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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25 views

Method for proving polynomial inequalities

Let $x\in\mathbb{R}$. Prove that $\text{(a) }x^{10}-x^7+x^4-x^2+1>0\\ \text{(b) }x^4-x^2-3x+5>0$ Possibly it can be proved in a few different ways, but I have first tried to prove it ...
2
votes
2answers
18 views

irreducibility of polynomials made by perturbation from a polynomial

Suppose $f(x)\in\mathbb{Z}[x]$ with $\text{deg}f=2n,n\in\mathbb{Z_+}$ and $f_m(x):=f(x)+ mx^n $ for each integer $m\in\mathbb{Z}$. Let us define a number $P_f$: ...
-1
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1answer
17 views

Maximum modulus of a holomorphic function on a disc within a certain sector

Given the polynomial $$f(z) = az^n + b \qquad (n \geq 2)$$ and a modulus $0 < \rho < 1$, can one find a modulus $0 < r < \rho$ such that there is a point $$w \in \{ |z| \leq r \} \cap \{ ...
1
vote
1answer
12 views

Monomial ideals and Dickson's lemma

I am currently revising for my exams and working on questions about monomial ideals and came across this question. Let I be the ideal of $\mathbb{R}[x,y]$ generated by all polynomials of the form ...
5
votes
4answers
56 views

Prove that $f=x^6+ax+5$ is reducible over $\mathbb{Z_7},\forall a\in\mathbb{Z_7}$

We have $f=x^6+ax+5\in\mathbb{Z_7}$ and we have to show that it is reducible on $\mathbb{Z_7}$, $\forall a\in\mathbb{Z_7}$. Here are all my steps: For $a=0$ we'll get $f=x^6+5\in\mathbb{Z_7}$. But ...
-1
votes
1answer
41 views

Fraction modulo integer in sage [on hold]

I'm working on a sage script right now, I have some polynomials coefficients that are rational, and I want to apply a congruence on these coefficientss, for example: $p = 1 + (7/2)x$ the function ...
0
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1answer
39 views

Are Zero Degree polynomials Considered monics?

DO zero degree polynomials that is constant polynomials considered monic polynomials? Example F(x)=16 Does it Matter the Field or the Integral region where i take the coeficients from?Sorry if the ...
0
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0answers
92 views

Is there a field in which every rational polynomial has a root (other than the obvious fields)?

Let $\mathbb{A} \subset \mathbb{C}$ denote the field of numbers algebraic over $\mathbb{Q}$. Is there a proper subfield $F$ of $\mathbb{A}$ such that every nonconstant polynomial $p(x) \in ...
0
votes
1answer
51 views

How is the degree of a polynomial defined? $a_1+a_2x^2+\cdots+a_nx^{n-1}$ has degree $n$ or $n-1$?

I have this polynomial: $$a_1+a_2x^2+\cdots+a_nx^{n-1}$$ or: $$a_0+a_1x^2+\cdots+a_{n-1}x^{n-1}$$ What is degree of those polynomials? $n$ or $n-1$, I'm little bit confuse... Thank you!
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0answers
12 views

What is the distance of vector to subspace U and ortogonal projection?

Find the orthogonal projection of a polynomial $1 + t - t ^ 3$ on the subspace $U = Ker D^ 3$, where Dp = p '. What is the distance of this vector to subspace $U$? $\langle p,q ...
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1answer
30 views

Show that this is the dot product in the vector space $P3(\mathbb{R})$ [on hold]

$$\langle p, q \rangle := \int_0^1 p'(t)q'(t)dt + p(0)q(0)$$ How one can start with this problem ? Interesting task I think.
2
votes
1answer
33 views

Localising polynomial ring $R[t]$, then for a non-maximal prime ideal $Q$, $(Q \cap R)S = Q$.

