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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Solving for Polynomials in a Differential Equation

Consider the following equation. P1(x) f'(x) + P2(x) f(x) = P3(x) P1(x) is a polynomial of degree m, P2(x) degree n, and P3(x) degree j. f(x) is a power series of degree N. This series is given: ...
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1answer
48 views

If one root is unsolvable, can there be a solvable root?

Suppose you have some polynomial $p(x)$ with rational coefficients in which at least one root is unsolvable by radicals, does this imply that all other roots of $p(x)$ are unsolvable by radicals? ...
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1answer
77 views

An operation with respect to which the set of prime numbers is closed

Like every (semi-)decidable set of natural numbers the set $P$ of prime numbers is diophantine, i.e. there are two polynomials $p(x)$, $q$ with natural coefficients and exponents – the first of ...
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Equation Theory: A polynomial with specific remainders when divided by specific divisors. What is the remainder when divided by BOTH divisors

Again, for my Equation Theory class, I have the subject question.$p(x)$ has a remainder of 3 when divided by $x-1$ and a remainder of 5 when divided by $x-3$. What is the remainder when $p(x)$ is ...
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How to find a polynomial with $f(1), f(4),f(9)$ prime and coefficients in $\{1,2,3…10\}$?

How to find a polynomial with $f(1), f(4),f(9)$ prime and coefficients in $\{1,2,3...10\}$? I can't think of any way on how to generate such types of polynomials? Also, would they have a minimum ...
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What is the condition for roots of conjugate reciprocal polynomials to be on the unit circle?

Given an Nth order complex polynomial $P(z) = \sum\limits_{n=0}^N a_nz^n$ such that $a_n = a^*_{N-n}$ i.e. conjugate reciprocal, Lakatos and Losonczi mention that a necessary and sufficient condition ...
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33 views

Zero of a polynomial and factors

If $(a-b)$ and $(a+b)$ are the zeroes of the polynomial $f(x)=2x^3-6x^2+5x-7$ find the value of $a$ and $b$. I solved this problem considering ${x-(a-b)} , {x-(a+b)}$ but I could not evaluate the ...
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2answers
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Motivational example for complex numbers

Years ago I was introduced to complex numbers. In class we had been talking about the cubic polynomial and its solutions. At one point we saw an example where, when using the formula, one had to stop ...
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334 views

Graphing: Given two points on a graph, find the logarithmic function that passes through both.

Is there such a method to do this? I would like to come up with a logarithmic function (a graph that looks like a square root graph) that passes through two given points. Haven't had any luck in ...
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54 views

Sign of the derivatives of a simple function

Consider the function $f(x)=x^b(1-x)^{1-b}$ defined on $[0,1]$, with $0 < b <1$. How can we prove that the even derivatives $f^{(2k)}$ have a constant sign on $(0,1)$? One can show that this ...
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111 views
+50

Existence of rational sequence such that a polynomial is split over $\Bbb{Q}$

Does there exist a sequence $(a_n)_{n\in \Bbb{N}}$ of rationals such that for all $n\in \Bbb{N}$, $a_n\neq 0$ and the polynomial $a_0+a_1X+\cdots+a_nX^n$ is split over $\Bbb{Q}$? I was asked this ...
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4answers
82 views

Polynomials that satisfy $(x-1)(p(x+1))=(x+2)(p(x))$ where $p(2)=12$?

I am taking a graduate class on Equation Theory and one of my homework questions asks me to "Determine all polynomials $p(x)$ such that $(x-1)(p(x+1))=(x+2)(p(x))$ and $p(2)=12$. A provided hint is to ...
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76 views

Finding all such polynomials under a gcd condition

Find all such polynomial $f(x)\in \mathbb{Z}[x]$ such that $$ \forall n\in \mathbb{N} \quad \gcd(f(n),f(2^n))=1$$ This is a problem from the Indian Team Selection Test. Can someone give me a solution ...
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258 views

A polynomial determined by two values

From a St. Petersburg school olympiad, 11th grade. Prove or disprove: a non constant polynomial $P$ with non-negative integer coefficients is uniquely determined by its values $P(2)$ and $P(P(2))$.
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32 views

Formula alteration

is there any way to transform the formula$ \frac {1-x}{x-3}$ into something that can be easily sketched, or which will help eliminate $x$ from the denominator?
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168 views

Can this result be simplified?

