18 lines
498 B
TeX
18 lines
498 B
TeX
\beginsection{2.3}
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Show that $(2+\sqrt2)^{1/2}$ does not represent a rational number.
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\medskip
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$(2+\sqrt2)^{1/2}$ is a solution to the equation
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$x^2-2=\sqrt2$.
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This can be rewritten as
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$$(x^2-2)^2=2$$
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which expands to
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$$x^4-4x^2+4=2$$
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Hence $(2+\sqrt2)^{1/2}$ is a solution to the polynomial
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$$x^4-4x^2+2=0$$
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By the Rational Zeroes Theorem the only possible rational solutions
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are $\pm1$ and $\pm2$, neither of which is an actual solution.
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Therefore $(2+\sqrt2)^{1/2}$ is not a rational number.
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