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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1answer
287 views

How to check that given polynomials form a Groebner basis

I am wondering if some polynomials are given, how do we know whether they form Groebner basis or not. Note that it is not necessary that given poly's form a reduced Groebner basis. I know how to find ...
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110 views

When is an intersection of varieties finite

Consider the general Bezout's theorem: If $p_1 \ldots p_n$ are polynomials with degrees $d_1,\ldots, d_n$ in $\mathbb{R}[x_1,\ldots,x_n]$, with $V = \{a=(a_1,\ldots, a_n) | p_i(a) = 0, \forall i\}$ ...
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1answer
127 views

Determining power series for $\frac{3x^{2}-4x+9}{(x-1)^2(x+3)}$

I'm looking for the power series for $f(x)=\frac{3x^{2}-4x+9}{(x-1)^2(x+3)}$ My approach: the given function is a combination of two problems. first i made some transformations, so the function looks ...
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1answer
185 views

Polynomial $P(a)=b,P(b)=c,P(c)=a$

Let $a,b,c$ be $3$ distinct integers, and let $P$ be a polynomial with integer coefficients.Show that in this case the conditions $$P(a)=b,P(b)=c,P(c)=a$$ cannot be satisfied simultaneously. Any hint ...
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51 views

Vandermonde matrix and polynomials

Question attached as image, deals with polynomials of order N and determinant of Vanderbilt matrix.
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1answer
98 views

Solve a cubic polynomial?

I've been having trouble with this question: Solve the equation, $$5x^3 - 24x^2 + 9x + 54 = 0$$ given that two of its roots are equal. I've tried methods such as Vieta's formula and simultaneous ...
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2answers
109 views

Prove for $p(x) \in \mathbb R_n[x]$ there exist unique coefficients

For all $n\in\mathbb N$ define $$\mathbb R_n[x] =\{p(x) \in\mathbb R[x]: \deg(p(x)) \leq n\}.$$ Let $n\in\mathbb N$. Suppose polynomials $p_0(x),p_1(x),\dots,p_n(x)\in\mathbb R_n[x]$ have degrees ...
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1answer
93 views

Irreducible and separable polynomial

Let $f(x)$ be an irreducible polynomial over $F[X]$, with $char(F)=p$ prime number. We know that $ \exists t \in \mathbb{N} | f(x)=g((x^p)^t)$. We shall prove that $g(x)$ is irreducible and separable. ...
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1answer
30 views

Quadratic fit check

I've performed LS fit to data in order to fit the following quadratic function: $$f(x,y) = A~x^2 + B~y^2 + C~x~y + D~x+E~y +F$$ Now, I would like to check that the fitted function looks like a ...
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1answer
1k views

Finding the sum of the coefficients of polynomial of degree 21

Problem : Find the sum of the coefficients of the polynomial$ p(x) =(3x-2)^{17}(x+1)^4$ Solution : $ p(x) =(3x-2)^{17}(x+1)^4 $ $= (a_0+a_1x+....a_{17}x^{17})(b_0+b_1x+....b_4x^4)$ for some $a_i; ...
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1answer
1k views

Finding the value of f(6) when f(x) of degree 5 with leading coefficient

Problem : Suppose $f(x)$ is a polynomial of degree $5$, and with leading coefficient $2009$. If further that $f(1) =1; f(2)=3, f(3)=5, f(4)=7, f(5)=9$. What is the value of $f(6)?$ My work : Let ...
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1answer
65 views

Composite Polynomials

let $p > 2$ be a prime number. For each a that belongs to the field $\mathbb{F}_p$, we define the following polynomial: $(x^2 - a)$ over the polynomials vector space in $\mathbb{F}_p$ ($p$ ...
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1answer
143 views

Polynomials with integer coefficients, with value close to $0$, in the interval $[-1,1]$

Are there some interesting properties of polynomials with integer coefficients of degree $2^n$ which satisfy $\mid P(x) \mid \le \frac{1}{2^k}$ ? I know that their coefficients are bounded and the ...
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1answer
95 views

Solution of a polynomial in interval $(0,1)$

Let $\displaystyle a_0 + \frac{a_1}{2} + \frac{a_2}{3} + ... + \frac{a_n}{n+1} = 0$, where $a_i$'s are some real constants. How can we prove that the equation $a_0 + a_1x + a_2x^2 + ... +a_nx^n = 0$ ...
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3answers
417 views

Minimum degree of a polynomial passing through points

If $P(x)$ is a polynomial such that $P(a_{1})=b_{1}, P(a_{2})=b_{2}, \ldots , P(a_{k})=b_{k}$, how can I find the polynomial which has minimum degree and for whom the relations above are true?
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1answer
214 views

Galois Group of $f(x)=x^4 - 10 x^2 + 1$

I am trying to calculate the Galois group of $f(x)=x^4 - 10 x^2 + 1\in\mathbb Q[x]$ over $\mathbb Q$. In my notes it says that the four roots are $\pm\sqrt 2\pm\sqrt 3$. So the splitting field of ...
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0answers
27 views

How to find $r(t)$ such that $r(t_i)=0$ for given $t_i$.

