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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How to prove that the evaluation map is a ring homomorphism?

This is a really easy question, but I'm stuck in the logic of it... Let $F$ be an integral domain and $F[x]$ its polynomial ring. Let $a\in F$ fixed, define $\phi: F[x]\to F$ as ...
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4answers
20 views

Find all complex and real roots of higher degree polynomials, given one root

$2+3i$ is a zero of $f(x)=x^4-4x^3+17x^2-16x+52$, find all of the zeros of $f(x)$ thanks!
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How to efficiently check whether two cubics are equivalent

I have a very long list of cubic polynomials in $N$ variables, with $N$ ranging from $2$ to $19$. For my purposes, any two cubics which are related by a rational change of basis in the $N$ variables ...
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$K[x,y]/(xy-1) \cong K[t,t^{-1}]$

In an exercise I found stated that, given a field $K$, $$K[x,y]/(xy-1) \cong K[t,\dfrac{1}{t}],$$ where $K[x,y]$ is the polynomial ring in two variables on $K$ and $( \cdot )$ indicates the generated ...
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45 views

Let a be largest real value of $x$ for which $x^3 - 8x^2 - 2x + 3 = 0.$

Let $a$ be largest real value of $x$ for which $x^3 - 8x^2 - 2x + 3 = 0$. Determine the integer closest to $a^2$. How I tried to do this: This is a third-degree polynomial, thus there are 3 ...
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Find the value of polynomial. [duplicate]

If the value of $x$ is $2+2^{\frac23}+2^{\frac13} $ than what is the value of $x^3-6x^2+6x$ ?
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1answer
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Roots of a complex polynomial with leading coefficient larger than absolute sum of rest

Suppose I have an $N^{\text{th}}$ degree polynomial $P_N(z)=\sum_{i=0}^N a_i z^i$ where $\{a_i\}_{i=0}^N$ are complex numbers such that $|a_N|> \sum_{i=0}^{N-1}|a_i|$, can I claim that all its ...
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Find the value of $f(x)$ for $x = 2 + 2^{2/3} + 2^{1/3}$

If $x = 2 + 2^{2/3} + 2^{1/3}$, then find the value of $f(x)=x^3 - 6x^2 + 6x$. I am unable to get to the answer - end up with more than one term. Please help me solve this!
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24 views

Roots of a quintic function

I need some pointers in the right direction for this question: Three of the roots of the equation $ax^5+bx^4+cx^3+dx^2+ex+f=0$ are $-2$, $2i$ and $1+i$. Find $a$, $b$, $c$, $d$, $e$ and $f$. I ...
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29 views

Linear polynomials relatively prime iff $ad-bc \ne 0$

Two nonzero polynomials $a+bx$ and $c+dx$ are relatively prime in $\mathbb{R}[x]$ if any only if $ad-bc \ne 0$. It's not too hard to show this on a case-by-case basis by enumerating each possible ...
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42 views

Real roots of a quintic polynomial with constraints

This is a question on the edge of math and programming. I pondered about the best way to state the problem: should I provide context, or get straight to the point of the question? Given various ...
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3answers
46 views

Prove that $a+b+1 = 0$

The polynomials $x^2+ax+b$ and $x^2+bx+a$ have common factors.prove that $a+b+1=0$. My attempt- I could do nothing other than dividing the polynomials to get $x^2+bx+a$=$x^2+ax+b+bx-ax+a-b$.Please ...
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27 views

There is a unique polynomial interpolating $f$ and its derivatives

I have questions on a similar topic here, here, and here, but this is a different question. It is well-known that a Hermite interpolation polynomial (where we sample the function and its derivatives ...
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Find bases and coordinates

Let Poly2 denote the vector space of polynomials (with real coefficients) of degree less than 3. Poly2 = {a1t^2+ a2 t+ a3 |a1; a2; a3 €R} You may assume that {1,t; t^2}is a basis for Poly2. (1) ...
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Polynomial Functions/Remainder Theorem Challenge Problems

Please any help would be greatly appreciated! Refer to the photo for the question below. I know how to do it when given only one factor. How should I do it for this case? Thanks in advance for the ...
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1answer
49 views

Intersection of two polynomial ideals

In the 4-dimensional affine space $\mathbb{A}^4$ with coordinates $x,y,z,t$, consider $X$ as the union of the planes $$ X'=\{x=y=0\} $$ and $$ X''=\{z=x-t=0\} $$ (I'm working on a algebraically ...
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1answer
25 views

A question about quadratic polynomials with complex roots.

Let $f(x) =x^2+p^x+q$ be a second degree polynomial, all of whose coefficients are real numbers (but not necessarily real algebraic numbers). If $f(x)$ has no real roots, can the (smallest) field F ...
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3answers
37 views

Need help solving this question (Remainder Theorem)

I know, it's probably an easy question for most of you people, but I really need help and if any one could explain step by step how to do this, that'd be great, Question One: The expression x² + ...
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1answer
29 views

Polynomial and super-symmetric tensor

A quadratic function uniquely determines a symmetric matrix. Ok that’s easy. Now a homogeneous polynomial function $f(x)$ also uniquely determines a super-symmetric tensor. My question is how do I ...
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15 views

Polynomial systems - conditions for real solution

I was working on the computation of equilibrium points for dynamical systems and arrived in the following system of $n$ polynomials in the variables $(x_1,\ldots,x_n)$ \begin{equation*} ...
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Find $f$ such that $f'(x)=ax^2+bx$, given the values of $f'(1)$, $f''(1)$, and $\int_0^2 f(x)\,dx$

The question is : Find the solution $f(x)$ if $f'(x)=ax^2+bx$, and (i) $f'(1)=6$, (ii) $f''(1)=18$, (iii) $\int_0^2 f(x)dx=18$. My solution is: According to $(i)$, we know $6=a+b$, and ...
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1answer
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Rational function in both $k(X)[Y]$ and $k(Y)[X]$

If I have a rational function in $X$ and $Y$ and it can be written as both a polynomial in $Y$ with coefficients being rational functions in $X$ (that is, an element of $k(X)[Y]$) and as a polynomial ...
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1answer
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Lebesgue Measure of the set of roots of a complex exponential equation

In the following equation $\{\beta_i\}_{i=1}^N$ and $\{\alpha_i\}_{i=1}^N$ are non-zero complex numbers: $\sum_{i=1}^N \beta_i e^{\alpha_i t} = 0$. I would like to know if the Lebesgue measure of the ...
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1answer
33 views
+50

Sign of Laguerre root finding iteration

I'm trying to understand the method by Laguerre for polynomial root finding. However, I have some difficulties to understand one sentence of the book Applied Computational Complex Analysis (vol. 1) by ...
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1answer
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Find unknown from Polynomials [on hold]

The polynomial $P(x)= x^4 - 4x^3 + hx^2 - 6x + 2$ has a factor in the form $(x-m)^2$, where $m \in \mathbb{N}$. Find the values of $m$ and $h$.
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Does such a polynomial map always exist?

First question: given $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}=\frac{P_3(f(x_1,x_2,\dots,x_n))}{P_4(f(x_1,x_2,\dots,x_n))}|\det (J(f(x_1,x_2,\dots,x_n)))|$$ where $P_i$ is ...
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Root of polynomial and field extension

I would be really thankful for any input. I am facing following problem. Given extension of finite fields $L/K$ of degree $3$, prove that every polynomial of degree $3$ with coefficients in $K$ does ...
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114 views

Show that every maximal ideal in $ \mathbb{Z}[x, y] $ contains a prime number [on hold]

Let $\mathfrak{M} \subseteq \mathbb{Z}[x, y]$ be a maximal ideal. Show that $ \exists\ p \in \mathbb{Z}$, $p$ prime such $p \in \mathfrak{M}.$ Thanks for the answers. I'd be interested in a proof ...
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Finding a cubic equation from the relation between the roots

I'm trying to solve this problem: $ x^3 - x^2 - 3x -10 = 0$ has roots α,β,γ. Let u = −α+β+γ. Show that u+2α=1, and hence find a cubic equation having roots −α+β+γ, α−β+γ, α+β−γ. I was able to ...
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Prove a sum formula by induction

I am to prove through induction that $$\sum_{k=1}^n (2k-1)^2 = \frac{n(2n-1)(2n+1)}{3}$$ And well, my method seems to be working, but I get stuck when I'm nearly done. First I prove the formula work ...
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If $x^2+5x+p$ and $x^2+3x+q$ have a common factor, then $(p-q)^2=$?

If $f(x) =x^2 + 5x + p$ and $g(x)= x^2 + 3x + q$ have a common factor, then $(p - q)^2$ = ? Source: BMA's Talent & Olympiad Class X Maths Ch.2 Q.23, Second Edition 2007 Given answer: ...
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1answer
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Remainder of Polynomials

A polynomial $P(x)$ of degree $n \geq 2$ has a remainder of $9$ when it is divided by $(x+2)$ and a remainder of $-1$ when it is divided by $(x-3)$. Find the remainder of $P(x)$ when it is divided by ...
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Cubic resolvent of quartic.

Where does the cubic resolvent of quartic come from? I would like to know its derivation since I am having a hard time memorising the formulas.
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1answer
40 views

Writing roots of f(x) as f(a) for some a

I was solving a problem when this random thought popped into my head. Suppose you have a function, say $f(x)=x^2-1=(x-1)(x+1)$. The roots of this function are $-1$ and $-1$. We can write these roots ...
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1answer
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Possible for variable to cancel in a product of multivariate rational expressions?

Let $f,g,p_i,q_i$ be polynomials over some field with $\gcd(p_i,q_i)=1$ and $q_i$ are not constants for $i=1,2$. Assume that one or more of $p_i$ or $q_i$ has a term containing a variable $x$ not ...
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A relation between polynomials

I've been stuck with a proof for a few days and would require any ideas on what information can be extracted from an equality. I have multivariate polynomials F,G,P and I know that they satisfy the ...
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2answers
79 views

Discrete set of zeroes of polynomials must be finite?

Let $F:\mathbb C^n\to\mathbb C^n$ be a polynomial mapping (i.e. $n$ polynomials in $n$ variables). Suppose that $Z = \left\{z \in \mathbb C^n : F(z) = 0\right\}$ is a discrete set (all points are ...
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Is $(\log(n))!$ a polynomially bounded function?

Is the following statement true? How would you prove it? i.e. Is it a polynomially bounded? $$ \lceil \lg(n) \rceil ! \in O(n^k) $$ How about $$ \lceil \lg \lg(n) \rceil ! \in O(n^k) $$ Thanks a ...
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GCD between a polynomial with terms of even degree and a polynomial with terms of odd degree.

We are given a polynomial $p(z)=a_0z^n+b_0z^{n-1}+a_1z^{n-2}+b_1z^{n-3}+\dots=P_1(z)+P_2(z)$, where $P_1(z)=a_0z^n+a_1z^{n-2}+\dots$, $P_2(z)=b_0z^{n-1}+b_1z^{n-3}+\dots$. Let ...
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Rayleigh quotient iteration and root finding

I'm trying to find the roots of a polynomial by finding the eigenvalues of its companion matrix. I understand that it is possible to use QR algorithm as the matrix happens to be in Hessenberg form ...
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2answers
53 views

Proof by contradiction, there is no rational number $r$ [closed]

Use a proof by contradiction to show that there is no rational number $r$. For which $r^3+r+1=0$
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37 views

Relation between the roots and the coefficients of a polynomial

I have studied that: For the polynomial $ax^3+bx^2+cx+d=0$, with roots $\alpha, \beta, \gamma$: We have: $$\begin{align} & \alpha + \beta + \gamma = -\frac ba \\ & \alpha\beta + \beta\gamma ...
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2answers
78 views

Method of dominant balance and perturbation

Approximate the solutions of $$\epsilon x^4 + (x-1)^3=0$$ I can't perform a singular perturbation because if I let $\epsilon=0$ then I lose a root. My professor suggests The Method of Dominant ...
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30 views

O-notation: composing functions

Big-oh and little-oh notation make things much simpler, and there are convenient rules for combining functions, for example, the ones mentioned here. One rule conspicuously missing from the above ...
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Remainder of dividing $x^{137}+x+1$ by $x+5$

In $\mathbb{Z}_7[x]$, what is the remainder of dividing $x^{137}+x+1$ by $x+5$? I can not find how to solve this problem of modular arithmetic. Anybody could tell me only as I proceed to solve this ...
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34 views

Formula for the $n^{th}$ positive integer that is not divisible by 2 or 3.

The first few terms of this sequence are 1,5,7,11,13,17,... The numbers increase by alternating adding 4 or 2. From what I remember from Algebra II, since the second level of differences is constant, ...
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2answers
77 views

How do you maximize a polynomial over an integer domain?

I am working on maximizing the polynomial $f:\mathbb{R}^N \rightarrow \mathbb{R}$ $$f(v):=\prod_{i=1}^{N}( v_i+\alpha_i^2)$$ over integer $n$-partitions of $P$, $n\leq N$: $$\left\{v \left|\, v_i \in ...
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Kernel of homomorphism $A[X] \to B$ between integral domains [duplicate]

Let $A \leq B$ be integral domains, where $A$ is integrally closed and $B/A$ is an integral ring extension. Let further $\varphi : A[X] \to B$ be some homomorphism of $A$-algebras. Is the kernel ...
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36 views

A question about the roots of irreducible polynomials.

Let f(x) be a polynomial of at least the second degree, all of whose coefficients are rational numbers and which is irreducible in the field of rational numbers. Let a+bi be any root of the polynomial ...
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Ordering of Polynomial Derivatives

Take a point $x \in [0,1]$. Now take a polynomial, $p$ of degree $n$ If I know that $\max(p'(x), p''(x),\dots, p^{(n-1)}(x)) = p'(x)$, what can I say about $p(x)$? So far I haven't spent too much ...