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Questions tagged [polynomials]

This tag is used for both basic and advanced questions on polynomials in any number of variables, including, but not limited to solving for roots, factoring and checking for irreducibility.

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0
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1answer
29 views

Adding the polynomials (bytes) in the ring $Z_2 [x]/(x^8+x^4+x^3+x+1)=GF(2^8)$

I need to solve few similar exercises. One of them is: Add the polynomials (bytes) in the ring $Z_2 [x]/(x^8+x^4+x^3+x+1)=GF(2^8)$ a) ‘57’+’02’ b) ‘03’+’03’ c) ‘FF’+’0F’ I try to learn it by ...
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0answers
6 views

Derive recurrence relation from polynomial.

Let $f(x) = x^3 −s_1x^2 +s_2x−s_3 = (x−α)(x−β)(x−γ) ∈ \mathbb{Q}[x]$ where $α, β, γ ∈ \mathbb{C}$. Denoting $σ_i = α^ i +β^ i +γ^ i$ for $i ≥ 0$, show that $σ_0 = 3, σ_1 = s_1$ and $σ_2 = s^2_1 −2s-2.$...
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1answer
14 views

Generating Reducible polynomials of a particular type

I am working on a paper and I need to know if there exists any REDUCIBLE bivariate polynomials of the following form: $Ax^4+2Ax^3+2Bx^2y-(2C+1)y+D=0$ Where A,B,C,D are all positive integers I ...
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2answers
16 views

Find values $m$ and $n$ if $x - 2$ is a common factor of $ x^2(m+n)x - n$ and $2x^2 + (m-1)x + (m+2n)$

If $x - 2$ is a common factor of the expressions $$ x^2(m+n)x - n$$ and $$2x^2 + (m-1)x + (m+2n),$$ find the values of $m$ and $n$. I've gotten $x=2$ and replaced the value with all the $x$ but ...
1
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1answer
13 views

Find a parameter $m$ of a second order polynomial for which the polynomial is injective on a given interval.

I am given the function: $f : \mathbb{R} \rightarrow \mathbb{R}$ $ f(x)=x^2-mx+2$ $ m \in \mathbb{R}$ And I am asked to find $m \in \mathbb{R}$ for which the function is injective on the interval $...
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0answers
13 views

Show that if m is any positive integer, then the polynomial $x^{p^m} − x$ has no multiple root in any extension of fields $L : \mathbb{F}_p$.

As the title says, I'm trying to show that if m is any positive integer, then the polynomial $x^{p^m} − x$ has no multiple root in any extension of fields $L : \mathbb{F}_p$. I know that if I could ...
1
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1answer
32 views

Find all polynomials f that satisfy the following property

Let $n \geq 2$ a positive integer. Find the polynomials $f$ with complex coefficients that satisfy the following property: $$f(z^n)=f^n(z)$$ for all complex numbers z. My trial was to denote $ f(z)=...
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1answer
9 views

Can this nearly-symmetric, dim-0, degree-2 polynomial system be linearly solved, or transformed to quadratic form?

A common problem from chemical thermodynamics is to determine the vector $x_i$ with dimension $N$ that solves $$n_i-x_i = s_i \frac{x_i}{\sum x_i}$$ for some constant vectors $n_i$ & $s_i$, ...
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1answer
12 views

Find all the possible integer values of $a$ such that the equation about $x$: $(a+1)x^2-(a^2+1)x+2a^3-6=0$ has all integer roots

Find all the possible integer values of $a$ such that the equation about $x$: $(a+1)x^2-(a^2+1)x+2a^3-6=0$ has all integer roots. I've been given this as a homework problem and haven't been able to ...
20
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2answers
593 views

A polynomial with nowhere surjective derivative

Let $ P:\mathbb {R}^2\rightarrow \mathbb {R}^2$ be a polynomial map. It is given that the Jacobian of $P$ is everywhere not surjective . Must the following be true: There exists polynomial maps $ f:...
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5answers
35 views

Find the range of $y=\sqrt {x^2+2x+3}$

I want to find the range of: $$y=\sqrt {x^2+2x+3}$$ I would like to know if we can solve this by writing $x$ in terms of $y$ and then finding the domain of that? If so how?
5
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1answer
31 views

