write an expression for the area of the walkway only. • Big area =L x W • =(2x+8)(x+8) • =2x2+24x+64 • Walk area = big area – small area • W(x)=2x2+24x+64-2x2 • =24x + 64

  22. d)You have enough gravel to cover 1000ft2 and want to use it all on the walk. How big should you make the garden? • Area of the walk = 24x+64 • Gravel area = 1000 ft2 • 24x+64 < 1000 • 24x/24 < 936/24 • X=39 • The width must be 39 ft. • The length must be 78 ft. (1000-64)=936

  23. Reference • http://www.coolmath.com/algebra/03-polynomials/index.html • http://www.mathsisfun.com/algebra/polynomials.html

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