Factor out and then look at values of the resulting expression for very large x.
Let f (x) be a monic polynomial of odd degree. Prove that for some A < 0,
f (A) < −1 and for some B > 0, f (B) > 1. Deduce that every polynomial of odd degree has a real root.
Suppose f(x) =x^(2n+1)+a2n x^(2n)...+a1x+a0, but I've no clue how to go from here. Could anyone please give me some hints? Any help is appreciated!
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For very large positive x, the expression in the brackets is very close to 1, right? And the is a very large positive value.
For very large negative x, the expression in the brackets again is very close to 1 and the is a very large negative value, since the exponent is negative.
So you know that for large enough positive x, f(x) > 0 and for large enough negative x, f(x) < 0. Now apply the intermediate value theorem.