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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How to obtain a specified set of coefficients of a multivariate polynomial

Let $x_i \in \mathbb{C}$ and $b_{ij} \in \mathbb{R}$ for all $i,j$ between $1$ and $n$. I have the following polynomial and the corresponding expansion (correct me if I am wrong): $\prod_{i=1}^n (1+ ...
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Find $\prod\limits=(\alpha_1+1)(\alpha_2+1)…(\alpha_n+1)$ where $\alpha_i$ are complex roots of a complex polynomial

The complex roots of a complex polynomial $P_n(z)=z^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0$ are $\alpha_i$, $i=1,2,...,n$. Calculate the product $(\alpha_1+1)(\alpha_2+1)\cdots(\alpha_n+1)$ By the ...
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Given that $f(x,y,x)$ is a factor of $g(x,y,z)$:

Suppose $f(x,y,z)=(x-z)^2+(x-y)^2+(z-y)^2$ and $g(x,y,z)=(x-z)^n+(x-y)^n+(z-y)^n$ Then prove that if $f(x,y,z)$ is a factor of $g(x,y,z)$, that $n$ is not divisible by 3. Please no solutions. I'm ...
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gcd of $x$ and $2$ in $Z[x]$

In $Z[x]$, $x$ and $2$ has gcd $1$. But they cannot be expressed as the linear combination of two polynomials. Then assuming that $1=2.f(x)+x.g(x)$ we are supposed to arrive at a ...
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Fastest way to perform this multiplication expansion?

Consider a product chain: $$(a_1 + x)(a_2 + x)(a_3 + x)\cdots(a_n + x)$$ Where $x$ is an unknown variable and all $a_i$ terms are known positive integers. Is there an efficient way to expand this?
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Polynomial with real roots

Consider the polynomial: $$f=X^4+4X^3+6X^2+aX+b$$ We know that $f$ has four real roots. Let $x_1,x_2,x_3,x_4$ be the roots of this polynomial. How can one compute ...
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1answer
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Why mentioning monic is important for gcd and lcm?

Let $F$ is a field and $F[x]$ be the polynomial ring over $F$. Now in the definition of the gcd or lcm of any two polynomials $g(x)$ and $f(x)$ it is mentioned that the ...
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24 views

Rotated parabola 2d vertex

I'm implementing an application where I need to get the vertex of a parabola, the parabola might be tilted; so it can have an angle with the x-axis not necessarily vertical or horizontal. Can I get ...
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Asymptotic running time for multiplying multivariate polynomials using Schönhage/Strassen

Question: I would like to ask the community where my following suggestion for an asymptotic bound for the running time of multiplying two multivariate polynomials using theorem $8.23 $ recursively ...
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1answer
57 views

Interpolating a Polynomial with a Subset of Interpolation Points

Consider we has a polynomial $P=(x-\beta)g(x)$, where $\beta \leftarrow \mathbb{Z}_p$, $p$ is a large prime, and $g(x)$ is a non-zero polynomial. Here degree of $P$ is fixed $n$. We evaluate $P$ at ...
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System with arbitrary function of an unknown

How can I solve the following system $$ (u_x)^2 - (u_t)^2 = 1 \\ u_{xx} - u_{tt} = f(u) $$ where $f$ is an arbitrary function of $u$, $u$ and $f$ to be determined. I don't know any approach, ...
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How to solve equation to the third power

I have the information that: $$ x^3 − x^2 −1 =0 $$ Has a "positive real root" of: $x \approx 1.4655\ldots$ My questions are, please: 1) What is a "positive real root". 2) How one gets from the ...
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How can $f(x,y)= x^4+x^3y+x^2y^2+xy^3+y^4$ be factorized into a product of two polynomials?

Let $x,y$ be 2 coprime integers. I assume the following polynomial:$$f(x,y)= x^4+x^3y+x^2y^2+xy^3+y^4$$ is not irreducible. So there must be at least 2 other polynomials of degree $\leq 4$ such that: ...
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How to find the value of $(a+b+c)(a+b+d)(a+c+d)(b+c+d)$ from the following equation?

I have a question about polynomial. Given a polynomial: $$x^4-7x^3+3x^2-21x+1=0$$ Given too that the roots of this polynomial are $a, b, c,$ and $d$. Find the value of ...
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A particular polynomial - 2

Is there a homogeoneous polynomial in $\Bbb Z[W,X,Y,X]$ that contains only coefficients from $W^4,X^4,Y^4,Z^4,W^2X^2,W^2Y^2,X^2Z^2,Y^2Z^2,WXYZ$ that factorizes into unequal quadratic forms? What is a ...
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53 views

A particular polynomial

Is there a homogeoneous polynomial in $\Bbb Z[W,X,Y,X]$ that contains only coefficients (which may be $0$) from $W^4,X^4,Y^4,Z^4,W^2X^2,W^2Y^2,X^2Z^2,Y^2Z^2,WXYZ$ that factorizes into unequal ...
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Composition of polynomial and multiplicative is multiplicative .

