Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.

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Splitting a multiplication into multiple smaller steps, reaching the same result

Suppose I have a number, x, which should be doubled every second. If one had a function which is called exactly once every second, the solution would be simple: All you would have to do was ...
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2answers
164 views

Polynomial's coefficient

Is there any general way to find out the coefficients of a polynomial. Say for e.g. $(x-a)(x-b)$ the constant term is $ab$, coefficient of $x$ is $-(a+b)$ and coefficient of $x^2$ is $1$. I have a ...
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4answers
375 views

How to interpret the phrase “transforms under the irreducible representation”?

I'm reading Robert Gilmore's "Lie Groups, Physics, and Geometry," and trying to understand his brief presentation of Galois theory. I think I get the gist of the method, but would be grateful for help ...
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2answers
532 views

Polynomial passing through two points with specific tangents

How can I calculate a Polynomial that passes through the origin and a given point (P) having given tangents at the origin (Ot) ...
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1answer
86 views

Distinguishing vector distributions induced by polynomials

I am given two sequences of multivariate polynomials $\overline{p}=(p_1,p_2,\dots,p_k)$ and $\overline{q}=(q_1,q_2,\dots,q_k)$, all of them on the variables $x_1,\dots,x_n$ over some finite field ...
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2answers
206 views

Intersections of roots of polynomial in a field

I tried to prove some property of fields, but I could not, and I hope someone can help me with that. I have a question about fields and roots. If I have an arbitrary family (each of them is a set of ...
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1answer
140 views

Vandermonde and curve interpolation

I hesitate here because of an understanding with a calculation problems. I want to calculate an interpolation using the Vandermonde matrix. see: http://en.wikipedia.org/wiki/Vandermonde_matrix My ...
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0answers
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General questions about level sets of polynomials of two variables

Sorry if I'm being too general here, but here it goes. I'm trying to find out more about levels sets of polynomials of two variables of degree $d$ $$ C = \{ (x,y) \ : \sum_{1 \leq i + j \leq d} ...
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160 views

Solve a polynomial of degree d

In my Artificial Intelligence class, I encountered this equation where I have to solve for b, the effective branching factor of a tree: ...
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1answer
404 views

Solving a real cubic using trigonometric methods

The problem is to figure out how to solve a real cubic equation of the form $x^3 + px + q = 0$ using trigonometry. The first step is to prove the identity $$ 4\cos^3 \theta - 3\cos \theta - \cos ...
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1answer
501 views

How to solve polynomials?

Using Galois theory we can effectively compute whether or not a polynomial is solvable in radicals - technically this means you can build a chain of field extensions by adding $n$-th roots of ...
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662 views

Describe the locus in the complex plane of the zeros of a quartic polynomial as the constant term varies

(Diagram and setup from UCSMP Precaluclus and Discrete Mathematics, 3rd ed.) Above is a partial plot of the zeros of $p_c(x)=4x^4+8x^3-3x^2-9x+c$. The text stops at showing the diagram and does ...
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1answer
246 views

Polynomial mappings of half planes and disks on the complex plane

Main problem: Let $\mathcal{L}=\left\{z\in\mathbb{C}:Re(z)<0\right\}$ be the left open half-plane of the complex plane and $\mathcal{C}=\left\{z\in\mathbb{C}:|z|<1\right\}$ be the open unit disk ...
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619 views

How are polynomials mod m reduced?

How do you reduce polynomials that are mod m? For example if I have 10x + 5 (mod 3) can I just reduce that to x + 2 (mod 3)?
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2answers
136 views

Computation of characteristic polynomial fails for me

For a matrix $A\in\mathbb{K}^{n\times n}$ where $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ the characteristic polynomial is defined as $$\chi_A(\lambda) := \text{det}(A-\lambda I_n) = \sum_{k=0}^n c_k ...
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2answers
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Need help in simplifying the root of a cubic

I have this not so great looking cubic polynomial in $x$: $$(a - b)^6 + 3 (a - b)^4 (-a^2 c - b^2 d) x + 3 (a - b)^2 (a^4 c^2 - 7 a^2 b^2 c d + b^4 d^2) x^2 + (-a^2 c - b^2 d)^3 x^3$$ where ...
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A quick question on polynomial division in fields of characteristic p

Suppose that $L$ is a field of characteristic $p$, $E$ is a field extension of $L$, a is a pth root of an element of $L$ such that a is not in $E$. Consider the polynomial $p(x):=x^p-a^p.$ Question: ...
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How do I pick a polynomial of some degree k-1 such that P(0) = specific number (mod m)?

