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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$R$ has nonzero nilpotent elements…

Let $R$ be a commutative ring with 1. a) Suppose $R$ has no nonzero nilpotent elements (that is, $a^n=0$ implies $a=0$). If $f(X)=a_0+a_1X+\cdots+a_nX^n$ in $R\left[X\right]$ is a ...
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467 views

Number of integral roots of a polynomial

Let $p(x)$ be a polynomial with integral coefficients. Let $a$, $b$, $c$ be three distinct integers such that $p(a) = p(b) = p(c) = -1$. Find the number of integral roots of $p(x)$.
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101 views

Absolute value of cubic polynomial roots lower than 1

Assume we have a cubic polynomial $ x^3 +bx^2+xc+d=0 $, with $b,c,d$ real numbers. Let $x_1, x_2, x_3 $ be the roots, either real or complex. What is the relation of the coefficients $b,c$ and $d$ ...
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Show that $\alpha^2 + \alpha - 1$ is a zero divisor in $R$

Studying for my algebra exam and looking through old exam exercises I came across the following problem Let $f = X^4 + 1$, $g = X^2 + X - 1 \in \mathbb{F}_3[X]$ and $\alpha = X + \langle f \rangle ...
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556 views

In EMI calculations, how to calculate “Rate” if EMI, Principal and Time are given

In EMI (Equated Monthly Instalments) calculations, the inputs are- Principal-P, Rate-r, and ...
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About Jordan Form

For a $A\in M_n({\bf C})$ with a minimal polynomial $m(x) = (x-c)^n$ then we have a Jordan form wrt some basis $$ A=\left( \begin{array}{ccccc} ...
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56 views

Finding the coefficient of a variable in a polynomial

Find the coefficient of $x^8$ in the polynomial $(x-1)(x-2)\cdots\cdots(x-10)$. How do I approach such problems?
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How to show this polynomial is less than zero?

$$ f(y)=Ny(y-1)^2-y-y^{2N+3}+y^{N+1}\left (N^2y^3+(-2N^2+N+1)y^2+(N^2-2N)y+N+1 \right ) $$ where $N$ is an integer bigger or equal to 2, and $y>1$.How to show $f(y)<0$? Any hint? Thanks a lot. ...
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906 views

Hardest question IMO I had ever seen!

Let $\alpha$ and $\beta$ and $\gamma$ be 3 real numbers. Prove that there exist only one polynomial $P(x)$ of the second degree such that $$\begin{cases}P(1)=\alpha \\ P(2)=\beta \\ ...
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2answers
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If $f$ is entire and $\exp(f(z))$ is a polynomial, then $f$ is constant.

In a recent question that was just deleted, @danielfischer gave at the end of his answer the following exercise: for entire $f$, $$e^{f(z)} \text{ is a polynomial} \iff f \text{ is constant}$$ I was ...
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theory of equations finding roots from given polynomial

If the equation $x^4-4x^3+ax^2+bx+1=0$ has four positive roots then $a=\,?$ and $b=\,?$ $\textbf{A.}\,6,-4$ $\textbf{B.}\,-6,4$ $\textbf{C.}\,6,4$ $\textbf{D.}\,-6,-4$ we can ...
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1answer
50 views

If singular set is finite then the ideal is radical

Let $F\in K[X,Y]$ and if the zero set $V(F,\frac{\partial F} {\partial x},\frac{\partial F} {\partial y})$ is finite then $\sqrt {(F)} = (F)$. I don't see the relation between $\frac{\partial F} ...
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A polynomial in two variables with perfect square values

How can I find for which X and Y the values of the polynomial $ 100X^2+100Y^2+160X+80Y+81$ are perfect square?
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$F$ is a polynomial, $\deg F = 3$, and $(x^2 - 1)(x^2 - 2) | F(F(x)) - x$. Prove that $F$ exists