I made the following problem a while ago but I can't solve it (also I don't think it's extremely hard ) : Let $f$ be a non-constant completely multiplicative function over $\mathbb{Z}$ . Assume ...
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37 views

Why does $\frac{x^n}{n^x}$ stop growing at the approximate value of $\pi (n)$?

I noticed while playing around with these functions that $n^x$ will start slow and then speed up really fast in its growth rate. While $x^n$ grows more slowly, but faster than $n^x$ at the start. ...
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2answers
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Finding coefficients of $x^n$ and $x^{n+r}$ in an expansion

I have to find the coefficients of $x^n$ and $x^{n+r}$ $(1 < r < n)$ in the expansion of: $$(1 + x)^{2n} + x(1 + x)^{2n - 1} + x^2(1 + x)^{2n - 2} + ... + x^n(1 + x)^n$$ How do I solve it?
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Legendre symbol identity

I try to solve the following problems ($p$ is an odd prime) Find the sum $$\sum_{a=1}^{p-1}a \cdot \left (\frac{a}{p} \right)$$ Find the sum $$\sum_{a=1}^{p-1} 2^a \cdot \left (\frac{a}{p} \right)$$ ...
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Approach for optimization problem with polynomial constraints?

I have a problem where the objective function is linear and constraints have polynomials (in one variable). So, my question is what are the main approaches to this issue? I can construct a small ...
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How to factor $ s^2LC + sRC + 2$

or $$ s^2+s\frac{R}{L}+\frac{2}{LC}=0 $$ Is there any way? I can't find out. Thanks in advance.
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How do I solve an equation like this?

How do I solve following equation for $X$: $$ AX^n + BX^{n-1} + CX^{n-2} + \dotsb + YX + Z = 0, $$ where $A,B,C,\dotsc,Z,n$ are known?
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1answer
45 views

Sum of digits modulo a polynomial

I made the following problems a while ago but I can't solve them (though I don't think it's too hard) 1.Let $s(n)$ be the digits sum of $n$. Let also $f(n)$, $g(n)$ $\in Z[X]$ . Assume that: ...
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1answer
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Durand-Kerner with derivative in denominator

The correction term for Durand-Kerner root finding method is $w_k = -\frac{f(z_k)}{\prod_{j\not=k}(z_k - z_j)}$ Wikipedia Talk page mentions that it is also possible to use derivative in ...
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2answers
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Contest problem in functional equations.

Let n be a positive integer with $f(n)= 1! +2! +3!+... +n!$ and P(x), Q(x) be polynomials in $x$ such that $f(n+2)=P(n)f(n+1)+Q(n)f(n)$ for all $n \geq 1$, then which of the options is/are ...
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Conditions for a unique root of a fifth degree polynomial

Fifth degree polynomials cannot generally be solved analytically, but at least one solution always exists. Given the normal form $$ax^5+bx^4+cx^3+dx^2+ex+f=0,$$ is it possible to find sufficient ...
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Reference for a Dickson Determinant Polynomial

For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation} ...
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Questions from an olympiad on number theory [closed]

The sum of the infinite series: $$ \frac{1}{2} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \frac{8}{64} + \frac{13}{128} + \frac{21}{256} + \frac{34}{512} +....$$ I am able to find the general term ...
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Divisors of zero in polynomial ring

I have the following theorem: McCoy: Let $R$ be a commutative ring with identity. If $f=\sum_{i=0}^na_iX^i$ is a zero divisor in $R[X]$, then there exists a nonzero $c$ in $R$ such that $cf=0$. ...
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Proof for the coefficient of $x^n$ in $(x^0 + x^1 + \dots + x^n)^n$

Theorem: The coefficient of $x^n$ in $(x^0 + x^1 + \dots + x^n)^n$ is $\binom{2n-1}{n-1}$. How to prove this? Multinomial theorem produces the following $$ \left(\sum_{k=0}^{n} x^k \right)^n = ...
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Solving a system of polynomial equations

How can I solve a system of polynomial equations like this one Maybe I'm missing a very basic trick... Can anybody suggest me an approach?
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A question in perturbation of $P(\lambda )$

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
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Tricks for a Specific System of Polynomial Equations

I'm looking for all the complex solutions to the following 3 equations (and for this consider $a$ to be some given constant, so that there are really just 3 unknowns in solving): ...
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homogeneous polynomials over finite fields

Let $F$ be a finite field and $p(X_1,\dots,X_n)\neq 0$ an homogeneous polynomial with coefficients in $F$. Is it possible that $p(x_1,\dots,x_n)=0$ for every $(x_1,\dots, x_n)\in F^n$?
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Polynomial Interpolation and Security

Let polynomial $P$ be $P(x)=g(x).(x−β)$, where $g$ is a polynomial and $\beta \leftarrow \mathbb{F}_p$. We evaluate $P$ at some $\textbf{x}=(x_1,..,x_n)$. This gives us $\textbf{y}=(y_1,..,y_n)$. ...
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Exercise in arithmetic of a finite field

I am in difficult in resolving this exercise in Galois Theory : "in $GF(2^5)$ calculates the product $(1,1,1,0,1)(0,1,0,1,0)$ , generator of $GF(2^5)^*$ ". I don't know how to proceed.. thank you
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Product of a Finite Number of Matrices with a Cosine Entry

Does any one know how to prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} 2\cos\frac{2j\pi}{n} & a \\ b & 0 \end{pmatrix}\right)=2 $$ when $n$ is ...
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Express roots in polynomials of equation $x^3+x^2-2x-1=0$

If $\alpha$ is a root of equation $x^3+x^2-2x-1=0$, then find the other two roots in polynomials of $\alpha$, with rational coefficients. I've seen some other examples [1] that other roots were ...
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Polynomial Interpolating; When $y_i$'s are Changed

This is a comlpementry question to the one posted in: Polynomial Interpolation And polynomial Roots Given $\{(x_1,y_1),...,(x_n,y_n)\}$, we can interpolate a polynomial $P$. Assume polynomial $P$ has ...
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Is there a linear decomposition of the Hadamard inverse of the sum of two matrices?

Let the matrix $$\Gamma = \alpha A + (1-\alpha)B$$ where $B$ is a square symmetric matrix, $A = c\ ee'$, where $e$ is a vector of ones, and $c$ a positive constant and $0 < \alpha < 1$. The ...
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1answer
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How to estimate the error on the position of the point x where y is maximal in quadratic relationships?

I would like to estimate the elevation at which species richness is expected to be maximal. The relationships between species richness ($y$) and elevation ($x$) follows a second order polynomial ...
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Linear Algebra Change of Basis problem

So, $\mathbb{P}_2$ is the vector space of all polynomials with degree less than or equal to 2 and that $E=\{1,t,t^2\}$ is a basis for $\mathbb{P}_2$ We define $p_1(t)=1+2t$ $p_2(t)=t-t^2$ ...
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81 views

Polynomial Interpolation And polynomial Roots

Given $\{(x_1,y_1),...,(x_n,y_n)\}$, we can interpolate a polynomial $P$. Assume polynomial $P$ has some roots including an specific root $\beta$. Consider we change one of $y_i$ to $y'_i$. Given ...
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How to solve this polynomial problem

$(3x-1)^4=a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$ Value required to be found :- $a_4+3a_3+9a_2+27a_1+81a_0$ I can find the value of $a_4+a_3+a_2+a_1+a_0$.Then I don't know how to continue.Please help.
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Factorization of a polynomials in complex number.

Factorize this expression: $$a^2+b^2+c^2-ab-bc-ca.$$ The result is $$(a+b\Omega+c\Omega^2)(a+b\Omega^2+c\Omega)$$ How I can get $\Omega$ here?What's the approach?
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Construct bivariate symmetric (polynomial) nonnegative functions (distributions) over the unit square with certain properties

Construct bivariate symmetric polynomials $f(x,y) = f(y,x) \ge 0$ over $[0,1]^2$, with $f(1,y) = f(x,1)=0$, such that the univariate marginal distributions are both proportional to $$(1-u^2)^4$$, ...
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Polynomial must be monotone between its extrema

Suppose that the polynomial function $f(x)=x^n+a_{n-1}x^{n-1}+\cdots +a_0$ has $k_1$ local maximum points and $k_2$ local minimum points. Show that $k_2=k_1+1$ if $n$ is even, and $k_2=k_1$ if $n$ is ...
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33 views

Existence of a splitting ring

Let $R$ be a commutative ring and $f\in R[X]$ be a monic non-constant polynomial. How can one show that there exists a commutative ring $S$ so that $R$ is a subring of $S$ and $f$ can be written as a ...
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1answer
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Transform to flatten a parametric curve (polynomial)

Given a polynomial parametrized by $p(t)=(x(t),y(t))$ such that $y(t)=p(t)$, $x(t)=t$, and $p(t)= \sum_{i=0}^na_it^i$, for real coefficients $a_i$, is there some transformation I can take such that ...