This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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1
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1answer
23 views

When does a binomial have repeated roots mod p?

Given a polynomial $f(x)=x^n+a$, and I have that $p$ does not divide $an$, can I show that $f(x)\pmod p$ has no repeated roots? I'm not sure how to proceed.
1
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1answer
56 views

How many partial derivatives does a multivariate polynomial have?

My motivation for this question is from the following toy example; define the (nondeterministic) finite state machine generated by the polynomial $f(x_0 , ... , x_n) \in \mathbb{Z} [x_0 , x_1 , ... , ...
1
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0answers
16 views

Coprime polynomials in $k[x,y]$ are also coprime in $k(y)[x]$

Let $f_1,...,f_n \in k[x,y]$ be polynomials with no common factor. Prove that when viewed as elements of $k(y)[x]$ they still do not have a common factor. Say we have only two polynomials $f=\sum ...
1
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1answer
40 views

Stirling's approximation from Euler-Maclaurin formula

I try to derive Stirling's approximation from Euler-Maclaurin formula with form: ...
0
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0answers
10 views

Determining positivity of a blackbox multivariate polynomial

Is there a way to check the positivity (or non-negativity) of a multivariate polynomial $f: \mathbb{R}^n \to \mathbb{R}$, of a given degree $d$, by querying the value of $f$ at finitely many points?
1
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1answer
23 views

Higher degree polynomial with complex roots

I'm working on the following problem: $$ r^4 - 3r^2 -4r = 0 $$ I factor out one $r$ and leaving me $ r(r^3 - 3r -4) = 0 $. One real root is $r=0$, and I'm unable to find the other ones. I tried ...
0
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0answers
8 views

Proof of $x ≤ \max { (k | c_{m1} |^{1/m1} , k | c_{m2} |^{1/m2} , . . . ,k | c_{m2} |^{1/mk})} .$

Can you please give a proof of this lemma : Let $P (X)$ be an univariate polynomial of degree n $: P (X) = X^{n} + c_{1} X^{n-1} + . . . + c_{n}$ with $ c_{n} \neq 0$. Let $c_{m1} , c_{m2} , . . . , ...
-2
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1answer
97 views
+100

About two polynomials $f,g$ such that $f=\pm g$

Let $R$ be an infinite commutative ring with unit and with characteristic zero. Assume that $f,g\in R[x_1,...,x_n] $ are nonzero and such that $f(x_1,...,x_n)=s(x_1,...,x_n) g(x_1,...,x_n)$, where ...
0
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1answer
26 views

Find a recurrence relation and the Fourier-Legendre Series

Rodrique's Formula for the $n$th Legendre Polynomial is $$P_n\left(x\right)=\dfrac{1}{2^nn!}\dfrac{d^n}{dx^n}\left(\left(x^2-1\right)^n\right)$$ The Fourier-Legendre series of a function f is ...
2
votes
1answer
33 views

If $P$ is an integer polynomial with $P(1)=P(2)=0$, then some coefficient is less than $-1$

Let $P (x)$ be a polynomial with integer coefficients. It is known that the numbers $1$ and $2$ are its roots. Prove that there exists a coefficient that is less than $-1$. My work so far: Let ...
3
votes
1answer
17 views

Discriminant of a trinomial $x^n+ax^m+b$

I am trying to compute the discriminant of the trinomial $x^n+ax^m+b$. I have tried using resultants but cannot see how to approach it. Any hints?
5
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3answers
2k views

The Degree of Zero Polynomial.

I wonder why the degree of the zero polynomial is $-\infty$ ? I heard that, it is $-\infty$ to make the formula $\deg(fg)=\deg(f)+\deg(g)$ hold when one of these polynomials is zero. However, if that ...
0
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0answers
17 views

Integral computation with Mathematica and Sympy differ

To compute the integral: $I = \int_{0}^{+oo} ue^{Au^{2}+Bu}du$ where $A<0$ and $B>0$ I have tried both Mathematica and Sympy but they yield different results: Mathematica yields: $ I = ...
1
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0answers
19 views

Polynomial division in factored form

Given two factored polynomials of the same degree $N$: $$ \begin{align} P(x) &= \prod_{k=1}^{k=N} (x - p_k) \\ Q(x) &= \prod_{k=1}^{k=N} (x - q_k) \end{align} $$ Due to $P$ and $Q$ having ...
1
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0answers
28 views

Lower bound on the difference between max. and min. values of a polynomial over $[-1, 1]$

Problem: $P(x)$ be a monic, n-degree polynomial with real coefficients. Prove that it is not possible that for all $t \in [-1, 1]$, $$\frac{-1}{2^n} < P(x) < \frac{1}{2^n}$$. I tried it to put ...
4
votes
2answers
63 views
+50

Roots of Sum of Two Polynomials (with Known Roots)

I am writing a piece of software and I'm trying to avoid root finding polynomials for efficiency purposes. I have two polynomials with complex coefficients, where the roots of both polynomials are ...
4
votes
1answer
35 views

Graphically solving for complex roots — how to visualize?

