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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Polynomial derivative from evaluated points

If we evaluate a polynomial on some points, denoted by x1,x2,..xn, to obtain some value denoted by y1,y2,..yn. Can we obtain dth derivative of this polynomial by knowing only the values of y1,...yn ? ...
3
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2answers
73 views

Prove that $\prod_{k=0}^{n-1}(z-\mathrm{e}^{2k\pi i/n})=z^n-1$

Prove that $$ \prod_{k=0}^{n-1}(z-\mathrm{e}^{2k\pi i/n})=z^n-1. $$ In my some problem I have used $$ \prod_{k=0}^{7}(z-\mathrm{e}^{2k\pi i/8})=z^8-1. $$ I have verified this. So I think in general ...
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3answers
72 views

Why is the polynomial $f(x)=x^3+x^2+x+1$ monotonic?

I have to argue why the polynomial $f(x)=x^3+x^2+x+1$ has a reverse function $f^{-1}$ which is defined in on the whole of $\mathbb R$. I'm certain the argument would simply be that because $f(x)$ is ...
0
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3answers
55 views

Proving a function has real roots

I am not interested in finding roots but interested in proving that the function has real roots. Suppose a function $f(x) = x^2 - 1$ This function obviously has real roots. $x = {-1, 1}$ How could ...
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1answer
12 views

Decompose the set of polynomials on $[0,1]$ in two subsets each not separating measures

It is well-known by the Stone-Weierstrass theorem that if we consider the set of finite measures $\mathcal{M}_f([0,1])$ on $[0,1]$ the following is true for $\mu_1, \mu_2 \in \mathcal{M}_f([0,1])$: ...
3
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1answer
68 views

Does there exist a finite set of polynomials which do not have roots over any prime field?

The polynomial $x^2 + 1$ has a root in $Z_p$ if and only if $p \not\equiv 3 \mod 4$, and the polynomial $x^2 + x + 1$ has a root in $Z_p$ if and only if $p \not\equiv 2 \mod 3$. So each of the ...
0
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2answers
29 views

Stuck finding the zeros of a polynomial (complex and real)

Stuck finding the zeros of this polynomial (complex and real): $$x^4+2x^2+1$$ I am not sure how I would factor this. The constant value is really throwing me off. I just need a hint on how to get ...
0
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1answer
44 views

A formula for polynomial derivative

Does the following elementary result have a name (or a reference to)? Given a field $K$, and a polynomial $P(x) \in K[x]$, divide the polynomial $P(x) - P(y)$ by $(x - y)$ in $K[x][y]$: $P(x) - P(y) ...
2
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1answer
74 views

Prove that $f(x)=x^4+8x^3+x^2+2x+5$ is irreducible in $\mathbb Q[x]$

Prove that $f(x)=x^4+8x^3+x^2+2x+5$ is irreducible in $\mathbb Q[x]$. I've tried many methods: Eiseinstein's criterion doesn't apply here. I've tried to project the polynomial over $\mathbb ...
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3answers
36 views

How to solve this polynomial word problem?

It's probably easy for a lot of you, just a question in my book I couldn't understand, can anyone explain how will we do this step by step. Thanks! Given that $2x^2 + 3px -2q$ and $x^2 + q$ have a ...
0
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1answer
46 views

How to solve the quadratic matrix equation

Given $\mathbf{A}$ and $\mathbf{B}$ two $m \times n$ real matrices, is there a closed form for the matrix equation \begin{equation} \|\mathbf{X}\|^{2}_{F} - 2\cdot trace(\mathbf{X}^T\mathbf{A}) ...
0
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3answers
64 views

Rearranging the polynomial $x^3-23x^2+142x-120$ prior to factoring it

In the example 15: They are saying that, $$x^3-23x^2+142x-120 = x^3-x^2-22x^2+22x+120x-120$$ From where did $22x^2$ and $22x$ come and also $120x$. Please help me clear my confusion.
0
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4answers
44 views

The divisibility of the values of quadratic polynomials in $x$, for integer $x$

I would like to know method of finding validity of the statement by proofs. 1) $8$ does not divides $x^2 - 7$ for any integral value of $x$? 2) For any odd integer $x;$ the term $(x-1)^2$ is always ...
0
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0answers
17 views

Solving a quadratic system of equations for a single variable

I have a quadratic system of $n$ equations that looks like: $$ (A_{j}^{i}y + B_{j}^{i})x_{j}=0 $$ For $i=0...n$. $A_{i,j}$ and $B_{ij}$ are integer matrices and sums over $j$ are implied. $j$ runs ...
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5answers
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Is it true that $n^2+3n+13$ is prime for all $n\in\mathbb ℤ^+$?

