Question

Hng---

Can someone /actually/ help me?

(7,0) (-4,0) (-13,0)

Using those points as zeros construct the polynomial function f(x) that will be the path of your rollercoaster. Show all of your work.

Answer 1

FACTOR THEOREM: If (x-a) is a factor of f(x), then f(a)=0

CONVERSE OF FACTOR THEOREM: If f(a)=0, then (x-a) is a factor of f(x)

Answer 2

You'll want to use the converse of factor theorem.

Answer 3

So: think of what "factor" of a polynomial means.
Take those 3 factors and multiply them out.
Try this here and now, please.

Answer 4

Example: (x-1)(x+2) = x^2 + x -2

Answer 5

f(a)= 7-a
f(a)= -4-a
f(a)= -13-a

Answer 6

(x-1)(x+2) = x^2 + x -2 is an example of what I'm asking you to do: multiply out the factors, using the FOIL method of multiplying out binomials.

Answer 7

(7-1)(7+2)= 49+14-7-2

Answer 8

(7,0) (-4,0) are 2 of the 3 horizontal intercepts of the graph of the polynomial you're looking for.

Using the guidance that Kenny has given you, the corresponding factors of this poly are
(x-7)(x+4).

Please multiply that out now.

Answer 9

(x-7)(x+4)= x^2+4x-7x-28

Answer 10

Very nice. Would you please simplify that by combining any like terms.

Answer 11

x^2-3x-28

Answer 12

Great. This is the product of the first 2 factors of your polynomial, the factors based upon (7,0) and (-4,0). The third horizontal intercept is

Answer 13

(x+13).

Multiply out: (x+13)(x^2-3x-28).

Answer 14

x^3-3x^2-28x+13x^2-39x-364
x^310x^2-67-364

Answer 15

x^3+10x^2-67-364
@mathmale

Answer 16

Nice, but wouldn't that be 67x instead of just 67?

Answer 17

Still here?
Do you happen to know the technique of synthetic division?

Answer 18

Yeah, sorry. Yeah, it's be 67x.

Uhm, I do know the technique of synthetic division. But it's kind of scratchy, why?

Answer 19

Once you've multiplied out your 3 factors to obtain a polynomial, you could use synth div to determine whether any of the original roots (7, -4 or -13) are actually roots of OUR new polynomial.

Answer 20

I'll leave it to you whether or not you want to try that.

I've made several attempts to check the known root -4, but so far haven't gotten confirmation that our polynomial is correct.

Answer 21

I have no idea, man...

Answer 22

Another way of checking your answer might be wolframalpha.com. Please go to this link:

http://www.wolframalpha.com/input/?i=factor+x%5E3%2B10x%5E2-67x-364

What would you conclude about your polynomial, based upon the information there?

Answer 23

That it's correct.

Answer 24

Cool. wolframalpha.com is a powerful tool for checking your results, making quick and accurate graphs, etc., but not a substitute for knowing how to carry out math operations yourself. Congrats. Why not move on to the next problem?

Answer 25

Sure. c: