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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Lower bound of a polynomial

Prove the following statement: Let $1<\beta<\sqrt{2}$ be a rational number, for any non-zero vector $(a_{n-1}\,,a_{n-2}\cdots\,,a_1\,,a_0)$(where $a_i\in\{-1\,,0\,,1\}$) and any $n\geq 3$, we ...
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A question about digital sum of polynomials over $\mathbb Z^+$

Given a polynomial with positive integer coefficients , let $a_n$ be the sum of the digits in the decimal representation of $f(n)$ , $n∈\mathbb Z^+$ , then is it true that there is a number which ...
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75 views

square system of polynomial equations having infinite number of solutions

Suppose we have a system of $n$ polynomial equations in $n$ unknowns over $\mathbb{C}$ and suppose that the corresponding ideal generated by these equations is not the unit ideal $(1)$. Under what ...
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Calculating a polynomial

$$ \sum_{x=1}^{2010 {1\over 2}} (4x^3 + x^2 + 2x + 1)^{[7]} = ? $$ where $f(x)^{[n]}= f(x)f(x-1)\cdots f(x-n+1)$ 2010 $1\over2$ = ${4021 \over 2}$ I couldn't compute this summation I would be ...
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Writing a polynomial as a linear combination of other polynomials

I'm currently working on writing $3(x)_4 - 12(x)_3 + 4(x)_1 - 17$ as a linear combination of $(x)_4,\ldots,(x)_0$ and am having difficulty understanding where the conversion comes from. I have the ...
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Prove/disprove: if $d \mid f$ and $d \nmid g$ then we can not know if $d\mid (f+g)$ or $d \nmid (f+g)$

Given three polynomials $f,g,d \in \mathbb F[x]$, we need to prove or disprove the following assumption: if $d \mid f$ and $d \nmid g$ so we can not say for sure if $d \mid (f+g)$ or $d \nmid ...
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9answers
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Prove that $x-1$ divides $x^n-1$

In algebra & polynomials, how do we prove that $$x-1 \mid x^n -1?$$
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Small question about a proof of Hilbert's Basis Theorem

I am currently going going through the proof of Hilbert's Basis Theorem: http://www.maths.usyd.edu.au/u/de/AGR/CommutativeAlgebra/pp806-850.pdf (it starts on slide 832) On slide 836-837 he makes the ...
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0answers
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Give a generating set for $Syz(h_1,h_2,h_3,h_4)$

I have a question which asks for a generating set of Syz($h_1,h_2,h_3,h_4$) where $h_1=x^2y+z^2$ $h_2=zy^2+yx^3$ $h_3=xz-y^2$ $h_4=y^4+yx^4$ I know that it is formed by... ...
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2answers
245 views

Holomorphic functions with polynomial real part

$f:\mathbb{C}\rightarrow \mathbb{C}$, $f(x+iy)=u(x,y)+iv(x,y)$ is a holomorphic function, its real part $u$ is a harmonic polynomial, i.e. $u\in \mathbb{R}[x,y]$ and $\frac{\partial^2 u}{\partial ...
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2answers
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Find a solution of the polynomial

Given $1<\beta<2$ and positive integer $n\geq2013$, then can we find a non-zero vector $(a_n\,,a_{n-1}\cdots\,a_1\,a_0 )$ where all $a_i\in\{-1\,,0\,,1\}$, such that ...
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1answer
51 views

$f(x)$ can't be factorize in $p(x)q(x)$ where where p and q are of degree $\le 3 $

Let $f(x)=x^4+26x^3+52x^2+78x+ 1989$ $f(x)$ can't be factorize in $p(x)q(x)$ where where $p\ and\ q$ are of degree $\le 3 $
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roots of the polynomial equations and relation among the coefficients

If the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$ ($a,b,c$ are real numbers) has no real roots and if at least one root is of modulus one, then what is the relation between $a,b$ and $c$?
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3answers
990 views

Show that this quadratic factor is positive for all real value of x

Let $p(x) = 4x^3 - 4x^2 + 5x + 4$, how can I show that the quadratic factor of $p(x)$ is positive for all real value of x ? I already found the factor of p(x) and it is $(2x+1)$ and $2x^2-3x+4$, but ...
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2answers
73 views

Quadratic equation and Proof [duplicate]

For rational numbers $a$ and $b$, the quadratic equation $x^2 - ax - b = 0$ has two solutions according to my professor. How can I Prove that if one of these is solutions is rational, the other must ...
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0answers
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Galois group of special polynomials

I checked the Galois groups of the polynomials $f(m,n) := mx^{(n-m)}+(m+1)x^{(n-m-1)}+...+(n-1)x+n$ for $0 < m < n$, and I only found one polynomial whose galois group is NOT the symmetric ...
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4answers
160 views

Irreducibility of a polynomial over rationals.

