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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3
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1answer
671 views

Characteristic Polynomial of a Linear Map

I am hoping for some help with this question from a practice exam I am doing before a linear algebra final. Let $T_1, T_2$ be the linear maps from $C^{\infty}(\mathbb{R})$ to ...
0
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3answers
133 views

Prove that if a is a root of $f(x)$ then $\overline{a}$ is also a root

Given a polynomial $ f \in \mathbb R[x] $, how do we prove that: if $a \in \mathbb C$ is a root of $f$ then also $\overline{a}$ is also a root of $f$ ?
1
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1answer
80 views

special polynomials

I'm searching for a polynomial $f$ of degree 4 with the following property: $f$ and all its derivatives have the maximum number of integer roots. Concretely formulated: $$\begin{eqnarray*} f(x) ...
2
votes
0answers
79 views

irreducibility conjecture

I tried to prove that every polynomial of the form $f(m,n) := m\cdot x^{n-m}+(m+1)\cdot x^{n-m-1}+\cdots+(n-1)\cdot x+n \quad \text{with} \quad 0 < m < n$ is irreducible over the rationals for ...
2
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2answers
71 views

Infinite Integral Domains

Let $D$ be an infinite integral domain, and let $g,h \in D[X]$. Show that if $g(x) = h(x)$ for all $x \in D$, then $g = h$. I understand this means a one-to-one correspondence, but how do I go about ...
2
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1answer
70 views

Minimal value of a polynomial

I do not know the following statement is true or not: Given $1<x_0<2$, there exists $\delta>0$ such that for any n, define $A=\{ f(x)=\sum\limits_{i=0}^{n}a_ix^i\}$ where $a_i\in\{0\,,1\}$, ...
0
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1answer
34 views

How to calculate the quotient and the reminder when $F=\mathbb Z_5[x],f(x)=3x+1,g(x)=x^3+2x+1?$

Division Algorithm says that, for any field $F$ and for $f(x),g(x)(\neq 0)\in F[x]$$~\exists$ unique $q(x),r(x)\in F[x]$ such that $f(x)=g(x)q(x)+r(x)$ where $\deg r(x)<\deg g(x)$ or $r(x)=0.$ I'm ...
3
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1answer
319 views

Polynomial Condition Number

I have a question, from "Applied Numerical Linear Algebra"(James W. Demmel), that I don't know how to do. Consider $\mathbb{R^{d+1}}$ as the set of polynomials of degree $\leq d$ and $S_a$ the set of ...
0
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1answer
52 views

$T:P_4(\mathbb{R})\rightarrow P_4(\mathbb{R})$ such that $N(T) = P_1(\mathbb{R})$ and $R(T)=P_2(\mathbb{R})$

So, I'm asked to give an example of a linear map $T:P_4(\mathbb{R})\rightarrow P_4(\mathbb{R})$ such that $N(T) = P_1(\mathbb{R})$ and $R(T)=P_2(\mathbb{R})$. As far as I understand, I'm trying to ...
2
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4answers
71 views

Show that $1$ and $2$ are zeros of the following polynomial

Show that $1$ and $2$ are zeros of the polynomial $P(x)=x^4-2x^3+5x^2-16x+12$ and hence that $(x-1)(x-2)$ is a factor of $P(x)$
2
votes
1answer
47 views

Creating and using calibration factors

Perhaps simple question, but I (the simple) need some guidance. The following applies to a project ongoing and is a challenge in that I am not a math whiz! As example, I wish to measure temperature ...
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0answers
52 views

Linear Algebra: Linear transformation and eigenvalues [duplicate]

Hi could some one please help. I am having problems proving this. Let $A$ be an $n \times n$ matrix with complex entries and let $f (t) =\det(A - tI)$ be its characteristic polynomial. Prove ...
0
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1answer
63 views

Finding a function which fits this data?

I need to find a polynomial (or other continuous elementary function) on the interval [70, 180] such that it passes through the points (70, 0) (this is a relative min), (105, 17) (this is a relative ...
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2answers
231 views

Proving injectivity of a general polynomial function

I'm currently trying to prove that a certain linear mapping is injective, but am stumped as to how I would go about doing so. I'm attempting to prove that a linear transformation $$T: P(\mathbb{R}) ...
0
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1answer
59 views

subproof from theorem of Polya

Suppose we have polynomial: $$f(z) = z^n + b_{n-1}z^{n-1} + \cdots + b_0$$ It is a complex polynomial of degree $\geq 1$ with leading coefficient $1$. Associate with $f(z)$ set: $$C := \{z ...
3
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1answer
75 views

Find all integer solutions of $35x^{31} + 33x^{25} + 19x^{21} \equiv 1 \pmod{ 55}$

Find set of all integers x for which the following holds: $35x^{31} + 33x^{25} + 19x^{21} \equiv 1 \pmod {55}$ Since $55 = 5\cdot 11$, simultaneous congruences: $35x^{31} + 33x^{25} + 19x^{21} ...
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1answer
41 views

Does there exist a degree $n$ polynomial in $n$ variables which vanishes at the vertices of the unit cube?