So recently we've been doing the complex roots of quadratics, cubics and polynomials in general in school. But my question is, is there a way to see where these roots are, just like you can see where ...
0
votes
1answer
27 views

Kernel of a polynomial with matrix, $ker(p(A))$

Let $A\in Mat(3,3,\mathbb R)$ a matrix and $\chi_A(x)=p_1(x)\cdot p_2(x)$ the characteristic polynomial. Evaluate $ker(p_1(A))$.$$A=\begin{pmatrix} 0 & 0 & 2 \\ 1 & 0 & 1\\ 0 & ...
6
votes
1answer
29 views

Iteration of polynomial has only positive roots

Let $P(x)$ be a real polynomial with a positive leading coefficient, and $k\geq 2$ an integer. Suppose that $Q(x)=P(P(\dots(P(x))\dots))$, where there are $k$ iterations of $P$'s, has at least one ...
4
votes
5answers
107 views

Roots of $x^{101}-100x^{100}+100=0$

I do not know how to prove that $x^{101}-100x^{100}+100=0$ has exactly two positive roots. Some can give me hint for solving this please. Thanks for your time.
5
votes
4answers
432 views

Algorithm to find the exact roots of solvable 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 ...
0
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0answers
53 views

Find a polynomial such that this proposed root finding algorithm fails.

Is this polynomial root finding algorithm below known, and under what conditions for the choice of polynomial coefficients does it find at least one root? Description of the algorithm: Consider the ...
1
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1answer
38 views

Is expression $(1/x)/(2/x^2)$ is fraction expression or rational expression?

A rational expression is nothing more than a fraction in which the numerator and/or the denominator are polynomials. Here are some examples of rational expressions. $$\dfrac{6}{x-1}, ...
0
votes
0answers
17 views

Loops around 0 of polynomial restricted to the unit circle [duplicate]

Given a polynomial with coefficients in C, consider the image of the polynomial restricted to the unit circle (That is plugging in only things with absolute value one). How many loops around 0 can ...
1
vote
1answer
48 views

Universal Equation For $x$

For: $ax+b=0\;;\; x= \frac{-b}{a}\;\;\;\;\;$ and for:$$ Ax^2 +bx+c = 0\;; \;x = \frac{-b\pm\sqrt {b^2-4ac}}{2a}$$ And for $$ Ax^3+bx^2+cx+d =0$$ Is there a constant transformation from equations ...
4
votes
3answers
8k views

The degree of a polynomial which also has negative exponents.

In theory, we define the degree of a polynomial as the highest exponent it holds. However when there are negative and positive exponents are present in the function, I want to know the basis that we ...
13
votes
3answers
524 views

What is the degree of the zero polynomial and why is it so? [duplicate]

My teacher says- The degree of the zero polynomial is undefined. My book says- The degree of the zero polynomial is defined to be zero. Wikipedia says- The degree of the zero ...
2
votes
2answers
2k views

What if a polynomial is identically zero?

From Barbeau's Polynomials: (a) Is it possible to find a polynomial, apart from the constant $0$ itself, which is identically equal to $0$ (i.e. a polynomial $P(t)$ with some nonzero ...
0
votes
0answers
24 views

A simple Lagrange interpolation-type identity

I am unable to prove an identity that looks very much like the Lagrange interpolation identity, Problem: Given $f(x)$ is a monic, $n-1$ degree polynomial and $a_1, a_2, \cdots a_n$ distinct real ...
1
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2answers
63 views

Factoring a degree 4 polynomial without power of 2 term

For my hobby, I'm trying to solve $x$ for $ax^4 + bx^3 + dx + e = 0$. (note there's no $x^2$) I hope there is a simple solution. I'm trying to write it as $(fx + g)(hx^3+i) = 0$ It follows that ...
1
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1answer
30 views

Multiplicity of roots of polynomial with rational coefficients decidable?

From the standpoint of intuitionistic logic, multiplicity of roots of generic polynomial is uncomputable due to the inability to compare two real numbers. Even though the roots themselves are ...
1
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0answers
12 views

finding generating function of orthogonal polynomials through their moments

I was studying a method to find the generating function of Orthogonal Polynomials through its moments. Please refer to the paper Use of Hermite's method to obtain generating functions for classical ...
3
votes
3answers
66 views

Need help solving $x^4-3x^3-11x^2+3x+10=0$

Solve $x^4-3x^3-11x^2+3x+10=0$ I have tried to solve this equation using 'general formula from roots' from https://en.wikipedia.org/wiki/Quartic_function. $$ax^4+bx^3+cx^2+dx+e=0$$ $$x_{1,2}=-\frac ...
0
votes
1answer
13 views

How does variable ordering in expressions work when creating functions from an equation?