$F$ is a polynomial, $\deg(F) = 3$, and $(x^2 - 1)(x^2 - 2) | F(F(x)) - x$. Prove that: a) $F$ exists b) There are at least 10 such polynomials What I've tried to do: $(x^2 - 1)(x^2 - 2) \mid ...
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Transcendental extension, rational functions and field tower

I would like to solve this exercise (Lang, Algebra): Let $E=F(x)$ where $x$ is transcendental over $F$. Let $K \neq F$ be a subfield of $E$ which contains $F$. Show that $x$ is algebraic over ...
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1answer
62 views

Roots of $z^n+a_1z^{n-1}+\ldots+a_n$ lie inside $|z|\leq 1$

Let $P(z)=z^n+a_1z^{n-1}+\ldots+a_{n-1}z+a_{n}$, and suppose $|a_1|+\cdots+|a_n|\leq 1$. Find the least $R>0$ for which all the roots of $P(z)$ always lie inside $|z|\leq R$. $P(z)=z^n-1$ has ...
6
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1answer
141 views

Point on unit circle such that $|(z-a_1)\cdots(z-a_n)|=1$

Let $a_1,\ldots,a_n$ be points on the unit circle. Let $P(z)=(z-a_1)\cdots(z-a_n)$. Prove that there exists a point $b$ on the unit circle such that $|P(b)|=1$. My solution: $|P(0)|=1$, and $P$ ...
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Factor ring of polynomial

$F[x]$ is a polynomial ring over a certain field $F$. $J$ is an ideal of $F$, $J = (f(x))$. I need to prove that if the polynomial $f(x)$ has a multiple root the factor ring $F[x]/J$ is not a field. ...
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1answer
53 views

The height of a prime ideal in the $\kappa[[X]][Y]$

Let $\kappa$ be a field and $S=\kappa[[X]]$ be the ring of power series which depends on the indeterminate $X$. Now consider the ring $S[Y]$, the ring of polynomials with coefficients in $S$ and ...
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54 views

Express non-negativity contraint on polynomial with coefficients on unit interval

Given a polynomial of order $N$ with coefficients $\{c_0,\ldots,c_{N-1} \}$: $P(x) = \sum_{i=0}^{N-1} c_i x^i$ with $x \in [0,1]$, what is the most general way to express the non-negativity ...
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Polynomial proof qustion [closed]

A Polynomial $P(x)=x^3 - Lx^2 + Lx - M$ has zeroes $a$, $1/a$ and $b$. $$a + \frac1a +b = L$$ $$a + ab +\frac ba = L$$ Show that either $M = 1$ or $M = L-1$. Can somebody "show" this please?
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$f(X^p)$ irreducible or $p$th power if $f$ irreducible

An exercise in Bourbaki: Let $K$ be a field of characteristic $p>0$ and $f$ irreducible monic polynomial of $K[X]$. Show that in $K[X]$ the polynomial $f(X^p)$ is either irreducible or the ...
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1answer
85 views

Show that T is a linear transformation and find a, b, c

I'm having trouble understanding this question and the proper way to solve it. I don't understand the solution given and why this was the right way to answer it. Problem: For the vector space ...
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1answer
52 views

Find all polynomial such for $x\in Z$ then $f(x)\in Z$

Find all polynomials $f(x)$ such that $\deg(f)=4$ and such for all $x\in Z$ then $f(x)\in Z$. My try: since $f(x)=x^4$ is such it because $$x\in Z\Longrightarrow x^4\in Z$$ and ...
4
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1answer
216 views

Determine Galois groups of polynomials

Determine the Galois groups of the following polynomials over Q. $f(x)=x^3−3x+1$. $g(x)=x^4+3x+3$. $h(x)=x^5+8x+12$. I have no way to find their Galois groups. I only obtained that since they are ...
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70 views

minimal polynomial of roots of irreducible polynomial

Let $f\in \mathbb{Z}[x]$ be irreducible over $\mathbb{Q}[x]$, the highest degree coefficient of $f$ is $1$. Let $\omega\in \mathbb{C}$ such that $f(\omega)=0$. Can we obtain that the minimal ...
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1answer
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Algebraic independence in $ k[x,y]$