I'm trying to work out the following past paper question and I've got stuck. $R$ is an integral domain and $S = R[t]$, the polynomial ring in one variable over $R$. We have that $Q$ is a prime ideal ...
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0answers
22 views

How to find the real polynomial, which, based on a given dot product, is the nearest to other polynomial

I have a subspace $$U := \{p \in P_2(\mathbb{R}): p'(0) = p'(1) =0\}$$ and a dot product: $$\langle p, q\rangle = p(-1)q(-1) + p(O)q(O) + p(l)q(l).$$ I would like to determine the shortest distance ...
0
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1answer
38 views

Good Triple Well Function [on hold]

I am looking for a good triple-well function with good control over the barrier height. Let's say that $y=9x^{2}-6x^{4}+x^{6}$. In this function there are three wells (even though only two wells are ...
0
votes
3answers
42 views

How do I factorise the following expression?

How do I go from the left expression to the right one? $$ (2-x)^2 \cdot (-2-x) - (-2-x) = - (x+2)(x-3)(x-1) $$ I'm guessing that I have to solve the third degree equation. What are the steps for ...
1
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2answers
54 views

Proof for quotient polynomial rings equivalent to field extension

I am predominantely looking for a proof, I have seen in my books and around but seem to have a hard time finding that if we let $\alpha_1,\alpha_2,...,\alpha_n$ be the roots of the minimal polynomial ...
4
votes
0answers
60 views

Irreducibility of $X(X-3)(X-\alpha)(X-\beta) + 1$

I'm trying to solve the following exercise: Show that for $\alpha,\beta\geq 3$, the polynomial $X(X-3)(X-\alpha)(X-\beta) + 1\in\mathbb Z[X]$ is irreducible. It is straightforward to check that ...
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votes
1answer
23 views

Minimal polynomial over $\mathbb{Q}$ of $\alpha$ in $\mathbb{C}$ has coefficients in $\mathbb{Z}$?

Let $m \in \mathbb{C}$ be integral over $\mathbb{Z}$. Prove that the minimal polynomial over $\mathbb{Q}$ has coefficients in $\mathbb{Z}$. The definition I use: $m\in \mathbb{C}$ is integral over ...
3
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0answers
37 views

Sum of zeros of $P(x)$

I asked this question here before too, but vaguely, hopefully, this time will be a better attempt: There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero ...
1
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1answer
16 views

Interpolation with nonvanishing constraint

Let $x_1,x_2,\ldots,x_n$ be distinct complex numbers. Let $y_1,y_2,\ldots,y_n$ be nonzero complex numbers, and let $K$ be a bounded subset of $\mathbb C$. Does there always exist a polynomial $P$ such ...
0
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0answers
36 views

Inequality on complex polynomial

For every $a\geq 0$, let $p_a(z)=1-z+az^3$. What is the maximal value of $a$ such that $$ p_a(|z|)\leq |p_a(z)| $$ for all $z\in \mathbb C$? EDIT: I claim that $a=\frac{4}{27}$ is the maximal value. ...
3
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1answer
26 views

How to prove this identity involving characteristic polynomials on both sides?

Suppose $A\in \Bbb C^{m\times n},B\in \Bbb C^{n\times m},m\ge n$, prove: $$\det(\lambda I_m-AB)=\lambda^{m-n}\det(\lambda I_n-BA)$$ I don't want to get into nasty determinant calculation. Instead, I ...
0
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2answers
25 views

Finding value of $m$ such that such that the polynomial is factorized

A polynomial $2x^2+mxy+3y^2-5y-2$ Find the value of $m$ much that $p(x)$ can be factorized into two linear factors
2
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1answer
56 views

How to find the recurrence relation from a given polynomial?

Consider the formal power series: $A(x)=\sum a_nx^n$. and $A(x)= \frac{8+14x-50x^2}{1-7x^2+6x^3}$ I am trying to derive a recurrence relation, Is there a general method for doing it? Please help, ...
0
votes
3answers
68 views

$(5x +1) ÷ (3x)$ is not a polynomial?