I am given the polynomial $x^4+1$ and I am asked to prove that it is irreducible in $\mathbb Q[x]$. I was just wondering if it is enough to show that $x^4+1$ does not contain a root in $\mathbb Q$ and ...
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2answers
165 views

Roots of cubic polynomial lying inside the circle

Show that all roots of $a+bz+cz^2+z^3=0$ lie inside the circle $|z|=max{\{1,|a|+|b|+|c| \}}$ Now this problem is given in Beardon's Algebra and Geometry third chapter on complex numbers. What might ...
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1answer
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Factorization of polynomial over arbitrary commutative ring

I have a somewhat silly sounding question: let $R$ be an arbitrary commutative ring with $1$. Let $f \in R[X]$. Do a) $f(1) = 0 \Rightarrow \exists g \in R[X]: f= (X-1)g$ b) for some $a \in R$: ...
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1answer
83 views

Factorization of the trinomial $x^{2n}+Dx^n+1$?

The following trinomials will factor for any $a$, $$1+a(-3+a^2)x^3+x^6 = (1+ax+x^2)(1-ax-x^2+a^2x^2-ax^3+x^4)\tag{1}$$ and similarly for, $$1+a(5-5a^2+a^4)x^5+x^{10}\tag{2}$$ ...
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$\{a^{k_1}\}=\{a^{k_2}\}=\{a^{k_3}\}$

Let $a\in\mathbb{R}\setminus\mathbb{Z}$. Prove or disprove that there do not exist three distinct $k_1, k_2, k_3\in\mathbb{N}$ such that $\{a^{k_1}\}=\{a^{k_2}\}=\{a^{k_3}\}\neq 0$, where ...
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203 views

Gröbner basis and generating set

I have come across the following past exam question... Define an ideal $J:=(z^2x+y^2-2y,x^3+y^3+z^3,x^2+2z^2) \subseteq \mathbb{Q}[x,y,z]. $ Compute a generating set for $J \cap \mathbb{Q}[y]$. ...
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2answers
252 views

How to find the number of real roots of a polynomial

no of real roots of the equation $(97-x)^{1/4} + x^{1/4}=5$ The options for the amount of real amounts-$1$ real root, $2$ real root, $3$ real root, $4$ real root. I got the answer as $2$ real roots ...
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2answers
506 views

Complex roots of polynomial equations with real coefficients

Consider the polynomial $x^5 +ax^4 +bx^3 +cx^2 +dx+4$ where $a, b, c, d$ are real numbers. If $(1 + 2i)$ and $(3 - 2i)$ are two roots of this polynomial then what is the value of $a$ ?
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1answer
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If $\mathbb F$ is algebrically closed then any polynomial in $\mathbb F[x]$ is a product of linear factors

I have came across this question in my homework and I have no clue what to do or how to start, I just can not connect the dots. If anyone can give me a direction, I'll be very thankful. Given an ...
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2answers
109 views

Prove that $x^5 - 5x + 1$ has no double roots in $\mathbb{C}[x]$

I am asked this question: Prove that $x^5 - 5x + 1$ has no double roots in $\mathbb C[x]$. Now here's what I said: $p(x) \in \mathbb C[x]$ has no double roots if and only if $gcd(p,p') = ...
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709 views

Number of distinct real roots [duplicate]

The equation $x^6 − 5x^4 + 16x^2 − 72x+ 9 = 0$ has (A) exactly two distinct real roots (B) exactly three distinct real roots (C) exactly four distinct real roots (D) six distinct real roots. ...
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3answers
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The way to prove that a polynomial is irreducible

Given: $x^3 + 2 \in \mathbb F_7[x]$ we have to prove that it is irreducible. Now, in finite fields, to prove that a polynomial is irreducible, is it enough to show that it has no roots within ...
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Roots and Coefficients

The polynomial $P(x)$ has integer coefficients $p+\sqrt{q}$ is a root of the polynomial, where $p, q$ are rational Prove that $p-\sqrt{q}$ must also be a root of the polynomial. (Assume that q is ...
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2answers
558 views

Zero-dimensional ideals in polynomial rings

I have the following past exam paper question, a similar sort of question seems to come up every year. And I'm completely lost with it... Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated ...
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1answer
95 views

Polynomials maps on $\phi \mathbb{A}^1_k \rightarrow \mathbb{A}^2_k$

I have the following question which seems to come up every year in exam papers, but just different numbers...Im really stuck on some parts... Define $\phi: \mathbb{A}^1_k\rightarrow \mathbb{A}^2_k$ ...
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Analogue of the Schwartz–Zippel lemma for subspaces

Let $f : \mathbb{R}^n \to \mathbb{R}$ be a nonzero multivariate polynomial of total degree $d$ over the reals, and $S \subset \mathbb{R}$ be finite. Pick a positive integer $k$, choose $y_1, \ldots, ...
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4answers
162 views

Interception with $x$-axis - not so trivial?

I want to find the interception with the x-axis of the following function: $f(x) = \frac{1}{4}x^4-x^3+2x$. So putting $0 = \frac{1}{4}x^4-x^3+2x$ I would get $0 = x(\frac{1}{4}x^3-x^2+2)$ but how to ...
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1answer
226 views

General proof that a product of nonzero homogeneous polynomials is nonzero (under certain conditions).

