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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Polynomial with arithmetic values

Can I find a polynomial in a second degree in two variables from the values of which can be found an infinite arithmetic progression? Thank you!
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28 views

Eigenvectors of the companion matrix

Suppose one has an Hermitian square matrix $A$ with $p$ is the characteristic polynomial $$ p(x)= a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + x^n ~, $$ and define the companion matrix of $p$ as $$ ...
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1answer
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what is f(x) < 0 asking for?

I'm trying to answer a question that says, State where $f(x)<0$ using any correct notation and I do not know what it is asking for. The question provides me a graph going from quadrant 2 to 4, and ...
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0answers
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Zero locus of 2-variate real polynomial are smooth curves

This seems like it should be an easy question, and probably already has already had answer in advanced mathematics, but I only know some basic calculus, so I would like to know how do I go about doing ...
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2answers
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$\frac{y-b}{r}=\frac{y}{s}$ to $y$ for finding the closest point on a line, from a point.

$$r=sy^2-sby$$ How do I get $y$ on one side? Originally I had: $\dfrac{y-b}{r}=\dfrac{y}{s}$
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2answers
33 views

Proof that the Runge Phenomenon occurs

Is there such a proof that states that the Runge Phenomena will always occur when interpolating with higher order polynomials or is this just observed empirically?
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1answer
42 views

For which elements $t$ in a finite field $\mathbb{F}_{p^n}$ is $t^2 - 4$ a square?

That is, how to characterize the elements $t \in \mathbb{F}_{p^n}$ for which there exists $x \in \mathbb{F}_{p^n}$ such that $t^2 - 4 = x^2$?
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1answer
41 views

Show that $(t^m-1)/(t^n-1)$ is a square if and only if $(\exists s \in \mathbb{Z})\ m=np^s$

I want to show the following lemma: Assume that the characteristic of the field $F$ is $p$ and $p>2$. Then $(t^m-1)/(t^n-1)$ is a square in $F[t, t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in ...
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Polynomial Interpolation and Data Integrity

This question is about polynomial interpolation and security. Please consider a scenario where we have a polynomial $f$, one of whose roots is $a$. We evaluate it at some ...
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46 views

Determining polynomial values

The polynomial $R(x)= x^4+Ax^3+Bx^2+10-1$ has a remainder of $-15$ when divided by $x+1$ and a remainder of $39$ when divided by $x-2$. Determine $A$ and $B$. Please help...
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1answer
82 views

Equality regarding Bernstein polynomials

The Bernstein polynomials are defined like this: $b_k(m,x)= {{m}\choose{k}} x^k(1-x)^{m-k}$, if $k<m$ I want to prove that $\sum\limits_{j=k}^m b_j(m,x) = m {{m-1}\choose{k-1} }\int\limits_{0}^x ...
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How would one define polynomials over the projective line $P_K^1$

May $K$ be a field. If I set $\varXi=(X:Y)$ as a "projective variable" and "projective coefficients" $a_k=(x_k:y_k)\in P_K^1$ - may I then write a polynomial map $P_K^1\longrightarrow P_K^1$ in a form ...
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1answer
57 views

Functional equation: Finding $f(100)$

A polynomial of degree 98 such $f (k)=1/k$ for $k=1,2,3...,98,99$ exists. How to find $f(100)$? What are the possible methods ?
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1answer
34 views

Decomposition of a homogeneous polynomial

Let $k$ be a field. Suppose I have a homogeneous polynomial $f$ in $k[x,y,z]$. If $f$ is reducible, does it always decompose as a product of homogeneous polynomials? Thanks!
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1answer
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Does every non trivial variety in $\mathbb{R}^n$ have empty interior?

By this question, we know that a non-trivial affine variety in $\mathbb{C}^n$ has empty interior. But the argument uses the (strong) fact that a holomorphic function vanishing in a non empty set $U$ ...
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1answer
64 views

Factoring $x^{15}−1$ into irreducible polynomials over $\mathrm{GF}(2)$

Factorize $x^{15}−1$ into irreducible polynomials over $\mathrm{GF}(2)$ The answer is $$(x+1)(x^2+x+1)(x^4+x+1)(x^4+x^3+1)(x^4+x^3+x^2+x+1)$$ but how would I find this? Please help.
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How does determining the area of rectangle relate to binomial multiplication?

So using the strategy to determine the area of the large rectangle I simply did $10\times10, 10\times2, 10\times4, 2\times4$ to get $168\mathrm{cm}$ total. The next question goes on to ask how ...
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1answer
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Number of polynomial factors of $a^n-b^n$?

This is a number theoretical problem that I discovered myself. Let $f(n)$ be the number of factors of $a^n-b^n$ with integer coefficients when its completely factored. For example: $f(1)=1$, because ...
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2answers
29 views

finding factors of a polynomial

In the math problem in the attached image, it explains how to find the factors of a polynomial whereby every possible factor of the function is of the form p/q, where "p is a factor of the constant ...
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37 views

Splitting even degree polynomials

I have an octic equation (degree $8$) and a sextic equation (degree $6$) in $\Bbb Z[x]$ with very large coefficients (size several hundred bits) that I know splits into two quartics and two cubics ...
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2answers
66 views

Find all the possible real values for $a,b,c,d$.

Let pairs $(a,c)$ and $(b,d)$ be roots of the equations $x^2 + ax - b = 0$ and $x^2 + cx + d = 0$ respectively. Find all possible real values for $a,b,c,d$.
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1answer
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How to split a quartic into two quadratics?

I have a quartic in $\Bbb Z[x]$ with very large coefficients that I know splits into two quadratics in $\Bbb Z[x]$. What is the best way to do find the quadratics?
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1answer
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Inverting power sum of symmetric polynomial

Suppose I have a set of power sum symmetric polynomial as $$S_p =\sum_i^N x^p_i ~~;~~~~~~~~p=\{1,N\}$$ and I have N of them $\{S_1...S_N\}$ Question is given this, can we find ${x_n=F(\{S_p\})}$? ...
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1answer
99 views

Solve $x^3+x+3=0$ by Galois's theory

Solve with radicals the following equation $x^3+x+3=0$, using Galois Theory. I'm just starting learning this and I do not have many ideas.
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1answer
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polynomial factorization when exponent is not given

How can I factorise this equation, given i already know some of its factors which are: $(a-b)(b-c)(c-a).$ Equation is : $$a^nb^{n-1} + a^{n-1}c^n - a^nc^{n-1} - a^{n-1}b^n - b^{n-1}c^n + b^nc^{n-1}$$ ...
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Matrix polynomial [on hold]

Suppose: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is a ...
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1answer
44 views

Lowest root of a quintic equation with 5 positive roots

I have a quintic equation $$ x^5-a_4 x^4+a_3 x^3-a_2 x^2+a_1 x - a_0=0 $$ with $a_n>0$ real coefficients, and I know that all 5 roots are real and positive (it is a characteristic polynomial). ...
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2answers
60 views

Finding roots of $2x^3-5x^2+18x+45$

solve $2x^3-5x^2+18x+45$ not exactly sure where to start on finding the zeros complex or real. There is one real zero and two complex I know that from graphing just cannot do it on paper to understand ...
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Maple not able to calculate Bernstein polynomial

Hope you can help me on this one. Please look at this simple Maple code: Obviously $B(1)=g(1)=4 \neq 0$. Why is Maple not able to compute this right? Am I doing something wrong? Kind regards PS: ...
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How to derive the Taylor expansion form of a polynomial expression?

I want to change this polynomial into the form $\sum_{k=0}^m a_k x^k$ $$q_m(x)=\sum_{k=0}^m(-1)^k\binom{2m+1}{2k+1}x^k(1-x)^{m-k}$$ I see no way to do this as I fear one might need intricate binomial ...
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1answer
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Quartic equation or Sextic equation? And how to solve it?

In this arxiv paper (p. 11, eq. (3.2)) the authors claim that equation (3.2) is ... a quartic equation [...] which can be solved explicitly. The equation in question is \begin{equation} ...
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Describing the graph of a function

For my Algebra II class one of the questions was: Describe the graph of the function $f(x) = x^3 - 18x^2 + 107x-210$. Include the $y$-intercept, $x$-intercepts, and the shape of the graph. And my ...
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1answer
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Prove the theorem of ideal (about g.c.d)

If $p_1,\ldots,p_n$ are polynomials over a field $F$, not all of which are $0$, there is a unique monic polynomial $d$ in $F[x]$ such that (a) $d$ is in the ideal generated by $p_1, \ldots, ...
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1answer
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Find $r$ in the next formula

Lets suppose I have the next values $$0, 7, 8, 5, 6$$ And the next formula $$4250 = \frac{0}{(1+r)} + \frac{7}{(1+r)^2} + \frac{8}{(1+r)^3} + \frac{5}{(1+r)^4} + \frac{6}{(1+r)^5}.$$ What is the ...
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1answer
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How would you divide a polynomial by another polynomial whose power is greater than its nominator? [on hold]

