Questions tagged [polynomials]
For both basic and advanced questions on polynomials in any number of variables, including, but not limited to solving for roots, factoring, and checking for irreducibility.
5 questions from the last 7 days
3
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1
answer
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Number of real roots of the n-th iteration of $f(x) = x^3 - 3x + 1$
Given, $$f(x) = x^3 - 3x + 1$$
I was solving a problem to find the number of distinct real roots of the composite function $f(f(x)) = 0$.
By analyzing the graph of $f(x)$, we can observe the local ...
- 964
2
votes
2
answers
143
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If the evaluation map $\varepsilon\colon R[X] \to R^R$ is injective, does every non-zero polynomial have only finitely many roots?
Let $R$ be a (commutative, unital) ring. We then have a homomorphism of rings $\varepsilon \colon R[X] \to R^R$, given by associating to a polynomial the function it induces.
EDIT: More precisely, for ...
- 99
2
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0
answers
170
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Finding all monic polynomials $P(x)$ such that $P(x+1)\mid P(x)^{2}-1$
Math olympiad polynomial problem:
Find all monic polynomials $P(x)$ such that
$$P(x+1)\mid P(x)^{2}-1$$
My attempts to solve this problem:
$P(x_{1}+1)=0\Rightarrow P(x_{1})=-1,1$
$x_{1}+1\mid P(x_{1}...
3
votes
0
answers
99
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Creative Alternatives to Vieta's formulas/Newton's identities
Vieta's formulas are well known.
$$\sum _{1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}\left(\prod _{j=1}^{k}r_{i_{j}}\right)=(-1)^{k}{\frac {a_{n-k}}{a_{n}}}$$
For example, the sum of the roots of ...
- 5,309
2
votes
1
answer
61
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How to prove $\frac{f(x)}{g(x)}=\sum_{i=1}^{n}\frac{A_{i} }{x-r_{i} } $ using Bezout Identity
(I'm actually learning calculus.)Before I started working on this problem,I went to read this proof:
Partial Fractions Proof
I think I understand what the proof tried to do(And I can complete some ...
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