Number of rational roots

Let $f(x) = a_0 + a_1 x + ...... + a_n x^n$ be a polynomial of degree n with integral coefficients. If $f(1), a_0, a_n$ are odd then number of rational roots are. My Try: Let $f(x)=(x-\alpha)g(x), \...
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0answers
23 views

Number of integral solutions of polynomial

Consider the equation $f(x) = x^4-ax^3-bx^2-cx-d=0,$ $a, b, c, d \in \mathbb Z^+,$ $ a≥b≥c≥d$ then number of integral solutions can be. I am unable to use Integral Root theorem and just reached to ...
1
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1answer
21 views

Finding the value of C in this function

If $$x^2 + (c-2)x -c^2 -3c + 5$$ is divided by $x + c$, the remainder is $-1$. find the value of c I replaced all the x value to -c and set it to an equation which equated to $-1$ I am confused ...
2
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0answers
19 views

Orthogonality of Legendre polynomials from generating function

Given the the Legendre polynomials generating function: $$G(x,t)=\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n$$ prove the relation: $$\int_{-1}^{1} (P_n(x))^2 dx = \frac{2}{2n+1}$$ My ...
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0answers
24 views

Generating a randomised polynomial

My question is related to this question on Cryptography SE. In the following, all the operations and polynomials are defined over a finite field of prime order, $\mathbb{F}_p$, where $p$ is a ...
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1answer
17 views

Find a formula that generates the number of possible combinations of roots for a given polynomial

Evaluate the number of possible combinations of roots for a given polynomial equation. For example for quadratic, you can have: 2 imaginary roots (conjugate) two real distinct roots or 1 repeated ...
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0answers
25 views

Roots of complex polynomial have modulus less than 1 [duplicate]

Let $0<a_0<a_1<...<a_n$, $a_i\in \mathbb R$. I need to show that if $$a_0+a_1z+...+a_nz^n=0$$ then $|z|<1$. Any hints? I don’t know how to begin. I can’t use Rouche’s theorem as it ...
2
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2answers
95 views

Factor $x^{35}+x^{19}+x^{17}-x^2+1$

I tried to factor $x^{35}+x^{19}+x^{17}-x^2+1$ and I can see that $\omega$ and $\omega^2$ are two conjugate roots of $x^{35}+x^{19}+x^{17}-x^2+1$. So I divide it by $x^2+x+1$ and the factorization ...
2
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1answer
24 views

How to find general solution of the following difference equation, which follows from a Markov chain?

How to find the general solution of the following difference equation? $$\alpha(x) = (1-p)\alpha(x-1) + p\alpha(x+2)$$ I constructed the following characteristic equation $$\lambda = (1-p) + p \...
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0answers
24 views

Polynomial interpolation: Construct basis functions such that $\mu_j(L_k)=\delta_{jk}$

We are looking for the solution of the Hermitian-polynomial-interpolation with the data points $x_0=x_1=0, x_2=2, x_3=x_4=1.$ Construct, analogous to the Lagrange-interpolation, basis functions $...
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1answer
50 views

Rationalize irrational equation

$$ \begin{cases} \sigma_{1,(2)}=x_1+x_2\\ \sigma_{2,(2)}=x_1x_2\\ \end{cases}\\ \color{red}{ \begin{align*} &&p_2&=\sqrt{x_1}+\sqrt{x_2}\\ &\Rightarrow&\left({p_2}^2-\sigma_{...
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1answer
42 views

Finding complex roots of polynomial by proving trigonometric identity

Consider the polynomial equation $5z^4-11z^3+16z^2-11z+5=0$, which has four complex roots with modulus one. Let $z=\operatorname{cis}\theta$. (a) Show that $5\cos2\theta-11\cos\theta+8=0$ ...
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0answers
37 views

Find the smallest constant $C$

Find the smallest constant $C$ such that for every polynomial $P(x)$ of degree $3$ that has a root in $[0,1]$, $\int_0^1 \vert P(x)\vert dx\leq C\max_{x\in[0,1]}\vert P(x)\vert$. Here's my ...
3
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2answers
117 views

