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Why must there always be at least one real root if the degree of a polynomial is odd?

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Please see the attached file.

Let us investigate the roots of the equation
g(x) = 4x3 + 3x2 − 6x − 1 = 0

We note first of all that this is a cubic equation. It can be seen that ...

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Roots of odd polynomials are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.