Question 428371
On planet ZOG, colored coins are used as money.  12 white, 9 red, 8 yellow, 4 blue, and no greens can be exchanged for 2 white, 1 red, no yellows, 1 blue, and 1 green.  Suppose that 1 green=n blues, 1 blue=n yellows, 1 yellow=n reds, and 1 red=n whites.

What is the whole number exchange rate for these coins?  Is one exchange rate possible?

Okay, so I started work and hit a dead end.  My work is as follows.

{{{ 12w+9r+8y+4b=2w+r+b+g }}}
therefore
{{{ 10w+8r+8y+3b=g }}}

We also know that g=bn, b=ny, y=nr, and r=nw.

Therefore {{{ 10w+8nw+8(n^2)w+3(n^3)w=(n^4)w }}}
or {{{ (n^4)-3(n^3)-8(n^2)-8n-10=0 }}} by subtracting everything to one side and dividing everything by w.

The problem is this: I am in Algebra I.  We haven't learned how to solve 4th degree polynomials, which makes me think that there is an easier way to solve this problem that I didn't pick up on.

Please help.  I don't necessarily want the answer, just an Algebra I way to solve this problem.

Thanks in advance.
 
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1st of all: There is a solution less than 10.
2nd: Let n = some value like n = 10.  You might
get a positive value.
Let n = some value like n = -10. You might
get a negative value.
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That mean there is a solution between n = -10
and n = 10.
---
Use that idea to pin the solution between 2
values.  Keep pinning it till you find the
solution.
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By the way, you have done a find job with
the problem.  Congratulations.
Cheers,
Stan H.
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