The Polynomial Project

Term 2Grade 11 math core project

Done By :Khloud Aleisaei.Shaikha Alkaabi.Fatima Alkathiri.Grade: 11-56

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.

Polynomials in our life :

Professional

Management

Health Care

Electronics Installation

Production

Task 1 Find the polynomial that gives the following values:

X -1 1 2 5

P(X) 10 -6 -17 82

𝑝( )= + ( − 0)+ ( − 0)( − 1)+ ( − 0)( − 1)𝑥 𝐴 𝐵 𝑥 𝑥 𝐶 𝑥 𝑥 𝑥 𝑥 𝐷 𝑥 𝑥 𝑥 𝑥( − 2) 𝑥 𝑥

a. Write the system of equations in A,B,C and D that you can use to find the desired polynomial

-6=A+2B

-17=A+3B+3C 82=A+6B+24c+72D

A=10

• 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)

b. Solve the system obtained from part a.

A=10

B=-8-6=10+2B -16=B 2B=-8

C=-1 -17=10-24+3C-3+c3C=-1

D=2 82=10-48-24+2D62+62=D72D=2

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=

2 -8=B

A=10

P(2)=A+3B+3C-17=10-24+3C-3/3 = C

C= -1

P(5)=A+6B+24C+72D144/72=D

2 = D

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)

The Bisection Method for Approximating Real Zeros

The bisection method can be used to approximate zeros of polynomial functions like ( )= 3+ 2−3 −3 𝑓 𝑥 𝑥 𝑥 𝑥

(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

Task 2: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

b. Find to the nearest tenth the third zero using the Bisection Method for

Approximating Real Zeros.

-2 -1 0 1 2 3 3.1 3.2 3.3 3.4 3.5 3.75 4

-9 10 7 -6 -17 -14 -12.468 -10.664 -8.576 -6.192 -3.5 4.65 15

3.5+4/2 = 3.75

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

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=3b. Write an expression for the area of the garden and walk.

Agarden+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

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