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### Odd-degree polynomials have roots (Intermediate Value Theorem) [duplicate]

Let $f(x)$ be a monic polynomial of odd degree. Prove that $\exists A\in \mathbb{R}$ s.t. $f(A)<0$ and there exists $B \in \mathbb{R}$ such that $f(B)>0$. Deduce that every polynomial of odd ...
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### If $r(x)$ is an $n$th degree polynomial of the form $r(x) = a_0+a_1x+\cdots+a_nx^n$ with $n$ odd, prove that $r(x)$ has at least one real root [duplicate]

If $r(x)$ is an $n$th degree polynomial of the form $r(x) = a_0+a_1x+\cdots+a_nx^n$ with $n$ odd, prove that $r(x)$ has at least one real root. In what case does equality hold? Attempt: We have that ...
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### Prove that the odd polynomial equation $p(x)=0$ has at least one real solution (use intermediate value theorem) [duplicate]

Let $a_0,a_1,....a_n$ be real numbers and $p(x) = a_nx^n+a_{n-1}x^{n-1}+...a_1x+a_0$ for $x \in R$ I am trying to prove that the odd polynomial equation $p(x)=0$ has at least one real solution by ...
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### Proof of a theorem about polynomial [duplicate]

I have known a theorem from the link Derivation of the general forms of partial fractions The theorem is that a polynomial of odd degree has a root.But I don't know its proof.Can anyone prove it with ...
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### Proof that every odd polynomial has a real root - NOT a duplicate

I've done a search for this first because it is a common question, however I have yet to find one I follow/like. I ask this because I actually have an idea for how I might do it, I just don't know ...
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### Every polynomial of odd degree $\ge 3$ over $\mathbb{R}[x]$ is reducible over $\mathbb{R}$

I need to prove that every polynomial of odd degree $\ge 3$ over $\mathbb{R}[x]$ is reducible over $\mathbb{R}$. If $p(x)$ is my polynomial, then I just have to prove that $p(x)$ has one real root, ...
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I need some help with this problem. Here is the link. Can you please tell me if there is an easier way to show that cubic polynomials have a real root? The question is in an analysis book from the ...
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### What are the sufficient conditions for the existence of non-real roots for real polynomials?

What are the sufficient conditions for the existence of non-real roots for the following equation: $$a_0x^{n} +a_1x^{n-1}+...+a_n=0$$ where $a_0, ... , a_n$ are real numbers.
Takumi Murayama says "Every polynomial in $\mathbb R[x]$ of degree at least 3 has a real root, and therefore is not irreducible". I think I understand why it is not irreducible, but what's the real ...
I'm studying complex dynamics and I am struggling to verify the following facts. Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be a complex-valued polynomial of degree $d\geq 2$. That is, define \$f(z)=a_dz^...