Let $k$ be a field, then $x$ and $y$ are algebraically independent in polynomial ring $k[x,y]$, so I would guess that 2 is the maximal number of algebraically independent elements in $k[x,y]$ But I ...
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1answer
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Solve : $x^4 + 6x^3 -3x^2 + 2 = 0$

$x^4 + 6x^3 -3x^2 + 2 = 0$ To find the zeros, I tried this by Ferrari's method but got stuck where a value of 'lambda' has to be obtained.
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1answer
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Showing $f(x^{p_1}) \mid f(x^{p_1 p_2})$ Given that $f(x) \mid f(x^{p_1}), f(x^{p_2})$

Hypothesis: Let $f = a_0 + a_1x + \ldots + a_nx^n \in \mathbb{Z}[x]$. Suppose $f(x) \mid f(x^{p_1})$ and $f \mid f(x^{p_2})$ for $p_1$ and $p_2$ two positive prime integers. Goal: Show that ...
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1answer
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Defining and describing a field extension being normal

My notes by Jens Carsten Jantzen (department of Mathematics at the University of Aarhus) defines a field extension as normal if: $N\supset K$ is a normal field extension if for each $\alpha\in N$ ...
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Given $f \in \mathbb{Q}[x]$ irreducible. How many and which roots of $f$ are contained in $\mathbb{Q}[x]/(f)$?

It is a fact that struggle me for a while. When working with irreducible polynomial over $\mathbb{Q}$ it is natural to build the extension ${\mathbb{Q}[x]}/{(f)} $ in which "lives " one root of the ...
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1answer
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Polynomial division with reference to CRC32

The polynomials we're using will always have coefficients of either 1 or 0. We have a divisor polynomial which will always remain the same, we'll call this polynomial $D$. We have a dividend ...
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1answer
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About a polynomial in two variables

It is well known that the polynomial $ AX^2+BX+C $ is a perfect square when its discriminant $ D=B^2-4AC=0 $ Is there a similar algorithm to check when the polynomial $ AX^2+BXY+CY^2+DX+EY+F $ ...
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rank of formal derivation

Let $K$ be a field, $n \in \mathbb{N}_{>0}$ and $K[x]_{ \leq n} $ he space of polynomials above $K$, that have a maximum degree of $n$. We define the formal derivation as follow: $\frac{d}{dx}= ...
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Minimal polynomial of $\alpha^2$ given the minimal polynomial of $\alpha$

Given that $\alpha$ is a root (in the field extension) of the irreducible polynomial $X^4+X^3-X+2\in\mathbb{Q}[X]$, I have to find the minimal polynomial of $\alpha^2$. I am thinking about this for a ...
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Does $f \mid f(x^{p_1}), f(x^{p_2}) \implies f \mid f(x^{p_1 p_2})$?

Hypothesis: Let $p$ be a prime positive integer. $f = a_0 + a_1x + \ldots + a_nx^n \in \mathbb{Z}[x]$ Suppose that $f \mid f(x^{p_1})$ and $f \mid f(x^{p_2}) \in \mathbb{Z}[x]$ Question: Does $f ...
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1answer
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Prove that $K\times K[X]/(X^7-1)\cong K\times \dots \times K$

Given that $K$ is a finite field of order $q\equiv1\text{ mod } 7$, I have to prove that $$K\times K[X]/(X^7-1)\cong K\times \dots \times K\ (8 \text{ times } K).$$ It's the same to prove that ...
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show that $X^4-4X^2-21$ is solvable by radicals

show that $$X^4-4X^2-21\in\mathbb{Q}[X]$$ is solvable by radicals. $\mathrm{Def}$: Let $f(X)\in K[X]$ and let $\Sigma$ be a splitting field for $f(X)$ over $K$. We say $f(X)$ is solvable by ...
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2answers
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Find $c_1,c_2,c_3\in\mathbb{Q}$ such that $(1+\alpha^4)^{-1}=c_1+c_2\alpha+c_3\alpha^2$ in $\Bbb Q(\alpha)$.

