- #1
r0bHadz
- 194
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Homework Statement
Use proof by contradiction to show there is no rational number r for which r^3+r+1 = 0
Homework Equations
The Attempt at a Solution
Assume there is a rational number r for which r^3+r+1=0. Then r = (a/b) with a,b ∈ℤ and b ≠ 0, and a/b is in lowest terms
Then a/b is a root, => (a/b)^3 + (a/b) + 1 = 0
=> [itex]\frac {a^3}{b^3} + (a/b) + 1 = 0 => a^3 + ab^2 + b^3 = 0 [/itex]
since a/b is in lowest terms, one must be odd, the other even. Let's examine a odd b even
let any letter variable you see from here on be ∈ ℤ
[itex](2p+1)^3 + (2p+1)(2f)^2 + (2f)^3 = 0
=>
[8p^3 + 12p^2 + 6p + 1] + [8f^2p + 4f^2] + 8f^3 = 0
=> 8p^3+8f^3 + 8f^2p + 12p^2 + 4f^2 + 6p + 1 = 0
=> 2[4p^3 + 3f^3 + 4f^2p + 6p^2 + 2f^2 + 3p] + 1 = 0
[/itex]
here we have the contradiction that 0 is an odd number.
now a even b odd:
[itex] a^3 + b^3 + b^2a = 0 =>
(2y)^3 + (2x+1)^3 + (2x+1)^2(2y) = 0 =>
8y^3 + [8x^3 + 12x^2 + 6x + 1] + [8x^2y + 8xy + 2y] = 0
=>
8y^3 + 8x^3 + 8x^2y + 12x^2 + 8xy + 6x + 2y + 1 = 0
2[4y^3 + 4x^3 + 4x^2y + 6x^2 + 4xy + 3x + y] + 1 = 0
[/itex]
as we can see, again zero is not an odd number.
So there is a contradiction. Since a/b is in lowest form, a/b cannot be a root.
I've been having problems with proof by contradictions guys. Every other proof method that I've learned so far is really easy for me but this one always kicks my butt. Does my proof seem okay to y'all?