1. The Polynomial Project
Term 2 Grade 11 math core project
The Polynomial Project
Done By :
Khloud Aleisaei.
Shaikha Alkaabi.
Fatima Alkathiri.
Grade: 11-56

2. Introduction:
polynomial functions are functions with x as an input variable, made up of several terms, each term is made up of two factors, the first being a real umber coefficient, and the second being x raised to some non-negative integer exponent. polynomial equation can help model different markets. Such as how much one will make if x amount of items sells.

3. Polynomials in our life :
Professional
Health Care
Production
Management
Electronics Installation

4. Task 1 Find the polynomial that gives the following values:X -1 1 2 5
P(X) 10 -6
-17 82
𝑝(𝑥)=𝐴+𝐵(𝑥−𝑥0)+𝐶(𝑥−𝑥0)(𝑥−𝑥1)+𝐷(𝑥−𝑥0)(𝑥−𝑥1)(𝑥−𝑥2)

5. A=10 -6=A+2B -17=A+3B+3C 82=A+6B+24c+72D 10 = A -6 = A + BC( 1+1 )
a. 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)
A=10
-6=A+2B
-17=A+3B+3C
82=A+6B+24c+72D

6. A=10 B=-8 C=-1 D=2 -6=10+2B -17=10-24+3C 82=10-48-24+2D 62+62=D -16=B
b. Solve the system obtained from part a.
A=10
B=-8
-6=10+2B
-16=B 2
C=-1
-17= C
-3+c 3
D=2
82= D
62+62=D 72

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
-10-6=
-8=B
A=10
P(2)=A+3B+3C -17= C -3/3 = C
C= -1
P(5)=A+6B+24C+72D 144/72=D
2 = D

8. 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)
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 2: P(x)=2x3-5x2-10x+7 P(-2)=-9 P(-1)=10 P(0)=7 P(1)=-6 P(3)=-14
a. 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(-1)=10
P(0)=7
P(1)=-6
P(3)=-14
P(4)=15
Mid point -1.5
Mid Point 0.5
Mid point 3.5

11. b. Find to the nearest tenth the third zero using the Bisection Method for Approximating Real Zeros.-2 -1 1 2 3
3.1
3.2
3.3
3.4
3.5
3.75 4 -9 10 7 -6
-17
-14
-8.576
-6.192
-3.5
4.65 15
3.5+4/2
= 3.75

12. Use a graphing program to graph the polynomial found in task 1

13. 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.
Choose any value for the width of the walkway w that is less than 6 ft.
w=3
b. Write an expression for the area of the garden and walk.
Agarden+walkway=(2x+6)(x+6)

14. A walkway = (2x+6)(x+16)-2x.x= (2x+6)(x+6)-2x2
c. Write an expression for the area of the walkway only.
A walkway = (2x+6)(x+16)-2x.x= (2x+6)(x+6)-2x2
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<1000
-2x2(2x2+12x+6x+36) -4x4-24x3-12x3-72x2
-4x4-36x3-72x2