Background, Notation, Definitions: Given a set $X$, I define the set $M(X)$ of monomials with $X$-indeterminates to be the set of elements of $\omega^X$ having finite support. Given $m_0,m_1\in M(X)$, ...
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2answers
99 views

express Pochhammer symbol $(x)_n$ as a polynomial of order $n$ in $x$

Define $$(x)_{n}=x(x-1)(x-2)...(x-n+1)=\prod_{k=1}^{n} (x-k+1)=\sum_{k=0}^n a_k x^k$$ Q: what is the closed-form expression for $a_k$ ?
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1answer
168 views

Factoring a polynomial over real numbers (no real roots)

Polynomial is: $$4x^4+2x+\frac{15}{16}$$ I know that the degree of the highest irreducible polynomial over reals is 2, so it should be possible to factor this polynomial into two second degree ...
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1answer
63 views

Intuitive bernoulli numbers

Can somebody explain me or give me a link with a intuitive point of view of Bernoulli numbers? I mean, somebody just saw a typical sequence of numbers that appears in some taylor expansions, and them ...
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1answer
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Rouché's Theorem on $z^{10} + 10z + 9$

Please note: this question was asked before, but the solution provided does not work as far as I know; see How to find the number of roots using Rouche theorem? We have $f(z) = z^{10} + 10z + 9$ and ...
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0answers
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Unifromly discrete bound

Given $1<\beta_1<\beta_2<2$, and $k\in\mathbb{N^+}, $ $k\geq2 $ define $$ D_k=\{\sum_{j=0}^{k-1}A^jd_{i_j}:d_{i_j}\in\{(0,0)\,,(1,1)\}\} $$ where$ ...
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Simultaneous solution(s) to $a^2+4b^2+4ab=0$ and $a^2+4b^2+32+16a-8b=0$?

Could you tell me just how should I solve this system: $$ a^2+4b^2+4ab=0\\ a^2+4b^2+32+16a-8b=0 $$ I can't remember the proceeding and it's driving me crazy. Thanks a lot
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Given $ 0 \neq f \in \mathbb C[x]$ prove that it has no double roots if and only if $gcd(f,f') = 1$

Given $0 \neq f \in \mathbb C[x]$ prove that $f$ has no double roots if and only if $gcd(f,f')=1$ How do we approach this question?
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673 views

Number of coefficients of multivariable polynomial

Let $g \in \mathbb{F}[x_1, \dots, x_n]$ be a polynomial of degree $d$ with $n$ variables. Number of its coefficients is ${n+d \choose d}$ Is there an easy proof? It clearly holds for univariate ...
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0answers
105 views

How to solve an equation in three variables fixing two of the variables?

Also, I have the following equation, I want to solve it for $b$ keeping $a$ and $c$ fixed. $5b^5+(60-5a)b^4+(125+50c-80a)b^3+(594c-445a-775)b^2+(2324c-1005a-3270)b+3000c-750a-3000=0.$ Also how to ...
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1answer
45 views

Polynomials in the complex ring of 2 variables

Given $I = \left<x^2+y^2-1, x^2-y+1, xy-1\right>$ show that this generates $\mathbb{C}[x,y]$. I have tried pages and pages of writing a linear combination of these such that the combination is ...
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2answers
180 views

Idempotents in a Quotient Ring

Let $R=\mathbb{Z}_p[x]/(x^p-x)$. Show that $R$ has exactly $2^p$ elements satisfying $r^2=r$. I know that for $f,g\in\mathbb{Z}_p[x]$, we have $f-g\in(x^p-x)$ if and only if $f(a)=g(a)$ for all ...
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1answer
88 views

Reference request: Newton-Kantorovich hypothesis for polynomials of integral coefficients

Kantorovich's theorem states that the Newton method for finding the roots of a nonlinear function is guaranteed to converge if a parameter $h$, determined by the values of the function and its ...
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0answers
56 views

robust computation of Groebner basis

I am trying to solve numerically polynomial systems of equations, over the reals. I am coming across the following phenomenon: let's say that i have a system of 7 equations with 7 unknowns. I am using ...
3
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1answer
59 views

Computational Complexity of Algorithms

I want to know if the following proposition is correct or not? For any integer k, there exists an problem P for which, the minimum possible time complexity of any solution algorithm is ...
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3answers
384 views

Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$

For odd $n$ the Fermat equation $x^n + y^n = z^n$ factorises as $$(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n,$$ where $\zeta = e^{2 \pi i/n}$. I tried seeing this was true by multiplying ...
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1answer
71 views

Finding the value of polynomial at a particular value / constant is given

If $P(k)$ is a polynomial of degree $8$ and $P(k) = \frac{1}{k}$ for $k = 1,2 ,3,\ldots,9$ then find the value of $P(10)$. As we know the following : $f(x) =a_nx^n+a_{n-1}x^{n-1}+.....a_1x+a_0$ ...