Question

Algera 2 Solving Polynomial Equations?!

Answer 1

Welcome to Oakville Lake Amusement Park!
As one of the new roller coaster engineers, you have been tasked with developing a roller coaster that will intertwine with existing Oakville Lake Amusement Park structures. For one of the more thrilling sections, the roller coaster will dive down in-between buildings, plummet underground, pop back up, and coast over a hill before shooting back underground. There must be three distinct points where the roller coaster crosses the x–axis. Precise measurements and attention to detail are very important.

Answer 2

https://lti.flvsgl.com/content/thjh21ncis0q7cqq94p4fd5kc0/educator_algebraII_v13_2_gs/module04/images/04_05_assessment/04_05_assessment.jpg

Answer 3

First, here is the existing map of current structures. It is important that the roller coaster does not go through the foundation of any of these structures.

1st point: ___6___
2nd point:___-2___
3rd point: ___-7___

1. Using the points above as zeros, construct the polynomial function, f(x), that will be the path of your roller coaster. Show all of your work.


2. Using both fundamental Theorem and Descartes` rule of signs, prove to the construction foreman that your funtion matches your graph. Use complete sentences.


3. Solve for the y–intercept for your function, f(x), and then construct a rough graph of your roller coaster. If your y–intercept is off the graph, give the coordinates of the y–intercept.

Answer 4

@amistre64 Please help me!

Answer 5

ive actually work on this a few times before

http://openstudy.com/study#/updates/54888264e4b079b941e803cc

is one of the times

Answer 6

how do we use the zeros of a function to construct a poly?

Answer 7

Oh I'm not sure....

Answer 8

im not here to fight with a time frame ....

Answer 9

Ok so how do I do number 1?

Answer 10

given a set of roots (a,b,c,...,k) we can create a polynomial that has those roots by making a product of the roots subtracted from x

f(x) = (x-a)(x-b)(x-c)...(x-k)

Answer 11

Ok so I know how to do #1, but I don't understand #2 and #3

Answer 12

define for me

fundamental Theorem:
and
Descartes` rule of signs:

Answer 13

The fundamental theorem of a field of mathematics is the theorem considered central to that field. The naming of such a theorem is not necessarily based on how often it is used or the difficulty of its proofs.

In mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie, is a technique for determining an upper bound on the number of positive or negative real roots of a polynomial. It is not a complete criterion, because it does not provide the exact number of positive or negative roots.

Answer 14

@amistre64

Answer 15

was just reviewing the post i did, we covered the solutions yes :)

the fundamental thrm of algebra tells us that the degree of a polynomial defines how many roots it has.

how many roots do we have?

Answer 16

3?

Answer 17

yeah, so we know that given the 3 different roots, the poly that we formed must satisfy the fundamental thrm by default: and that there are three distinct points where the roller coaster crosses the x–axis.

Answer 18

gotta love the diamonds .... something about an encoding blurb in a copy paste

Answer 19

in order to determine the rule of sign stuff, we will actually need the polynomial to play with

Answer 20

Ok so now what? What would I say to prove to the construction foreman that my funtion matches the graph?

Answer 21

(x-6) (x+2)

x(x+2) -6(x+2)

xx+2x -6x+2(-6)

(x+7) (xx -4x -12)
^^^^^^^^
gotta wrap it for 'properness'


x(xx -4x -12) +7(xx -4x -12)

xxx -4xx -12x +7xx -4(7)x -12(7)

xxx +3xx -12x -28x -84

xxx +3xx -40x -84

Answer 22

You said that on the other question ^^

Answer 23

and i believe i was right :) so

f(x) = x^3 +3x^2 -40x -84

as far as the fundamental thrm goes, you will have to concult your material to see how it is better defined, it should basically say that 3 distinct roots form a polynomial of degree 3 by default.

Answer 24

Ok so f(x) = x^3 +3x^2 -40x -84 is the polynomial for #1, and for #2 how do I explain?

Answer 25

so descartes rule of sign is a little more involved, but its just counting really.

count the number of times the operation changes from + to - in the poly (the start is always a zero count):

f(x) = x^3 +3x^2 -40x -84
^ ^
0 1

by default we can only have 1 real positive root.

Answer 26

So I can just say that ^ for #2?

Answer 27

next we change x to -x and do the same thing

f(-x) = (-x)^3 +3(-x)^2 -40(-x) -84

f(-x) = -x^3 +3x^2 +40x -84
^ ^ ^
0 1 2

so there are at most 2 negative real roots, or by some odd course of nature, there might be no real negative roots and 2 complex roots.

Answer 28

Ok is that all for #2?

Answer 29

what you say is up to you, im just trying to convey some information that might help you construct your own answers.

Answer 30

all that is for descartes rule yes

Answer 31

Ok so how about #3? how do i do that?

Answer 32

how do we determine a y intercept?

Answer 33

the graph and stuff i believe in on the other post i did

Answer 34

I'm not really sure...It says to solve for the y–intercept for your function, f(x), and then construct a rough graph of your roller coaster. If your y–intercept is off the graph, give the coordinates of the y–intercept. I have no idea how to do that

Answer 35

what is our function?

Answer 36

f(x) = x^3 +3x^2 -40x -84

Answer 37

now recall that the y axis has another name, the x=0 axis

so to find the value of y intercept, we simply let x=0, what do we end up with?

Answer 38

0?

Answer 39

x^3 +3x^2 -40x -84, if x=0 this does not go to 0

Answer 40

What does it go to?

Answer 41

let x=0 and work the calculation ....

Answer 42

Wait but I got 0? How did you not get 0?

Answer 43

x^3 +3x^2 -40x -84

let x=0

0^3 +3(0)^2 -40(0) -84

0 +0 -0 -84 does not equal 0

Answer 44

so it's -84?

Answer 45

of course

Answer 46

So is that the answer to #3 or is there more?

Answer 47

make the rough graph of it ....

Answer 48

and then is that it? that's it for the whole thing after i make the graph?

Answer 49

other than organizing it into your own words ... yeah

Answer 50

How would you say it?

Answer 51

ive already said it :)

Answer 52

No but for #2 it says explain. You already did that? @amistre64

Answer 53

i explained it the best i could yes.

Answer 54

Did you get my message about the graphing website?

Answer 55

sketching a rough graph is simply makeing tics where the curve crosses the axises

Answer 56

if you wanted to use a graphing site, yes you would want to use the equation we came up with

Answer 57

Ok thank you so much! I really appreciate your help. Thanks for being so patient with me lol :)

Answer 58

youre welcome :) good luck and all