Number of possible polynomials

Let $a,b,c,d$ be four integers (not necessarily distinct) in the set $\{1,2,3,4,5\}$. Find the number of polynomials of the form $x^4+ ax^3 + bx^2 + cx +d$ which is divisible by $x+1$. My Try: Let $...
2
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1answer
23 views

If $L/K$ is a finite separable extension and $P \in K[X], Q \in L[X]$, then $P = Q^n \Rightarrow Q \in K[X]$

Problem statement: Let $L$ be a finite separable extension of $K$, $P \in K[X]$. Then, show that $P = Q^n$ with $Q\in L[X] \Rightarrow Q \in K[X]$. $P,Q$ are unitary. I think I've shown the result ...
4
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2answers
70 views

Show that $\lambda$ is a repeated root of $p(z)$ if and only if $p(\lambda) = p'(\lambda) = 0$

Suppose $p \in \mathcal{P}\mathbb{(C)}$ and $\lambda \in \mathbb{C}$. Applying the division algorithm, we know there exists $q \in \mathcal{P}\mathbb{(C)}$ and $r \in \mathbb{C}$ such that $$p(z) = ...
0
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0answers
10 views

How can I find a formula to fit a series, of: a value, and a pair of values?

I have a mechanism that can reach a set of x/y co-ordinates. At each x/y point, a certain piece of the mechanism will be at a certain angle. I would like to record the angle of the piece for a number ...
5
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8answers
234 views

An inequality for polynomials with positives coefficients

I have found in my old paper this theorem : Let $a_i>0$ be real numbers and $x,y>0$ then we have : $$(x+y)f\Big(\frac{x^2+y^2}{x+y}\Big)(f(x)+f(y))\geq 2(xf(x)+yf(y))f\Big(\frac{x+y}{2}\Big)...
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0answers
14 views

Nature of roots of self composed polynomial

Suppose there is a polynomial $p(x)$ with degree $n.$ The problem is to tell the nature of the roots of the equation $p(p(x)) = 0$ if the nature of roots of the equation $p(x) = 0$ is given. ...
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1answer
24 views

Maclaurin series solution to Legendre equation and general expression for coefficients

Consider a Maclaurin series solution $$y = (1−x^2)y′′ −2xy′ +α(α+1)y=0, −1<x<1.$$ Show that $$a_2 = \frac{-α(α+1)}{6}a_0$$ $$a_3=\frac{−(α−1)(α+2)}{6}a_1$$ and, for all $n≥2$, $$a_{n+2} = \...
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1answer
42 views

Cayley-Hamilton Theory [problem]

State the Cayley-Hamilton theorem, and verify it for the matrix $$ A = \begin{bmatrix} 6 & 2 & -1\\ -6 & -1 & 2\\ 7 & 2 & -2\\ \end{bmatrix} $$ I got the equation $x^3-5x^2+7x+...
1
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1answer
51 views

How to factorize the quintic polynomial $2x^5 + 6x^4 + 7x^3 + 21x^2 + 5x + 15$? [on hold]

The polynomial is $$2x^5 + 6x^4 + 7x^3 + 21x^2 + 5x + 15$$ I want to find out the easiest way I can do factorize. Please show me the steps.
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1answer
20 views

is there any quick method to solve the determinant to find characteristic equation?? [Linear Algebra] [on hold]

any quick method to find the eqn??you can click and view the image of question.
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1answer
25 views

How do you find the equation of a quartic with 3 inflection points?

I have 3 points where I would like inflection points. $(t0,f0),(t1,f0),(t2,f2)$ The roots give the following equation for the derivate. $y'=(x^3-(t0+t1+t2)*x^2+(t0*t1+t1*t2+t2*t0)*x-(t0*t1*t2))*g$ ...
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0answers
40 views

Polynomial Division of a “Special” Polynomial

Let $f(m)=(2n+1)((2n+1)^2-1^2)((2n+1)^2-3^2)\ldots((2n+1)^2-(2m-3)^2)/(2m-1)!$ for some Positive integers $n,m$ we have to find the coefficients of $t^{1-k}$ quotient on polynomial Division of $$...
1
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0answers
28 views