We have $t_i=\dfrac{2v_0^2\sin\theta\cos(\theta+\phi)}{g \cos^2\phi}$. Is there a way to find the function $r(t)$ such that $r(t_i)=0$? Were $\phi, \theta$ are constant and $r(t) \neq 0$ for $t \neq ...
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1answer
105 views

A zero for a homogeneous polynomial is a zero for the associated inhomogeneous polynomial

I am trying to prove a simple statement from Reid, Undergraduate Algebraic Geometry, pg 16. Let $F(U,V)$ be a nonzero homogeneous polynomial of degree $d$: ...
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3answers
108 views

Deduce a coefficient of a cubic polynomial.

So, I have this question which is still troubling me: Find the value of $k$ such that the equation $2x^3 + 3x^2 + kx - 48 = 0$ has two solutions equal in value but opposite in sign. I've had ...
2
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2answers
121 views

Prove the following:

$$\sum_{k=1}^{\infty }\frac{1}{(2k-1)^{2}}=\frac{\pi ^{2}}{8}$$ I don't really know how to prove this, will assuming that $$cos(x)=\sum_{k=0}^{\infty }(-1)^{k}\frac{x^{2k}}{(2k)!}$$ help?
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“Convex” polynomials

Let me define "convex" polynomials, as the smallest class $\mathcal{C}$ of functions $p:\mathbb{R}\rightarrow \mathbb{R}$ defined (inductively) as: UPDATED (case 0 was missing): 0) $p(x)=x$, i.e., ...
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1answer
97 views

Solving a polynomial

I've come across this page about partial passwords, and I am having difficulty trying to understand the polynomial equations in there. I can understand basic algebra but this page's explanation is ...
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1answer
53 views

What's the difference of naming a polynomial ring as $\mathbb{C}\{ x,y\}$ and $\mathbb{C} [x,y]$?

I sometimes see both notations and I am led (maybe misled) to believe that they are the same thing. What is the formal difference between both of them? Or there isn't any?
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2answers
51 views

Pointwise order on polynomials

I'm curious about the following problem. Given two polynomials $p,q:\mathbb{R}\rightarrow \mathbb{R}$ is it possible to determine automatically if $p(x)\leq q(x)$ for all $x\in\mathbb{R}$? I assume, ...
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1answer
788 views

Counting roots of a multivariate polynomial over a finite field

How many roots can there be of a polynomial $f \in K[x_1, x_2, \dots , x_n]$ where $K$ is a finite field and the maximum exponent of $x_i$ in any term is $m$ for all $i$, assuming not all coefficients ...
2
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1answer
86 views

Related To Polynomial Division

How to prove the following result Show how a polynomial with odd number of term will never be divisible by a divisor with $x+1$ as factor for modulo $2$ arithmetic. I don't have any idea.
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45 views

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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1answer
36 views

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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1answer
74 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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87 views

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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1answer
828 views

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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1answer
87 views

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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8answers
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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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47 views

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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27 views

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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1answer
50 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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1answer
166 views

Question about modular inverse

How could i prove this? Let $F$ be a finite field of characteristic $2$ and $g \in F[X]$ an irreducible polynomial. Splitting this polynomial in even and odd part we get $g(X)=g_0(X)^2+Xg_1(X)^2$. ...
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2answers
68 views

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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2answers
554 views

Verifying roots of radical function

Suppose you have a consistent radical function $R(x)$ that can be solved by finding the roots of an nth degree polynomial, assumed to be solvable using radicals. Some of the $n$ roots maybe ...
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138 views

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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2answers
70 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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64 views

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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1answer
82 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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135 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 ...
2
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2answers
157 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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4answers
689 views

How do you solve equations of any degree?

I have stuck solving this problem of financial mathematics, in this equation: $$\frac{(1+x)^{8}-1}{x}=11$$ I'm stuck in this eight grade equation: ...
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4answers
159 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 ...
2
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2answers
520 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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78 views

How to show an ideal is zero-dimensional? [duplicate]

Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated by $\{y^2-xy-2xz,y^3+z^2+1, x^2yz-yz\}$. Show that $J$ is zero-dimensional. How do I go about showing this?
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2answers
626 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. ...