Let ${\cal C}=\{0,1\}^n$ be the vertices of the unit cube. The polynomial $$ (1-x_1) x_1 x_2 \cdots x_n$$ has degree $n+1$ and vanishes at every point $(x_1, \ldots, x_n) \in {\cal C}$. Does there ...
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6answers
143 views

Proof that if $\forall a f(a) = g(a)$ then $f=g$

How do we prove formally that if: $\forall a f(a) = g(a)$ $=>$ $f=g$ when $f,g \in \mathbb F[x]$
6
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1answer
108 views

My proof that if for a k degree polynomial $P(x)$, for the matrix $A$, $P(A)=0$ then $A$ is invertible

Let $P(x)$ be a $k$-degree polynomial with with non-zero free coefficient. Prove that if for matrix $A$, $P(A)$=0, then $A$ is invertible and $A^{-1}$ is $k-1$ degree $A$ polynomial. Here's my ...
8
votes
1answer
116 views

Killing functions by successive differentiation.

It is clear that a polynomial function $f(x) = a_0 + a_1 x + \dots + a_n x^n$ has the property that some derivative of $f$ vanishes. (Of course, it's the $(n+1)$-th derivative.) One can also check ...
0
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0answers
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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1answer
76 views

Polynomials maps

Looking at past exam papers I have seen several question of this style but don't know how to answer it. Any help in the methods need to be uses to calculate would be a great help Consider $\phi: ...
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1answer
47 views

Number of solutions of $P(x_1, …, x_n) = 0$ in $\mathbb{F}_q^n$

I have this exercise : $q = p^r$ (it is not clearly precised in the exercise that $r = n$). and $\mathbb{F}_q$ is the finite field of cardinal $q$. Let's $P \in \mathbb{F}_q[X_1, ..., X_n]$ of degree ...
4
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1answer
787 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 ...
1
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1answer
32 views

Find $p$ values such that $p$ is prime and $f_p$ is divided by $g$ and $g \cdot x$

Let's consider the following polynomial over $\mathbb Z_p[x]$: $$f_p = \overline{20}x^4-x^2+\overline{3}$$ I need to: find the $p$ values so that $p$ is prime and $f_p$ can be divided by ...
0
votes
1answer
126 views

find all positive integers for a given diophantine equation involving 4 or 7 variables

Given equation: Ap + Bq + Cr + Ds + Et + Gu + Vg = K; (Eq. in 7 variables); suppose we have A, B, C, D initialize with = 1,2,5,10,20,50 and 100 respectively; and K = 50000; How do we solve it? ...
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4answers
87 views

Finding the value of a polynomial at zero

Given a polynomial $p_n(x)\,$ with $\, n \geq 0 \,$ such that $\, p_0(x)=1 \;,\; p_1(x)=x\;$ and $\; p_n(x)=xp_{n-1}(x)-p_{n-2}(x)\;$ , how can I find $\;p_{10}(0)\; $ ?
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0answers
77 views

Showing $f(x)=g(x^{p^a})$ over field of (prime) characteristic $p>0$.

Let $f$ be a non-constant irreducible polynomial over a field $F$ of (prime) characteristic $p>0$. I need to prove that $f$ can be presented as: $f(x)=g(x^{p^a})$, where $g$ is irreducible over ...
8
votes
2answers
477 views

Calculate the number of real roots of $x^8-x^5+x^2-x+1 = 0$

Calculate the number of real roots of $x^8-x^5+x^2-x+1 = 0$ My try: $$\left(x^4-\frac{x}{2}\right)^2+\frac{3}{4}x^2-x+1 = ...
1
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1answer
32 views

Solving for $f(n+1)$ when $f(k)$ is known for $k=0,1,…,n$

I posted earlier about polynomials but this is different type of problem I think. I seem to have an answer but I mistrust it.... A polynomial $f(x)$ where deg[$f(x)$]$\le{n}$ satisfies $f(k)=2^k$ ...
1
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1answer
73 views

If $f(X) = a_0 + a_1 X + a_2 X^2 \in \mathbb{F}[X]$ then show $f$ is uniquely determined by $f(x)$, $f(y)$, $f(z)$?

This is the exact question: It's part(ii) that I don't understand - what does it mean and what is it asking me to do? How would I go about constructing a proof? Any help would be much appreciated.
4
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2answers
339 views

finding the remainder of $x^{100}-2x^{51}+1$

I have never been great with polynomials. Here's my problem. Find the remainder of $f(x)=x^{100}-2x^{51}+1$ when $f$ is divided by $x^2-1$ This sounds easy right? Why can't I figure it out? My ...
2
votes
1answer
60 views

Any comprehensive material to revise the mathematics

I left school long back and so my mathematics knowledge also fades out. I am trying hard to re-collect the basics about log / permutaion / combination / probability / polynomial equations. I tried ...
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2answers
41 views

Get polynom from polynom, roots of second one are multiplication of first one.