Integral ring extensions: constructive proof of transitivity

If $R \to S$ and $S \to T$ are integral ring extensions of commutative rings, then it is known that the composite $R \to T$ is an integral ring extension as well. The usual proof reduces to the finite ...
1
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1answer
64 views

Measure theory based definition of general position

This is a soft question. I would like a cursory overview of how the precise notion of a "generic point" came to be in algebraic geometry. I don't know any algebraic geometry, so this maybe a hard ask. ...
1
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1answer
37 views

Find the explicit formula for a polynomial

I have this collection of ten polynomials of the same kind. It is possible, starting from these, to formulate an explicit expression for the n-th polynomial $ P_ {n} $? $P_{0}=1$ $P_{1}=1+X_{1}$ $...
-1
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0answers
15 views

Affine and projective geometry book question nr2. [on hold]

Question: Let $f(x)$ be a polynomial in $x$. Prove that the curve $y=f(x)$ has a singular point at infinity if and only if the degree of $f$ is at least $3$.
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2answers
22 views

Finding the value of $A$ and $B$

The polynomial $f(x) = A(x-1)^2 + B(x+2)^2$ is divided by $x + 1$ and $x - 2$. The remainders are $3$ and $-15$ respectively, I don't really know how to begin this with, help me with the steps
0
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2answers
22 views

Finding the value of k using the factor

Suppose $5x - 2$ is a factor of $x^3 - 3x^2 + kx + 15$. Find $k$. I've tried getting the $x$ value of the factor $5x - 2 = 0$ and got $x= \frac25$ and replaced all the other $x$s with $\frac25$ and ...
0
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2answers
35 views

How to use synthetic division when the denominator is $x^2 + 1$?

If the problem is to write the following with simplified polynomials $$\frac{x^2 + 5x + 6}{x^2+1}$$ Is it possible to do this problem with synthetic division? If so, how? I've tried googling, ...
2
votes
1answer
118 views
+50

Finding a polynomial with $f(i) = a _{i}$ $(i = 1, 2, \dots, n)$ which is monotonic increasing on $[1, n]$

Is there is a positive integer $m$, depending only on $n$, such that for any strictly increasing integer sequence $a _{1}, a _{2}, \dots, a _{n}$, there is some polynomial $f(x)$ of degree at most $m$ ...
1
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1answer
292 views

Polynomial equal to sum of squares of polynomials [duplicate]

Given a nonnegative polynomial $p(x)$ on $\mathbb{R}$, does there exist some $k$ such that for some polynomials $f_1,\ldots ,f_k$ we have $p(x)=\sum_{i=1}^k(f_i)^2$? I think yes, because of the ...
1
vote
0answers
47 views

Show that every nonnegative polynomial $P \in \mathbb R [x]$ can be written as a sum of squares of real polynomials

Define a real polynomial $P \in \mathbb R[x]$ to be nonnegative if $P(x) \geq 0$ for all $x \in \mathbb R$. Show that every nonnegative polynomial $P \in \mathbb R [x]$ can be written as a sum of ...
-1
votes
1answer
35 views

Can $(\frac{a+b}{2})^2+(\frac{a-b}{2})^2+c$ be written as $(\frac{a+b+\cdots}{2})^2+(\frac{a-b+\cdots}{2})^2$ [on hold]

Can $(\frac{a+b}{2})^2+(\frac{a-b}{2})^2+c$ be written as $$(\frac{a+b...}{2})^2+(\frac{a-b...}{2})^2$$ for $a,b,c \neq0$ ? If not are there any special cases?
0
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0answers
8 views

Hermite interpolation vs Hermite polynomial

Is there any connection between Hermite interapolation and Hermite polynomials?
3
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4answers
104 views

Three polynomials which have the same value for a variable

Let $P_1(x)= ax^2-bx-c$, $P_2(x)=bx^2-cx-a$, and $P_3(x)= cx^2-ax-b$ be three quadratic polynomials, where $a,b$, and $c$ are non-zero real numbers. Suppose that there exists a real number $k$ such ...
1
vote
1answer
67 views

Symmetric polynomials for Putnam

I was taking a look at Titu Andreescu's "Putnam and Beyond" book and came across a polynomial issue that involved the knowledge of unusual symmetrical polynomials. The places where I studied the ...