On the Mathwarehouse page on polynomial equations, it gives this expression as an antiexample, something that is not a polynomial: $(5x +1) ÷ (3x)$ However, it also says on the same page that if it ...
0
votes
0answers
22 views

Pointwise convergence of Bernstein polynomials for piecewise continuous functions

I know that $B_nf \to f$ uniformly if $f:[0,1] \to \mathbb R$ is continuous. But can anybody explain to me, why $B_nf \to f$ pointwise in every point where $f$ is continuous if $f:[0,1]\to \mathbb ...
3
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1answer
19 views

How to proceed with Euclidean algorithm for finding greatest common divisor of two polynomials.

I am trying to find \begin{equation*} gcd(x^4-x^3-4x^2-x+5,x^2+x-2). \end{equation*} I have done the first step of long division and found. \begin{equation*} x^4-x^3-4x^2-x+5=(x^2-2x)(x^2+x-2)-5x+5 ...
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votes
1answer
27 views

Polynomial of degree 3 [on hold]

f(x) is a polynomial of degree $3$. We are given that the coefficient of $x^3$ is $1$ and that $f(x)$ is divisible by $(x−3)$ and $(x+1)$. If $f(4)=30$, then what is $f(2)$?
4
votes
1answer
64 views

Questions about a topological proof of the FTA

I'm a high school student, curious about proofs of the Fundamental Theorem of Algebra. Specifically, I've been thinking about one of the topological proofs of the theorem, given in Courant's book, ...
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4answers
72 views

What is the decomposition of $x^4+x^3+x^2+x+1$. [on hold]

What is the decomposition of $$x^4+x^3+x^2+x+1.$$ It seems that there is a special way to decompose this, I couldn't find it. It will be great that if you help me about it, thanks. I am asking for ...
6
votes
2answers
137 views

Proof of an identity of $n!$

I came up (numerically) with an identity concerning n! and I was wondering about a proof of it. Here it is: \begin{align} \ n! &= \sum_{r=0}^{n} { \binom{n}{r} (-1)^r(k-r)^n } \quad \forall n ...
2
votes
1answer
19 views

Sequences formed by integer evaluations of polynomials modulo $ p^{k} $, where $ p $ is a prime number and $ k \in \Bbb{N} $.

I have the following question. Let $ p $ be a prime number and $ k $ a positive integer. Let $ (a_{n})_{n \in \Bbb{Z}} $ be a two-way sequence in $ \Bbb{Z} / p^{k} \Bbb{Z} $. Then is it true that ...
3
votes
2answers
90 views

Showing $r\in\mathbb Q\setminus \{0\}\implies e^r\notin \mathbb Q$

For a given $n>0$, let $\displaystyle J_n:x\to \frac{1}{n!}\int_{-x}^x(x^2-t^2)^ne^tdt$ a. Prove that there exists $A_n,B_n\in \mathbb R_n[X]$ such that $\forall x\in \mathbb R^+, ...
1
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1answer
15 views

Generalizing the Remainder Factor Theorem

Today, I spent most of my time developing a systematic procedure for finding remainder polynomial when higher degree polynomials are divided by some polynomial of degree $\leq$ the degree of the ...
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3answers
32 views

Simple questions about a polynomial ring

Reading Pinter's algebra, I'm little bit confused. In ch.24, the author says that x which appears in a polynomial is to be considered as a 'placeholder' for a moment... All right, then i was trying ...
0
votes
4answers
55 views

Find the remainder when $(x+1)^n$ is divided by $(x-1)^3$

Find the remainder when $(x+1)^n$ is divided by $(x-1)^3$ I know that \begin{equation*} (1 + x)^n = 1 + nx +\frac {n(n-1)}2!\cdot x^2 +\frac {n(n-1)(n-2)}3! \cdot x^3 +... \end{equation*} ...
5
votes
1answer
47 views

Remainders of quadratic trinomial

The problem is to determine, whether there exist a quadratic trinomial $f(x) = ax^2 + bx +c$ with integer coefficients (with $a$ not a multiple of 2014), such that the numbers $\ f(1), \ f(2),\, ...
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0answers
58 views

