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.
27,110
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AES S-box as simple algebraic transformation
The next matrix represents the Rijndael S-box according to wikipedia and other sources
$$\begin{bmatrix}s_0\\s_1\\s_2\\s_3\\s_4\\s_5\\s_6\\s_7\end{bmatrix} =
\begin{bmatrix}
1 & 0 & 0 &...
- 463
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Mapping polynomial into quadratic space
Given a parametric polynomial function $y = f(x,a)$, is there a way to find a quadratic equation that has the roots of the $N/2$'th and $N/2+1$'th (sorted) root of the original polynomial, regardless ...
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Are there natural numbers possible for (a/b)+(b/a)=-1? [closed]
Was trying to find integer values for a and b but was unsuccessful. Can there be any natural values? If not, how can we be sure?
We were asked to find a^3-b^3, for my school test, which is solved to ...
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How is this rough draft of a proof that $(x^2+10x-3=0)∉Qred?
I wanted to prove that the above quadratic is irreducible over Q without using common methods such as quadratic formula, discriminate, completing the square, rational root theorem, Eisensteins ...
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Hamilton-Cayley proof by Induction [closed]
Knowing the form of the chacartristic polynom for a 2x2 matrix is (1) Xa(t) = t^2-trace(A)*t+det(A)
If i evaluate Xa(A) this gives me zero. Can i prove the nxn case via induction with the hypothesis ...
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Number of real roots of a specific polynomial
I am working with the following polynomial:
$f_n := (x^2 + 2(n − 1)) \cdot (x − 2) \cdot (x − 4) \cdot (x-6) \cdots (x − 2(n − 2)) − 2$; $n\ge3$
I had to show that $f_n$ has exactly $n-2$ real roots.
...
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Does the Gauss-Lucas theorem generalize to arbitrary entire functions with a finite number of zeros?
The Gauss-Lucas theorem states that if $P(z)$ is a non-constant polynomial with complex coefficients, then all the zeros of $P'(z)$ lie within the convex hull of the set of zeros of $P(z)$.
Does this ...
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How to understand 'the problem of determining the exact number of monomials in P(x) given by a black box is #P-Complete'
In this paper:https://dl.acm.org/doi/pdf/10.1145/62212.62241, what's given is $P(x)$ a (sparse) multivariate polynomial with real(or complex) coefficients.
The author claimed two things.
It is known ...
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Optimal Value t for Subdivision of Cubic Bézier Curve and How to Calculate It
In Gabriel Suchowolski’s paper, “Quadratic bezier offsetting with selective subdivision”, he explains how the midpoint—or better said, a parameter $t$ of 0.5—is often not the optimal* point on a ...
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Given nonzero $p(x)\in\mathbb Z[x]$. Are there infinitely many integers $n$ such that $p(n)\mid n!$ is satisfied?
This problem comes from a famous exercise in elementary number theory:
Prove that there are infinitely many $n\in\mathbb Z_+$ such that $n^2+1\mid n!$.
I know a lot of ways to do this. A fairly easy ...
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Proof of Unique Factorisation of Polynomials over $\mathbb C$ by Identity Principle
Proposition: Any polynomial $p(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ can be expressed uniquely as
$$
p(x) = a_n(x-r_1)(x-r_2)\cdots(x-r_n),
$$
where $r_1, r_2,\ldots, r_n$ (not necessarily ...
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Find the natural number n for which the set $A_n$ has exactly 323 integers
the problem
For n a natural number we define $A_n=\{x\in \Bbb R \,|\,\, |x+n+4| \leq 3n-4\}$. The natural number n for which the set $A_n$ has exactly 323 integers is....?
my idea
i absolutely have ...
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solution-verification | solve $ x^4+x^2+1=2^y$
the problem
Find x and y, natural numbers, knowing that $x^4+x^2+1=2^y$
my idea
I took the cases 1) if x is even then $x^4+x^2+1$ is odd
if x is odd then $x^4+x^2+1$ is again odd
so $x^4+x^2+1$ is ...
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Add constraints to cubic or quintic polynomial [closed]
How can I add constraints to cubic or quintic polynomial such that the generated line is within a region. For example in the blue colored region below:
Image_Graph
For example if I generate a line ...
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Differences in meaning for notations $\alpha_i$ versus $\alpha(i)$ and meaning of $\beta(ij)$ for denoting axioms in monomials
The following are partly taken from Malik and Sen's Fundamentals of Abstract Algebra
Background
First we note that we can reconstruct the monomial $x^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ from $...
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