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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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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 ...
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 $σ_2 = s^2_1 −2s-2.$...
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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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 ...
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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 $n_i$ & $s_i$, ...
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 ...
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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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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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 ...
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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 $5\cos2\theta-11\cos\theta+8=0$ ...
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 ...
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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 $3$.
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
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 ...
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, ...
118 views
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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 $m$ ...
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 ...
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 ...
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?
8 views

Hermite interpolation vs Hermite polynomial

Is there any connection between Hermite interapolation and Hermite polynomials?
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 ...