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.

learn more… | top users | synonyms

0
votes
0answers
5 views

Is my proof showing that if products of powers of elementary symmetric polynomials are the same, then the power is same, correct?

Below proof is assuming this lemma, which can be proven easily: Let $\leq$ be a monomial ordering on $R[X_1,...,X_n]$. Let $f,g\in R[X_1,...,X_n]$ be nonzero polynomials such that $LC(f)$ is ...
2
votes
4answers
39 views

Proof that a polynomial has a minimum in $\Bbb R$

I have to prove to following statement and I am having a really hard time here. There it is: Prove that the following polynomial has a minimum in $\Bbb R$ $$p(x)=x^4 + a_3x^3 + a_2x^2 + a_1x + ...
1
vote
2answers
16 views

Show that every $f(x) \in K[x]$ can be represented as $g(x^{p^e})$

A (probably simple) question I encountered but I don't know how to approach: Let $K$ be a field of prime characteristic $p>0$. Show every $f(x) \in K[x]$ can be represented as $g(x^{p^e})$ ...
0
votes
0answers
17 views

Old and recent results concerning the number of real roots for a polynomial function

Let $$p(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots +a_0$$ be a real polynomial function with real coefficients. My question is: I want to make a list of old and recent results concerning the number of real ...
0
votes
2answers
46 views

The product of two of the four roots of $x^4 -20x^3+ kx^2 + 590x -1992 = 0$ is $24$ the find $k$. [on hold]

Please help. I tried to solve by taking sum of roots as $20$ and product as $1992$. No idea how to proceed. Thanks in advance
-2
votes
0answers
32 views

Sum of the cube roots of the roots of a cubic polynomial [on hold]

If $\alpha, \beta, \gamma$ are zeroes of a polynomial $$x^3+3x^2-2015x+1$$ then find $\sqrt[3]{\alpha}+\sqrt[3]{\beta}+\sqrt[3]{\gamma}$.
7
votes
1answer
82 views

Proving that a polynomial of the form $(x-a_1)\cdots(x-a_n) + 1$ is irreducible over $\mathbb{Q}$

I want to prove that for any set of distinct integers $a_1,\ldots,a_n$, the polynomial $$h = (x-a_1)\cdots(x-a_n) + 1$$ is irreducible over the field $\mathbb{Q}$, except for the following special ...
0
votes
0answers
31 views

What will happen if there is a way predicting at a least one root of $p_{n}(x)=0$ without calculator?

let $p_{n}(x)$ be a polynomial of degree $n$ defined as follow : $p_{n}(x)=x^n +a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+.....a_{0}$ which : $a_{n-1},a_{n-2},.....,a_{0}$ are non nul real numbers coefficients. ...
-1
votes
1answer
32 views

How to find one perpendicular to the basis of the set? [on hold]

In the set of real polynomials, consider the inner product given by $$\langle p,q\rangle = \int_0^1 p(x)q(x)dx$$ How do I find a polynomial perpendicular to both elements of the set $\{1 + t, t^2 - ...
-2
votes
1answer
28 views

How to determine A* (t + 1)? [on hold]

In the space of the real polynomials P1 (R) is given scalar product linear transformation is given by the rule:
-2
votes
6answers
63 views

What must be added to $(x^3-3x^2+4x-13)$ to obtain a polynomial which is exactly divisible by $(x-3)$? [on hold]

Please explain with details. I am not able to understand this question with examples. Please describe. than you everyone for your supportive questions . :)
1
vote
1answer
18 views

Quadratic Diophantine equations on the ring of polynomials

The set of solutions of quadratic equation $a^2+b^2=c^2$ on $\mathbb{Z}$ can be described by Pythagorean triples up to multiplication. Can I use similar results on the ring of integer coefficient ...
0
votes
2answers
50 views

Finding $n$th root of 2 is irrational using given polynomial

The polynomial $f(x)$ is defined by $f(x)=x^n + a_{n-1}x^{n-1}+ \cdots + a_{2}x^2+a_1x+a_0$ where $n \geq 2$ and the coefficients $a_0, \cdots, a_{n-1}$ are integers, with $a_0 \neq 0$. ...
1
vote
1answer
26 views

Why does it mean that $n$-th variable is removable?

I'm reading the proof for "the fundamental theorem of symmetric polynomials" and I have a trouble with it (http://en.m.wikipedia.org/wiki/Elementary_symmetric_polynomial) Let $P(X_1,...,X_n)$ be a ...
0
votes
1answer
14 views

For any monomial ordering, $1\leq m$ for any monomial $m$

Let $R$ be a ring. Let $\leq$ be a well-ordering on the set of (monic) monomials in $R[X_1,...,X_n]$. Then, $\leq$ is said to be a monomial ordering iff $mm_1\leq mm_2$ whenever $m_1\leq ...
2
votes
3answers
33 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 ...
3
votes
2answers
24 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
votes
1answer
23 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
57 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
42 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
votes
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 ...
-1
votes
0answers
97 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
52 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!
0
votes
0answers
21 views

What is the distance of vector to subspace U and ortogonal projection? [on hold]

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 ...
-1
votes
1answer
33 views

Show that the linear transformation is the dot product in the vector space of real polynomials $P3(\mathbb{R})$ [on hold]

$$\langle p, q \rangle := \int_0^1 p'(t)q'(t)dt + p(0)q(0)$$ How to find the A* (adjoint) transformation to the transformation A, which is given by the rule:
0
votes
0answers
27 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
votes
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
43 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
vote
2answers
58 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
1answer
73 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 $f = X(X-3)(X-\alpha)(X-\beta) + 1\in\mathbb Z[X]$ is irreducible. It is straightforward to check ...
-2
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
votes
0answers
39 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
vote
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
votes
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
votes
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
votes
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
votes
1answer
57 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
votes
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 ...
-2
votes
1answer
28 views

Polynomial of degree 3 [closed]

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
66 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, ...
-2
votes
4answers
72 views

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

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
138 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
vote
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 ...
0
votes
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*} ...