I have a polynomial $P$, with unknown roots $r_1,r_2, ... ,r_n$. My goal is to find a polynomial $X$ with roots $s_1,s_2, ... ,s_n$, where each $s_i = 2r_i$ I shall get $X$ with no need to find the ...
4
votes
2answers
193 views

Why is the concept of transcendental numbers linked with rational coefficients? Why not real nor complex coefficients?

I've read this: In mathematics, a transcendental number is a (possibly complex) number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with rational ...
2
votes
1answer
53 views

Show that exists $x$ and $y$ such that $P(x)*P(y) < 0 \ ; x,y\in \Bbb C$

I would like to ask how could I perform following proof: Prove(show) that exists such $x$ and $y$ that $P(x)*P(y) < 0$. Where $P(x)$ and $P(y)$ are polynomials. $\forall\ x,y\in \Bbb C$
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1answer
110 views

Polynomial-Why is the answer for removing parentheses and solving different

I am a 45 yo self-taught software engineer and I am finally studying mathematics. Pretty simple question: I am using the ALEKS software and during two different lessons on polynomials we get ...
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2answers
399 views

Roots of polynomials over finite fields

I've been trying to find the decomposition of $x^2-2$ to irreducible polynomials over $\mathbf{F}_5$ and $\mathbf{F}_7$. I know that for some $a$ in $\mathbf{F}_5$ (for example), $x-a$ divides $x^2-2$ ...
0
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2answers
367 views

Bezout's identity in $F[x]$

Let $F$ be a filed and $F[x]$ be a polynomial ring. Let $p(x)$ be an irreducible polynomial show that $\gcd(p(x),q(x)) = 1\Longrightarrow \exists q(x),s(x).\;r(x)p(x)+s(x)q(x) = 1$ I know the proof ...
2
votes
1answer
155 views

how to find bounds on (complex) coefficients from bounds on a polynomial?

I'm trying to prove the following two statements about a polynomial $p$ of degree $n$ with complex coefficients: If $|p(x)|\le1$ for all real $x$ with $|x|\le1$, then every coefficient of $p$ has ...
0
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1answer
155 views

approximating Lipschitz functions by polynomials?

If $f:[0,1]\rightarrow\mathbb{R}$ satisfies $|f(x)-f(y)|<|x-y|\ $ for all $x$, $y\in[0,1]$ and $\epsilon>0$ is fixed, why must there be a polynomial $p$ of degree less than ...
0
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1answer
52 views

Polynomials decomposition into irreduceables

I've been trying to find the composition to irreduceables of the following polynomials with no much success: X^2 +1 over the field F7 and X^2-2 over the field F5 Is there any method/algorithms I ...
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2answers
41 views

Polynomial Form for $f$ a Polynomial Such That $f(1)=0$

What general form, as, for example $ax+by=c$, does the polynomial whose various forms are all evaluation at $1$ to be $0$? $k_1\overbrace{(x-1)(x+a_1)(x+a_2)\cdots}^{\text{$n$ ...
0
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1answer
18 views

Uniformly discrete on the $\mathbb{Z}[\beta]$

Is the following statement true? Given $1<\beta<2$ define $\mathbb{Z}[\beta]$ with coefficients belonging to $0$ or $1$, then there exists $\delta$, such that for any $f\,,g\in ...
1
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1answer
71 views

Dimension Recovery of $S \subset P_n(F)$

How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that $f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset? Added from answer posted by Trancot on 18 Apr ...
3
votes
1answer
459 views

Prove that $D[x]$ is an integral domain if $D$ is one.

Prove if $D$ is an integral domain and $f,g\in D[X]$ are nonzero, then $fg$ does not equal $0$ and $\deg[f(x)g(x)]=\deg f(x) + \deg g(x)$. I do not know much about this since I just learned about it. ...
1
vote
1answer
73 views

Lower bound for polynomial with complex coefficient

Let $p(z)=z^{n}+a_{n-1}z^{n-1}+...+a_{1}z+a_{0}$ be a polynomial with complex coefficients. Define $R:=1+\sum_{k=0}^{n-1}|a_k|$. Show that $|p(z)| > R$ for all $z \in \mathbb C$ with $|z|>R$. ...
0
votes
1answer
28 views

Showing a polynomial $f\in\mathbb Q[x]$ is irreducible if it has rational coefficients?

I'm trying to figure out how I can do this for some arbitrary function. Say I find a monic associate of $f$ that we'll call $f_1(x)$. If I then apply Eisenstein's Criterion or Descartes' Rational Root ...
5
votes
1answer
125 views

Does the polynomial $P(x)$ have integer zeros?

The following is a homework question: Let $P(x)$ be a polynomial with integer coefficients and $P(x_1)=P(x_2)=P(x_3)=P(x_4)=P(x_5)=P(x_6)=P(x_7)=7$ where $x_i$ are distinct integers. Determine if ...
1
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0answers
32 views

Homogeneity of translated polynomial

I am currently trying to understand the very basics of complex algebraic curves and I came across the following statement in the book by F. Kirwan (Definition 2.9): The multiplicity of the complex ...