I was trying to solve the value of $x$ in $x^4-4x^2-2 = 0$ in terms of radical. The answer I got is $x=\sqrt{2+\sqrt{6}}$. How can this value be simplified even more, while still expressing it in ...
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27 views

AES polynomial binary division

Could some one just explain me how the binary division of this polynomial evalutes to the mentioned ans ? $$x^{13} + x^{11} + x^{9} + x^8 + x^6 + x^5 + x^4 + x^3 +1 \pmod{x^8 + x^4 + x^3 + x +1} ...
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2answers
39 views

Trigonometric solution of a 6 degree polynomial

How do i prove that $\sin^2 \frac{\pi}{13}$ is a root of the equation $$2^{12}.x^6-13(2^{10}.x^5-5.2^8.x^4+3.2^8.x^3-7.2^5.x^2+7.2^2x-1)$$? Any hints/answers would be appreciated.
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187 views

Let $I$ be an ideal generated by a polynomial in $\mathbb Q[x]$. When is $\mathbb Q[x] / I$ a field?

I was looking at my old exam papers and I was stuck on the following problem: Let $I_1$ be the ideal generated by $x^4+3x^2+2$ and $I_2$ be the ideal generated by $x^3+1$ in $\mathbb Q[x]$. If ...
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84 views

Polynomial P(x) such that [on hold]

Let $P(x)$ be a real polynomial with degree $n$ such that $|P(x)| \lt 1$ for all $|x| \le 1$. Prove that $P(2) \lt 4^n$.
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310 views

Checking if a System of Polynomial Equations is Consistent

I'm trying to determine whether any solutions exist to a system of $(n+1)$ polynomial equations in $n$ unknowns. For example: $$ \begin{align*} xy&=-2\\ x^2-1&=0\\ y^3-3y^2+2y&=0 ...
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Version of Chevalley's Theorem proving $\mathbb{F}_q$ is a $C_1$-field for $\mathbb{Z}/p^n\mathbb{Z}$?

Let $k=\mathbb{F}_q$ Chevalley's theorem states that if $f(x_i)\in k[x_1,...,x_n]$ is such that $f(0,...,0)=0$ and $deg(f)<n$, then there is a non-trivial $a_i\in k^n$ such that $f(a_i)=0$. Is ...
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3answers
53 views

Show that the complex closed line integral $\oint\frac{\mathrm{d}z}{p(z)}$ is $0$ ($p$ is polynomial)

Let $p$ be a polynomial of degree $n\geq2$ and has $n$ different roots $z_1,\dots,z_n$. Prove that $\oint\frac{\mathrm{d}z}{p(z)}=0$ where the closed path is large enough so that all roots are in the ...
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1answer
146 views

An integral and $\pi(n)$

Are there polynomials $P,Q\in \mathbb{R}[x]$ satisfying : $$\int_{0}^{\log n}\frac{P(x)}{Q(x)}\,\mathrm{d}x=\frac{n}{\pi(n)}\quad \text{ for infinitely many }n\in \mathbb{N}$$ Here $\pi(n)$ is the ...
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21 views

A nonlinear system of equation

In the real numbder set: $x,y,z$ are variable, $a_i,b_i,c_i,d_i$ is given ($i\in\{1,2,3\}$) What is the conditions for the following equations have solutions? $$a_1xy+b_1x+c_1y=d_1$$ ...
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394 views

Polynomial bounds?

Q1: Is the function $$\lceil{\lg n}\rceil!$$ polynomial bounded? Q2: Is the function $$\lceil{\lg\lg n}\rceil!$$ polynomially bounded? $$\lg = \log_2$$ Polynomially bounded: $f(n)$ is polynomially ...
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Descartes rule of signs extension

Let $V(\text{sequence})$ be the number of sign changes in the sequence, e.g. $V(-3,0,-2,9,0,1)=1$. Show that $V(a_0,a_1,...,a_n)\ge V(a_0,a_0+a_1,a_0+a_1+a_2,...)$. Furthermore, prove that if ...
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Using Descarte’s rule of signs to determine the number of positive roots.

Using the Descarte’s rule of signs to determine the number of positive roots. \begin{equation} f(q)=[(k_f+k_d+k_p*(1-q))(\lambda_b* \gamma ...
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zeros of a simple polynomial

For $x, y \in GF[2^n]$, consider the two-parameter polynomial $P(x,y) = x \cdot y + f(x) + g(y)$, where $f$ and $g$ are arbitrary polynomials on $GF[2^n]$. Can we say anything about the number of ...
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+250

How many pairs of nilpotent, commuting matrices are there in $M_n(\mathbb{F}_q)$?

As a follow-up to this question, I've been doing some work counting pairs of commuting, nilpotent, $n\times n$ matrices over $\mathbb{F}_q$. So far, I believe that for $n=2$, there are $q^3+q^2-q$ ...
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How to solve the polynomial equation $\sum_{i=1}^{i=m} \frac{l_i}{l_i - x} = n$?