Hi this is for the Shamir Secret Sharing algorithm. How do I create a polynomial of degree k-1 such that P(0) = s (mod m). where k-1 is just an integer, s is just an integer, m is a large prime.
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How to show that $\sqrt{2}+\sqrt{3}$ is algebraic?

In Abbot's Understanding Analysis I am asked to show that $\sqrt{2}+\sqrt{3}$ is an algebraic number. I have shown that those two are algebraic separately (that was simple), but I can't figure out ...
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3answers
912 views

How to solve polynomial equations in a field and/or in a ring?

I'm studying for my exam, and I stuck on solving polynomials in a field and/or in a ring. Let me give you some examples: (1) Solve equation $x^2+4x+3=0$ in field $\mathbb{Z}_5$, $\mathbb{Z}_8$ and in ...
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2answers
202 views

Given any polynomial p(x) over Z, can one construct a graph with characteristic polynomial p(x)?

Given any polynomial $p(x)$ over $\mathbb{Z}$, can one construct a graph with characteristic polynomial $p(x)$? [Edit: Title question added to post.} Further questions include: Are there classes of ...
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1answer
229 views

Finding the point where the tangent is the x-axis

Consider a simple example of a cubic: $px^3+qx^2+rx+s$ with $p,q,r,s\in\mathbb{R}$. I want to find the point $a$ in the following figure, where the cubic has a double root, as a function of the ...
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3answers
575 views

Roots of a cubic equation

When will the roots of a cubic polynomial $ax^3+bx^2+cx+d$ generate a cyclic group of order 3? I literally mean: the cubic equation ax3+bx2+cx+d will have three roots (possibly all three real, ...
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1answer
352 views

Adding and Multiplying Polynomials Recursively

What theorem can be used to recursively multiply two polynomials together? Is there another theorem that uses recursion to add together two polynomials recursively? I'm looking for something that ...
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552 views

Recursive method to evaluate a polynomial

I want to find a recursive way of evaluating any polynomial (I'm given the polynomial, and a value for x, and I need to replace the x in the polynomial with the value). The polynomial can be anything, ...
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2answers
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How to calculate a third degree polynomial function

I have this formula calculated by excel: $$y = -0.0001x^3 + 0.0294x^2 - 0.0567x - 68$$ This formula was calculated using this data: ...
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2answers
121 views

Why is a polynomial of form $g(X^{p^m})$ over a field of characteristic $p$ not necessarily inseparable?

A small proposition in Ash's Algebra states that over a field $F$ of prime characteristic $p$, an irreducible polynomial $f$ is inseparable if and only if $f'$ is the zero polynomial, or ...
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4answers
348 views

Problem: Sum of absolute values of polynomial roots

Can you please give me some hints as to how I might approach this problem? Thanks! Given the polynomial $f = 2X^3 - aX^2 - aX + 2, a \mathbb \in R$ and roots $x_1, x_2$ and $x_3,$ find $a$ such ...
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1answer
120 views

Bounding $p$-adic valuations in inequality

I'm developing an algorithm that comes across inequalities of the form \begin{align*} \operatorname{ord}_p(c(b)) > \alpha \end{align*} for some polynomial $c \in \mathbb{Q}[b]$, $c(b) = c_0 + c_1b + ...
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Infinite series representation for root of polynomials?

Given a polynomial $p(x)=a_nx^n+\dots+a_1x+a_0$, can every root of the polynomial be represented as $\sum_{k=0}^\infty b_k$ with the $b_k$'s being a function of $a_0,\dots,a_n$ using only elementary ...
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363 views

always irreducible polynomial?

There exists irreducible polynomials in $\mathbb{Z}[x]$ (e.g. $x^4-10x^2+1$) which is reducible modulo every prime $p$.(A proof can be found in J.S. Milne's "Fields and Galois Theory", page13. Here is ...
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1answer
162 views

Why is the content of an irreducible polynomial not a unit?

I'm reading a corollary in my book, but I don't follow one sentence. Corollary: Let $D$ be a UFD with quotient field $F$. If $f$ is a nonconstant polynomial in $D[X]$, then $f$ is irreducible ...
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When does a polynomial divide $x^k - 1$ for some $k$?

Given a monic polynomial $f\in\mathbb{Z}[x]$, how can I determine whether there is a $k\in\mathbb{Z}^+$ such that $f|(x^k-1)$? For example, $x^2-x+1$ divides $x^6-1$, but $x^2-x-1$ does not divide ...
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1answer
208 views

Efficient calculation of polynomial product

I have 2 polynomials $p_1(x_1,\ldots,x_n)$ and $p_2(x_1,\ldots,x_n)$, of which I have to compute the product, with a special property: The exponent of each variable is always either $0$ or $1$, where ...
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3answers
293 views

Is there an algorithm to find the roots of high-order polynomials?