Factorisation of large polynomials and Galois theory

As I understand it, one of the consequences of Galois theory is that there is no way of expressing the solutions to a general polynomial of degree 5 or higher in terms of radicals. Would a theory that ...
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3answers
71 views

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then $a=b=c=0$ [closed]

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then it is an identity i.e. it is true for all values of $x$ and $a=b=c=0$. What is a proof of this?
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1answer
54 views

what is the value of $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$?

if we have $a+b+c=1$ and $ab+bc+ac=\frac{1}{3}$ then what is the value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$$ and $$\frac{a}{b+1}+\frac{b}{c+1}+\frac{c}{a+1}$$. from the hypothesis we have ...
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0answers
24 views

Expressing in terms of partial fractions

Simplify $$\frac{7x^3+52x^2+97x+60}{x^4+8x^3+15x^2-4x-20}$$ into the form $$\frac{A}{x+p}+\frac{B}{(x+p)^2}+\frac{C}{x+q}$$ where $A,B,C,p,q$ are constants. Thank you for any help given. I have been ...
4
votes
2answers
50 views

Name for the following set of polynomials

I have the following set of polynomials defined by $$P_n(x) = \sum^n_{k = 0} \frac{n!}{k!} x^k, \quad x \geqslant 0.$$ The first few are \begin{align*} P_0 (x) &= 1\\ P_1 (x) &= 1+x\\ P_2 (x) ...
2
votes
3answers
63 views

Finding value of a quadratic

The polynomial \begin{equation*} p(x)= ax^2+bx+c \end{equation*} has $1+\sqrt{3}$ as one of it's roots and also $p(2)=-2$. Is there any way to know the value of $a$, $b$ and $c$? I tried but I can ...
1
vote
1answer
12 views

Is there a way to find a good lower bound on $\Vert p_n \Vert_\infty$ without finding the extrema?

Let $$p_n(x):=x^n+c_{n-1}x^{n-1}+ \cdots + c_0$$ be defined over some interval $[a,b]$. Is there a way to find a good lower bound on $\max_{x\in [a,b]} | p_n (x) |$ without actually finding the ...
2
votes
1answer
34 views

what is the decomposition of $(x^2+y^2+z^2)(x+y+z)(x+y-z)(-x+y+z)(x-y+z)-8x^2y^2z^2$?

I want to have a decomposition of this : $$(x^2+y^2+z^2)(x+y+z)(x+y-z)(-x+y+z)(x-y+z)-8x^2y^2z^2$$ I have tried all possible calculation which came to my mind,I will describe one of it which is ...
3
votes
1answer
39 views

Polynomials and Commutativity

Let $f(x)=2013x+1$. Suppose $g(x), h(x)$ are polynomials with real coefficients such that $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$. Prove that $g(h(x))=h(g(x))$. I tried to look at the coefficients of ...
1
vote
2answers
50 views

Find the lowest degree of the polynom $P$?

I have to determine the lowest degree of $P$ given by the following system : $\left\{ \begin{array}{l} P \equiv 2X \ \mod[X^2 -2X +1] \\ P \equiv 3X \ \mod[X^2 -4X+4] \end{array} \right.$ First, ...
0
votes
2answers
38 views

How to go about this proof for non zero polynomials.

How do I go about proving this? Let $\mathbb{F}$ be a field and $X$ an indeterminate, and consider the polynomial ring $\mathbb{F}[X]$. Let $f(X), g(X) \in \mathbb{F}[X]$ with $f(X), g(X) \neq ...
0
votes
0answers
26 views

When is ${(1-t^2)^{-N/2}}{\det(f_t(A))}$ expressible as polynomial?

Given a matrix valued function $f_t:\mathbb R^{N\times N} \mapsto \mathbb R^{N\times N}$. For $f_t(A)=B$ both $A$ and $B$ are symmetric. Which properties can be assigned to $f_t$, when the following ...
0
votes
3answers
51 views

A question about polynomials [duplicate]

How can we prove that the following expression is a polynomial? $$ \frac{1-x^{2^{n-1}}}{1-x} $$ I've asked this question just for learning the ways different from using ...