Let m and n be strictly positive integers, and a set of m real positive numbers $$l_{i, i \in \{ 1, m \}}.$$ I want to solve numerically: $$\sum_{i=1}^{i=m} \frac{l_i}{l_i - x} = n$$ finding the m ...
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1answer
18 views

Switching dependent for independent variable in polynomial

I've got this expression f(y) = $ax^2 + bnx + cn^2$ A basic polynomial. given that i know the coefficients abc and that I can keep n const too, I have been trying to switching x and y since I ...
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1answer
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Kantorovich Theorem example

I need to write in C a program that finds roots of a 6th order polynomial. I was thinking of using Kantorovich Theorem convergence of Newton's method to find when can I use Newton-Rhapson method. I'm ...
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Polynomial -definition-non-negative degrees

What is the rationale that the degree of a polynomial is non-negative? Can the degree be a fractional number. why the definition is only with the non-negative integers We are bounded by the ...
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1answer
26 views

Find the polynomial function

Anybody knows how to find the polynomial function with evaluated values, where if the degree is $n$ I have $n+1$ values of the function like $f(0) = a_0, f(1) = a_1, \ldots, f(n) = a_n$.
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1answer
163 views

What's the name of this set

Let's take field $(K, +,\times,0,1)$. Let's take set $S(K) \subset K$ such that: \[ \forall~e \in K,\exists~a \in S(K),\exists~p \in K: e = a \times p \times p \] \[ \lnot \exists~a, b \in S(K), ...
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213 views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
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Is there a computationally efficient way to find the part of a vector, which is of certain order in independent variable x?

Let $\vec{a}$ be an element of a vector space over the space of monomials, i.e. $$ \vec{a}\left(x\right)=\sum_{j=1}^{N}a_jx^{k_{j}}\vec{e_{j}} $$ Remark: For simplicity, here we operate with only ...
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Fast way to find the smallest root $\mod M$ of a polynomial

Suppose you're given a polynomial of degree $d$ with integer coefficients: $$ P(x) = \sum_{i=0}^{d}{a_i x^i} $$ Is there a fast way to find the smallest root modulo $M$, where $M$ is some composite ...
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2answers
44 views

Prove an inequality (Using Taylor expansion)

Prove: $\frac{x}{1+x} < \ln(1+x) < x$. I thought a good practice would be to prove it using Taylor Expansion. Here's my try: $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$ The n=1 ...
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5answers
411 views

Polynomials Shouldn't Have factors using Rational Root Theorem but it does!

I came across this polynomial $X^4 + X^3 + 2X^2 + X + 1$ I tried to factor it using Rational root theorem, but it seems there are no roots possible. 1 or -1 don't work. But I know for a fact ...
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1answer
20 views

Relation between the roots of a cubic equation and the coefficients

$ax^3 +bx^2 + cx + d= 0$ If the roots are $\alpha$ $\beta$ and $\gamma$, Is there any relationship between the sum of the squares of the roots and the coefficients of the quadratic equation.. In ...
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Finding double root of $x^5-x+\alpha$

Given the polynomial $$x^5-x+\alpha$$ Find a value of $\alpha>0$ for which the above polynomial has a double root. Here's an animated plot of the roots as you change $\alpha$ from $0$ to $1$ I'm ...
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50 views

Representing a higher degree polynomial as product of smaller degree polynomials?

Consider an equation $H(Z)=1+\frac 52Z^{-1}+2Z^{-2}+2Z^{-3}$ I want to write it as a product of a first degree polynomial and another polynomial, which will be...$$H(Z)=(1+2Z^{-1})(1+\frac ...
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2answers
77 views

Most Efficient Method to Find Roots of Polynomial [duplicate]

I am designing a software that has to find the roots of polynomials. I have to write this software from scratch as opposed to using an already existing library due to company instructions. I currently ...
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85 views

How to Show Polynomial Growth < Exponential Growth (Without L'Hopital!)

Can anyone offer me a way to show that exponential growth trumps polynomial growth, without using L'Hopital's Rule? When I learned function growth speeds in high school, the closest thing to a proof I ...
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30 views

AMM Polynomial equation

Solve the equation: $x^7+7px^5+14p^2x^3+7p^3x+q=0$ I've tried obvious things like factorization or maybe guessing a solution. I'd appreciate a solution not too far from high school level.
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423 views

geometric interpretation of quadratic equation with complex coefficients

When an equation has real coefficients and non-negative discriminant, the geometric meaning of it's roots is intersection of the function with the x-axis. I know how to get roots of quadratic ...
3
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3answers
108 views

Basis in the vector space of all polynomials

Let $V$ vector space of all polynomials $p(t) = a_0 + a_1t + \cdots + a_nt^n$,$\forall n \in\mathbb{N}$ and $a_0,\ldots,a_n \in\mathbb{R}$. How can I prove that $ \gamma = \{1,t,t^2,\ldots\}$ is a ...
2
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1answer
287 views

Define the Intersection Points of Polynomials

I am facing the following problem. Let’s consider that there are 2 points that are not known. $${(x_0,y_0) (x_1,y_1)}$$ I know that from these 2 unknown points a set of quadratics passes ...