Term 2 grade 11 core project

1. Term 2 grade 11 core project Done by : Hassan Taher Class : 11-05 ID : 1000823

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. Task 1: find the polynomial that gives the following values • In task 1 we have four things to do: • Write the system of equations in A, B, C, andD that you can use to find the desired polynomial. • Solve the system obtained from part a. • Find the polynomial that represents the four ordered pairs. • Write the general form of the polynomial of degree 4 for 5 pairs of numbers.

4. Write the system of equations in A, B, C, andD that you can use to find the desired polynomial. • 10=A • -6=A+2B • -17=A+3B+3C • 82=A+6B+24C+27D

5. Solve the system obtained from part a. • A=10 • B=-8 • C=-1 • D=2

6. 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) • =10+-8X-8+-1(X2-X+X+1)+2(X2-1)(X-2) • =2X2-5X2-10X+7

7. Write the general form of the polynomial of degree 4 for 5 pairs of numbers. • (PX)=A+B(x-x0)+C(x-x0)(x-x1)+D(x-x0)(x-x1)(x-x2)+E(x-x0)(x-x1)(x-x2)

8. The Bisection Method for Approximating Real Zeros The bisection method can be used to approximate zeros of polynomial functions like FX=x3+x2-3x-3 (To the nearest tenth)

10. Task 2: Find the zeros of the polynomial found in task 1. • In this task there is two things we have to do: • 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. • Find to the nearest tenth the third zero using the Bisection Method for Approximating Real Zeros.

11. In this task there is two things we have to do: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. F(-2)=-9 use: 2x3-5x2-10x+7 F(-1)=10 F(-0.5)=10.5 F(0)=7 F(1)=-6 F(0.5)=1 F(3)=-14 F(4)=15 F(3.5)=-3.5

12. Find to the nearest tenth the third zero using the Bisection Method for Approximating Real Zeros. F(3)=-14 use: 2x3-5x2-10x+7 F(4)=15 F(3.5)=-3.5 F(3.75)=4.65 F(3.625)=0.31

13. Task 3: Real World Construction • In this task we have to do only FOUR things and they are: -Choose any value for the width of the walkway w that is less than 6 ft. -Write an expression for the area of the garden and walk. -Write an expression for the area of the walkway only. -You have enough gravel to cover 1000ft2 and want to use it all on the walk. How big should you make the garden?

14. Choose any value for the width of the walkway w that is less than 6 ft. • 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. W=5

15. Write an expression for the area of the garden and walk. • F(X)=2X3-5X2-10X+7 • =2(5)3-5(5)2-10(5)+7 • =82

16. Write an expression for the area of the walkway only. • Area of walkway= total_grarden =(2x2+30x+100)-(2x2) =30x+10 ft2

17. You have enough gravel to cover 1000ft2 and want to use it all on the walk. How big should you make the garden? • 30x+10=1000 30x=990 X=33 width of the garden Area of the garden = 2x2 =2(33)2 =2178 ft2

18. The end