kabutjv7

2022-05-01

Let $f(x)$ be a monic polynomial of odd degree. Prove that $\mathrm{\exists}A\in \mathbb{R}$ s.t. $f(A)<0$ and there exists $B\in \mathbb{R}$ such that $f(B)>0$.

Deduce that every polynomial of odd degree has a real root.

There are questions that answer the final part, but they do not do so by proving the first part. I am fairly sure that this involves the intermediate value theorem, but not sure how to implement it in this case.

Deduce that every polynomial of odd degree has a real root.

There are questions that answer the final part, but they do not do so by proving the first part. I am fairly sure that this involves the intermediate value theorem, but not sure how to implement it in this case.

RormFrure6h1

Beginner2022-05-02Added 13 answers

For the final part. If you have $f(A)<0$ and $f(B)>0$ then by the IVT every value in [f(A),f(B)] is attained by f(x) for some x between A and B, and this includes 0.

To show the existence of the A and B show that for x large one has that the sign of f(x) is the sign of the leading coefficient. And, if the degree is odd for small x one has that the sign of f(x) is the opposite sign of the leading coefficient.

To show the existence of the A and B show that for x large one has that the sign of f(x) is the sign of the leading coefficient. And, if the degree is odd for small x one has that the sign of f(x) is the opposite sign of the leading coefficient.

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