Let $\alpha\in \overline{\mathbb{Q}}$ be a root of $X^3+X+1\in\mathbb{Q}[X]$. So this is the minimal polynomial of $\alpha$ because it's irreducible in $\mathbb{Q}[X]$. I had to find the minimal ...
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For the given ideals $I=\langle i\rangle$ and $J=\langle j\rangle$ find a suitable $f$ in $\mathbb{Q}[x]$ such that $I+J=\langle f\rangle$

Given $i= x^2+2x+3 , j= x^3+x+1$ we have to find some $f\in \mathbb{Q}[x]$ such that $I+J =\langle f\rangle.$ As I couldn't make any significant head way to the solution so I am leaving this ...
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Polynomial Interpolation

My professor gave the following question as a practice for study guide. Any assistance in terms of helping me to solve this would be much appreciated. Suppose that $f$ is continuous and has ...
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Does $f=g_1^{n_1}\cdots g_k^{n_k}$ imply $Gal(f)=Gal(g_1)\times\cdots\times Gal(g_k)$?

Let $f\in \mathbb{Z}[x]$ and $f=g_1^{n_1}\cdots g_k^{n_k}$ where $g_1,\cdots, g_k$ are distinct irreducible polynomials over $\mathbb{Q}$. Whether does it hold $Gal(f)=Gal(g_1)\times\cdots\times ...
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210 views

A 3rd degree polynomial $P(x)$ has three unequal real roots. What is the least possible # of unequal real roots for $P(x^2)$

I got that if P(x) is a 3rd degree polynomial then P($x^2$) must be a 6th degree polynomial. I don't know how to proceed from here.
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Show that the (Veronese-like) surface is given by zero locus of the following polynomials

Consider the following set in $\mathbb R^6$: $$ S= \bigl \{(x_1^2,x_2^2,x_3^2,x_1x_2,x_2x_3,x_3x_1) \mid (x_1,x_2,x_3) \in \mathbb R^3, \; x_1^2+x_2^2+x_3^2 = 1 \bigr\}. $$ If we denote by ...
6
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2answers
169 views

Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?

We know that $f(x)=x^4+1$ is a polynomial irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$. My question is: Is it true that if $f(x)$ has a linear factor over ...
2
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1answer
96 views

Question about chebyshev polynomial

chebyshev polynomials are defined as such: $T_n(x)=cos(n*arccos(x))$ I'm asked to show that $deg(T_j(x))=j$ and that $T_0,T_1,T_2,...,T_n$ are an orthogonal basis of $\mathbb R_n[x]$. I think I can ...
5
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2answers
140 views

Are the elementary symmetric polynomials “unique”?

The elementary symmetric polynomials are interesting in that they generate the set of symmetric polynomials, in the sense that every symmetric polynomial is some polynomial applied to the elementary ...
6
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2answers
118 views

Checking irreducibility of polynomials over number fields

Are there general methods for checking irreducibility of polynomials over number fields? For instance, letting $F = \mathbb{Q}(\sqrt{3})$, I want to know whether $x^3 - 10 + 6\sqrt{3}$ is irreducible ...
3
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1answer
42 views

Which of the following polynomials are separable?

Which of the following polynomials are separable? a)$\;t^4-8t^2+16\in\mathbb{Q}[t]$ b)$\;t^{17}-t\in\ \mathbb{F}_{17}[t]$ c)$\;t^{17}-X^{17}\in \mathbb{F}_{17}(X)[t]$ a) My first idea to use the ...
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3answers
97 views

f(x) and g(x) are two polynomials, then choose the right option…

If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $h(x)=xf(x^{3})+x^{2}g(x^{6})$ is divisible by $x^{2}+x+1$, then choose the correct option: $A. f(1)=g(1)$ $B. f(1) $ is not equal ...