It is not generally possible to determine the roots of a polynomial whose grade is bigger than 4 in terms of roots and basic operations. But I heard, that it is possible to give a criteria whether a ...
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1answer
163 views

Lower bound over a convex function

I will be to grateful if help me find a tight lower bound $g(x)$ over the following convex function: $$f(x) = \sqrt{1+4x^2} -1 + \log(\sqrt{1+4x^2}-1) - \log(2x^2) \geq g(x),$$ where $g(x)$ is ...
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Proving $\{P(a)\}$ (where $a$ is transcendental) is dense

So, I have read this problem, and it's bugged me since: Let $a \in (1,2)$ be a transcendental number 1) Let $Y = \{ P(a) : P \in \mathbb{Z}[X] \}$, show that Y is dense in $\mathbb{R}$. ...
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5answers
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Trick for roots of symmetric polynomials

Given a polynomial such as $X^4 + 4X^3 + 6X^2 + 4X + 1,$ where the coefficients are symmetrical, I know there's a trick to quickly find the zeros. Could someone please refresh my memory?
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1answer
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Polynomial equations with finite field arithmetic

there are given 3 equations (they are connected with cyclic codes): $$s(x)=v(x)+q(x)g(x)$$ $$g(x)h(x)=x^7+1$$ $$s(x)=v(x)h(x)\bmod(x^7+1)$$ I have following data (for $GF(8)$ with generator ...
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1answer
66 views

Can not 'see' how to get next line of a particular Sturm Sequence

Stu(0)P = X^4 + pX^2 + qx + r Stu(1)P = 4x^3 + 2px + q Stu(2)P = -[2px^2 + 3qx + 4r]/4 Should anyone know how to get from the 3 lines above to Stu(3)P shown here on next line:- Stu^3(P) = ...
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1answer
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From complex solution to solutions over finite fields

There are several ways (Hilbert's Nullstellensatz, model theory, transcendence bases etc.) to prove the following (amazing!) result: If $f_1,...,f_r$ is a system of polynomials in $n$ variables with ...
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2answers
485 views

Integrate in Mathematica takes forever

I'm trying to calculate the length of a curve from a polynomial in Mathematica and for some reason Integrate doesn't complete, it just runs until I abort the execution. The polynom: ...
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2answers
141 views

Euclids Algorithm for polynomials and a greatest common divisor

I have question about a problem I've encountered while attempting to solve an exercise (it's from an exercise in a homework series). Suppose we have two polynomials $f$ and $g$ (presumably over the ...
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0answers
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Efficient way to recompute weights when shifting range of Legendre polynomial bases

I am storing a 2D (Cartesian) density function as a 2D patch with known X/Y limits and a set of 11 coefficients of the third order 2D Legendre polynomial basis functions over that patch. This works ...
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Turning real roots into curves (for visualisation)

One can obviously map a set of real numbers $x_1, x_2, \ldots x_N$ to a curve in 2-D via $y=(x-x_1)(x-x_2)\ldots(x-x_N)$. Thinking about data visualisation, one can portray a set of $N$ observations ...
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2answers
74 views

bound on the leading coefficient of a polynomial

Given a polynomial with real coefficients, that satisfies $\forall x \in \left[ -1,1 \right]: \ |p(x)|\leqslant 1$, I have to show, that its leading coefficient, $a_m$ satisfies $a_m \leqslant ...
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4answers
161 views

Is there a usual method for finding the minimal polynomial of trigonometric values?

I've been thinking a bit about finding the minimal polynomials of side lengths of regular $n$-gons inscribed in the unit circle. For example, I recently wanted to find the minimal polynomial of the ...
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614 views

Existence of n distinct (real) roots of an orthogonal polynomial

I'm trying to get my head around the proof that an orthogonal polynomial ($P_n$ say) has at least n distinct roots. My understanding of the proof ...
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4answers
1k views

Finding double root. An easier way?

Given the polynomial $f = X^4 - 6X^3 + 13X^2 + aX + b$ you have to find the values of $a$ and $b$ such that $f$ has two double roots. I went about this by writing the polynomial as: $$f = X^4 - 6X^3 ...
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2answers
107 views

Rational function sequence product

I want to know a closed formula for $$\prod_{m = 0}^{n} \frac{m^2 + a}{m^2 + a + 1},$$ a being any given complex. When the exponent is 1, it's pretty trivial, because of cancellations, but with other ...