Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, dividing, factoring and solving for roots.
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What is the discriminant of $t^4 - 2(p+q)t^3 +(p-q)t -1$?
How is it possible to find that the discriminant of $$t^4 - 2(p+q) t^3 + 2(p-q)t -1$$ is equal to $\Delta = 27p^2q^2 - (q^2 - p^2 -1)^3$ ?
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0answers
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Can a linear combination of even Legendre polynomials have common real root(s) with a linear combination of odd Legendre polynomials?
I am using the following definition of Legendre Polynomials: $P_0(x)=1$, $P_1(x)=x$ and
$$P_{k+1}(x)=\left(\frac{2k+1}{k+1}\right)xP_k(x)−\left(\frac{k}{k+1}\right)P_{k−1}(x)$$
Let
...
1
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1answer
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affine variety of infinitely many polynomials can be represented as an affine variety of its finite subset
Let $f_1,f_2,\cdots$ be an infinite sequence of polynomials in $k[x_1,\cdots,x_n]$ and let $V(f_1,f_2,\cdots)=\{(a_1,\cdots,a_n)\in k^n:f_i(a_1,\cdots,a_n)=0$ for $i=0,1,\cdots\}$. Show that there is ...
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1answer
44 views
How come the Bernstein operator creates a polynomial of the same degree as its input function?
This is a copy of the following question on the Computer Science Stack Exchange: ...
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2answers
28 views
Division of a cubic equation by one of its factors [duplicate]
I'm trying to divide a cubic equation by a factor.
This is the equation:
$$ -\lambda^3 -\lambda^2 + 10 \lambda - 8 = 0$$
and this is the factor : $(\lambda - 1)$
I Googled about it and I found the ...
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4answers
67 views
Calculating a complex derivative of a polynomial
What are the rules for derivatives with respect to $z$ and $\bar{z}$ in polynomials?
For instance, is it justified to calculate the partial derivatives of ...
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3answers
93 views
Polynomial and distinct roots
Find all real m such that $x^3-2x^2-2x+m$ has 3 distinct rational roots.
Source: School exam paper. No idea why it seems so hard
I don't think rational root theorem works, since m is not necessarily ...
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6answers
159 views
Representing the function $\mathbb Z_9\to\mathbb Z_9$, $f(0) = 1$, $f(1) = \ldots = f(8) = 0$ as a polynomial in $\mathbb Z_9[x]$
Let $\mathbb Z_9=\left\{0,1,2,3,4,5,6,7,8\right\}$ be the set of integers modulo 9 and $f:\mathbb Z_9 \rightarrow \mathbb Z_9$ be a function.
Assume $f(0)=1$, $f(1)=f(2)=...=f(8)=0$. What is the ...
7
votes
1answer
203 views
Krylov-like method for solving systems of polynomials?
To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
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1answer
37 views
what if geometric sequence + geometric sequence
I wrote a program that basicly can find the formula of the sequence that created with any-degree equation.
For example if you give my program that sequence:
[1926, 2811, 833240, 28778265, 398155842, ...
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vote
2answers
142 views
Quadratic equations
Does anyone know how to find integer solutions of the quadratic equation
$$y^2+y+z=f$$
where $z$ is a fixed odd prime or $1$ and $f$ is a fixed odd prime greater than $3$?
This problem arose from ...
3
votes
1answer
939 views
Finding inverse of a difficult function
Hi I'd like to find the inverse of:
$$
y=(1/3)(x^{4} + 4x^{3})
$$
I have learned to do inverses using the following example:
$$y=2x-1$$
$$x=2y-1$$
$$x+1=2y$$
$$(x+1)/2=y$$
$$f^{-1}(x)=(x+1)/2$$
...
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2answers
51 views
How to prove an odd-degree polynomial starts and ends at values of different sign?
Consider an odd-degree polynomial. How to prove that it starts a value that has different sign from its end value?
Or
$$
\lim_{x\to -\infty} f(x) \lim_{x\to+\infty } f(x)<0
$$
Please don't use ...
6
votes
3answers
68 views
Is there a pattern of the factorization of a polynomial modulo $p$ as $p$ varies
Take $P\in\mathbb{Z}[X]$ and factorize it modulo $p$, where $p$ is a prime.
Modulo different $p$'s the factorization varies. Is there a pattern in this variation? I mean, for example, if $P$ is ...
1
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2answers
35 views
Solving cubic equations
I was trying to solve a cubic equation which is :
$ -\lambda³ -\lambda² + 10 \lambda - 8 = 0$
I googled about it and I found the Rational Root theorem which is takes time to do it, but I found that ...