I have a polynomial which is: $$\frac{(x^3-4x)}{(4x^2-4x+1)} = -10$$ Is there a way to do this? I have thought about doing long division which was not helpful...
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1answer
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Functions that are “balanced” on the support of a permutation

Let $F = GF(2^n)$. Let $P(x), Q(x) \in F[x]$ be such that $P(x)$ is a permutation, while $Q(x)$ is not a permutation. For $\lambda \in F^*$ define the function $g_\lambda(x) = Tr(\lambda Q(x))$. Let ...
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Runge Phenomena and Taylor Expansion

From The Weierstrass Approximation Theorem Vs The Runge's Phenomenon: We contrast this to polynomial interpolation: this is a specific method for generating a sequence of polynomials that ...
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1answer
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Notation for polynomials and equating coefficients

I am reading a paper that defines $P_k(s|t)$ as a polynomial of degree $k$ in $s$ given $t$. Does this mean that each term is of the form $f_{k}(t)s^{k}$? (What does "given $t$" mean?) The paper ...
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1answer
34 views

Understanding a simple proof about minimal polynomials

Let $T \colon V\to V $ be a linear operator, where $V$ is a vector space over $F$. Suppose that the minimal polynomial $M(t)$ of $T$ can be factored into the product of two coprime and monic ...
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Minima and maxima of the 6th degree polynomial are not expressible in radicals.

Question: Prove that there exists a polynomial $P$ with $\deg P \geq 6$ such that the minima and maxima are not expressible in radicals. I have the following proof: the minima and maxima of a 6th ...
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82 views

Prove an equality

If $a+b+c=0$ prove that $\frac {(a^4 +b^4 +c^4)}{2}=\frac {(a^2+b^2+c^2)}{2^2}^2$ I have expanded the right side and have got this far: $a^4+b^4+c^4+2(a^2b^2+a^2c^2+b^2c^2)$ I need $a^2=b^2=c^2$ ...
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Proving that $t^{p^r}-a$ is irreducible when $a\in k$ is not a $p$th power

Let $p$ be an odd prime, $F$ a field of characteristic $0$ and $a\in F$ with $a\neq 0$. Assume $a$ is not a $p$th power in $F$. Prove that for every positive integer $r$, $t^{p^r}-a$ is irreducible ...
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1answer
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Roots of cubic equation

If$\frac{1+\alpha}{1-\alpha},\frac{1+\beta}{1-\beta},\frac{1+\gamma}{1-\gamma}$ are the roots of the cubic equation $f(x)=0$ where $\alpha,\beta,\gamma$ are the real roots of the cubic equation ...
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To construct a power series such that the radius of convergence of the power series $\sum_{n=0}^{\infty} a_n b_n x^n$ is $2R$.

Let $\sum_{n=0}^{\infty} a_n x^n$ is a power series with radius of convergence $R(>0)$. To construct a power series $\sum_{n=0}^{\infty} b_n x^n$, other than $\sum_{n=0}^{\infty} (\frac x2)^n$, ...
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1answer
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Smooth Fano Polytopes and Hypersurfaces

This is a rather extended question, so I will try to make it as compact and readable as possible. I am trying to practice with the Macaulay2 software, in particular the polyhedra and ...
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1answer
50 views

Polynomial root finding: Bernstein vs Budan

Budan's and Vincent's theorems can be used to isolate the real roots of a real polynomial. I have read papers which compared it favorably to other root finding methods. However, roots can also be ...
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1answer
48 views

False positives with Descartes rule of signs

Descartes rule of sign can be used to isolate the intervals containing the real roots of a real polynomial. The rule bounds the number of roots from above, that is, it is exact only for intervals ...
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1answer
40 views

Polynomial GCD in the presence of floating-point errors

The crucial requirement for using root isolation methods based on Vincent's theorem is that the input polynomial does not have multiple zeros. One way to remove the multiple zeros is to use polynomial ...
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4answers
95 views

coefficient of $x^{17}$ in the expansion of $(1+x^5+x^7)^{20}$

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. find the coefficient of $x^{17}$ in the expansion of ...
3
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1answer
44 views

Proof of Cohn's Irreducibility Criterion

I was looking for an elementary (or involving introductory level abstract algebra/analysis) proof of Cohn's Irreduciblity Criterion: If $$ a_0, a_1, \dots, a_n \in \Bbb{Z} $$ and $$ 0 \le ...