I'm having a really hard time understanding some aspects of functions, i've tried looking around on Khan academy and haven't quite found something to answer my question, i'm sure i'm overlooking ...
3
votes
2answers
402 views

Finding characteristic polynomial of adjacency matrix

Short question im having a tad difficulty with. I'm trying to find the characteristic polynomial of a graph that is just a circle with n vertices and n edges. I think the adjacency matrix should ...
0
votes
4answers
39 views

Determine whether the set of vectors is a linear subspace

Let $V$ be the vector space of all polynomials $f\left(t\right)$ over $\mathbb{R}$ of degree at most $3$. I am trying to show that whether all polynomials $a+bt+ct^2+dt^3$ with ...
6
votes
3answers
3k views

Lagrange basis functions as bases of Polynomials Space

Suppose $L$ be a Vector Space of Polynomials of $x$ of degree $\leq n-1$ with coefficients in the field $\mathbb{K}$. Define $$g_i(x) :=\prod _ {{j=1},{j\neq i}}^n \frac{x-a_j}{a_i-a_j}$$ Show that ...
1
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0answers
20 views

Constant of Holder-type Inequality for Polynomial Function

Is anybody aware of an inequality in the following form $$ \Vert f \Vert_{L_p(\Omega)} \leq C(p) \Vert f \Vert_{L_q(\Omega)} $$ where $f$ is a polynomial function of degree $p$ on $\Omega \subset ...
2
votes
2answers
46 views

To show that the variables in the system are same in magnitude

I am stuck with this interesting problem, If for non-negative integers $a, b, \text{and} c$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are both integers then ...
-1
votes
0answers
25 views

Problem from complex analysis [duplicate]

How to solve such problem: Prove that polynomial $a_0 + a_1 z + ... + a_n z^n$ where $0 < a_0 < a_1 < ... a_n$ has $n$ roots in the circle $|z| < 1$
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1answer
41 views
0
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0answers
34 views

Is there an analytic solution to find zeroes of a polynomial plus sin()?

Is there an analytic solution to find the zeroes of an equation of the form: $$0 = at^2+bt+c+\sin(mt^2+nt+o)$$
2
votes
1answer
31 views

When are the limits of roots of a polynomial identical to the roots of the limit of the polynomial?

I have a univariate polynomial of degree $n$ (where $n$ is larger than $4$). The real-valued coefficients of the polynomial depend on a parameter $\psi$, i.e. $$p_\psi(x)=a_n(\psi) x^n+a_{n-1}(\psi) ...
5
votes
1answer
52 views

Polynomials bounded by $[-1, 1]$ iff argument is in $[-1, 1]$

Problem: $f(x)$ is a polynomial with complex coefficients, such that $-1 \leq f(x) \leq 1$ iff $-1 \leq x \leq 1$. Find all such $f(x)$. My observations: Now, its easy to see that coefficients are ...
3
votes
3answers
145 views

Solving the second taylor polynomial

So I've found myself in a predicament when trying to implement the second Taylor polynomial. Here is my question: Let $f(x) = \sqrt{x}$, find the second Taylor polynomial $P_2(x)$ for this ...
1
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1answer
22 views

A matrix couples two vectors, outputting a degree-two polynomial. How to transform that matrix so it acts on a projection of those vectors?

I have an expression of the form \begin{eqnarray} \left[ \begin{array}{c} x_{1}, x_{2}, x_{3} \end{array} \right] \left[ \begin{array}{ccc} M_{11} & M_{12} & M_{13} \\ M_{21} & ...
1
vote
1answer
46 views

Algebra question regarding polynoms [on hold]

Are there cases in which a polynomial cannot be written in a polynomial split? So can any P(x) be written in the form $P(x) = (-1)^n(x-a_1)^{k_1}...(x-a_p)^{k_p}$?
0
votes
1answer
71 views

When I know $a+b+c, a^2+a^2+b^2, a^3+b^3+c^3$, then how can I find the $a$ and $b$ and $ c$ [on hold]

When I know $$a+b+c = A$$ $$a^2+a^2+b^2 = B $$ $$a^3+b^3+c^3 = C$$ Then how can I find the $a$ and $b$ and $c$?
4
votes
3answers
104 views

Is this an equivalent statement to the Fundamental Theorem of Algebra?

Is the following equivalent to the usual statement of the fundamental theorem of algebra: Let $$f(z)=c_nz^n+\cdots+c_1z+c_0$$ be a polynomial with complex coefficients. For all but finitely many ...
0
votes
2answers
38 views

How to factor a third degree polynomial once you know one root?

Suppose $p(x) = 9x^3 - 30x^{2} + 29x - 8 $. If we wish to solve $p(x) = 0$, then we can observe that $x=1$ is a root of $p$. Then, we can write $(x-1)( \dots ) = 0$. How does one find the expression ...