The Polynomial Project Term 2 Grade 11 math core project The Polynomial 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 Health Care Production Management Electronics Installation

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)

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

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=10-24+3C -3+c 3 D=2 82=10-48-24+2D 62+62=D 72

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+72D 144/72=D 2 = D

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)

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

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

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 -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. 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)

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