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**1) Prove that the acute angle whose cosine is 1/10 cannot be trisected with straightedge and compass.**

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I worked it out and at the end found out that , if I can prove that the cubic polynomial 40x

^{3}- 30x -1 has no rational roots, then I am done.

Now, is there any way to prove (e.g. divisibility, elementary number theory) that 40x

^{3}- 30x -1 has no rational roots without using the rational root test? (since it is very long and tedious and no calculators are allowed)

I was trying to prove this by contradiction.

Suppose x=m/n is a rational root in lowest terms.

But I am stuck at arriving at a contradiction...how can I possibly do so?

Any insights?

Thanks for any help!