Linked Questions

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3answers
5k views

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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2answers
2k views

Prove that odd polynomials have at least one root. [duplicate]

Intuitively this is easy for me to understand, but I don't know how to start the proof. Can someone help me? Prove that any polynomial of the form $x^n+\sum_{i<n}a_{i}x_{i}$, with $a_{i}\in \...
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4answers
1k views

Show that in ℝ[x], no polynomial of odd degree > 1 is irreducible. [duplicate]

I think that logically, I understand the concept because no matter what polynomial you have you can always factor it into something with a x to a power plus or minus some real number, and that real ...
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1answer
390 views

Prove odd degree polynomials have roots. [duplicate]

Let $p:\mathbb{R}\rightarrow\mathbb{R}$ be a polynomial of odd degree. Prove that there is a solution of the equation $$p(x)=0, x\in\mathbb{R}$$ I am giving this question in an analysis textbook and ...
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2answers
219 views

Let $p(x) = a_0 + a_1 x + .. + a_n x^{n}$. Assume $a_n \neq 0$ and n is odd. Prove $\exists$ x such that p(x)=0? [duplicate]

I have no idea how to solve this question. I'm trying to show that given $p(x) = a_0 + a_1 x + .. + a_n x^{n}$ and further assuming that $a_n \neq 0$ and n is odd, there exists x such that p(x)=0. I'...
0
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2answers
96 views

How can I show that $\mathbb R [x]/(x^5+x-3)$ is not an Integral domain? [duplicate]

How can I show that $\mathbb R [x]/(x^5+x-3)$ is not an Integral domain? To prove that it is not an Integral domain at first I have show that $(x^5+x-3)$ this ideal is not a prime ideal as $\mathbb R ...
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0answers
130 views

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 ...
0
votes
1answer
63 views

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 ...
0
votes
1answer
67 views

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 ...
1
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4answers
734 views

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 ...
2
votes
1answer
715 views

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, ...
3
votes
2answers
117 views

Please check my work! Question about cubic polynomials

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 ...
3
votes
2answers
105 views

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.
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votes
1answer
205 views

A real polynomial of degree more than or equal to 3 is reducible, but does it necessarily have a real zero?

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 ...
2
votes
1answer
50 views

The escape radius of a polynomial and its filled Julia set

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^...

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