polynomial functions from word problem

[echoes]

Been looking at this problem for a few hours and think I have a start. Here the question.

Four computers working together can upload a large file in 5 hours. One computer can upload the file in 3 hours less than the slowest computer; another computer can upload the file in 2 hours less than the slowest computer; and another computer can upload the file in 1 hour less than the slowest computer. Your goal is to find the time required for each computer to upload the file without help from the other three computer.

1) write a polynomial function that models the work of all four computer uploading the file


So far I have came up with x^4 - 9x^3 + 23x^2 + 15x - 5
got this from X(x-3)(x-2)(x-1)=5 hours

I am not sure this is correct, just want someone to guide me in the right direction here please. Or if you could show the steps on how to get the proper equation for question 1. There are a total of 5 questions, and if I get question 1 I can do the rest but question 1 is giving me a fit.

thanks for any assistance!

 

[BigGlenntheHeavy]

\(\displaystyle 1) \ Rate \ * \ Time \ = \ Amt. \ of \ work \ done.\)

\(\displaystyle Hence, \ 5\bigg[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}\bigg] \ = \ 1 \ which \ is \ 100\% \ of \ the \ upload.\)

\(\displaystyle Ergo, \ 5\bigg[\frac{1}{t-3}+\frac{1}{t-2}+\frac{1}{t-1}+\frac{1}{t}\bigg] \ = \ 1\)

\(\displaystyle Solving \ for \ t \ on \ my \ trusty \ TI-89, \ I \ get \ about \ 21.56 \ hrs. \ which \ equals \ 21:33:45.\)

1st computer: 18.56 hrs. (by itself)

2nd computer: 19.56 hrs (by itself)

3rd computer: 20.56 hrs (by itself)

4th computer: 21.56hrs (by itself)

All computers working together can do the job in 5 hours.

\(\displaystyle f(t) \ = \ 2t^{4}-52t^{3}+202t^{2}-232t+60\)