Prove or disprove the statement: If $n\in\mathbb ℤ^+$, then $n^2+3n+13$ is prime. I am lost here. All I know is that $n$ is greater than or equal to one, since it is a positive integer.
0
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1answer
14 views

Interpolating Polynomial

I need help with this. Find a polynomial of degree 4 of the form f(x) = ax4 + bx3 + cx2 + dx + e Plot points (1, 7),(2, 2),(3, 9),(5, 1), and (7, 5). f(x)=? Thank you
4
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0answers
63 views

Writing the roots of a polynomial with varying coefficients as continuous functions?

Consider the monic polynomial $$p_{\zeta}(z) = z^n + a_{n-1}(\zeta)z^{n-1} + \dots + a_0(\zeta), $$ where the $a_{i}$'s are continuous functions defined over $\mathbb{C}$. As is well known, the ...
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2answers
51 views

How to solve for X in Cubic poynomial

I've been given a Polynomial (Cubic) $$k=\frac16x\cdot(x+1)\cdot(2x+1)$$ If $k$ is given, is there any way to solve for $x$?
3
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2answers
30 views

Existence and uniqueness of weights for the rule $\int_a^b f(x) \ = \ \sum_{0 \leq k \leq n} w_k f(x_k)$

I want to establish this statement: If $a<b$ and $\{x_0,x_1, \cdots x_n\} \subset \mathbb{R}$ distinct, then there is one and only one set of weights $\{w_0, \cdots w_n \} $ such that $\int_a^b ...
0
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1answer
25 views

do all real polynomials include constant poly?

Do all real polys include constant polys? or 0 ? and I just want to make sure is integral of 0 always 0? I think it is yes for both questions that I mentioned above. I just want to double check ...
0
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0answers
34 views

Coefficients of polynomials.

Let $F=\sum\limits_{i=0}^m a_i x^i, G=\sum\limits_{i=0}^n b_i x^i$ be polynomials such that $a_i,b_i$ are $1$ or $100$ and $F$ divides $G$. Prove (or disprove) that all coefficients of the ...
6
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1answer
74 views

Coincidence? : $d(ax^2+bx+c)/dx=\pm \sqrt{\Delta}$

As the title says, is it just a coincidence that $d(ax^2+bx+c)/dx=\pm \sqrt{\Delta}$? (where $\Delta=b^2-4ac$, i.e. discriminant of the quadratic). We can get this easily from rearranging the ...
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1answer
26 views

SAGE: Is it possible to extract the irreducible factor of a polynomial for the purpose of constructing a Number Field?

I'm in the middle of making a program that tests a certain fact for many number fields. At this current step I get say a hundred polynomials, which are reducible. I want to factor them (over Q), take ...
1
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1answer
29 views

Find the max volume using polynomials with the sum of the height and perimeter less than 100cm

I have to find out which shape of packaging for a fragile object has the most volume to fit the object and styrofoam packing. The sum of the height and the perimeter must be less than 100cm. There is ...
0
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1answer
13 views

How can I calculate a polynomial trend line where `y` always increases as `x` increases?

Assume the following given coordinates: [0,0],[12,200],[24,2000]. The following equation generates a second order polynomial trend line (at least that's what excel ...
0
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29 views

Functions commute with a given polynomial

Given a polynomial $f(x)\in \mathbb{C}[x]$,how to find(describe) functions(smooth or continuous or polynomial) that are commute(under composition) with $f(x)$? There are trvial ones :$x,f,f\circ ...
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54 views

Relation between Galois theory and Fermat primes

I am curious about a possible relation between Galois theory and Fermat primes. There is a general solution to any polynomial equation of degree less than or equal to $4$. The only Fermat primes (of ...
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1answer
42 views

Best program for multiplying many multivariable polynomials

I want to make a table with two columns: The first column will consist of many(possibly hundreds) of polynomials in two variables. The second column will be a function applied to all of the ...
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0answers
26 views

Find roots of this third-degree polynomial given some information. [duplicate]

I have this polynomial: $x^3−8x^2−2x+3=0$ The question is asking: Let $a$ be the largest real value of $x$. Find the integer closest to $a^2$. I took this polynomial's derivative, and ended up with ...
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Improved Gröbner basis algorithm

I'm just learning about Gröbner bases and the Buchberger algorithm. I have seen chapters in several pieces of literature that deal with improving the Buchberger algorithm, but they never seem to ...
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1answer
28 views

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 ...
0
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4answers
31 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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29 views

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 ...
0
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2answers
33 views

$K[x,y]/(xy-1) \cong K[t,t^{-1}]$

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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4answers
80 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 $x^3-6x^2+6x$ ?
2
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1answer
21 views

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 ...
2
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2answers
79 views

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!
0
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1answer
34 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 ...
0
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0answers
31 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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0answers
62 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 ...
2
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3answers
55 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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1answer
55 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
66 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
35 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
40 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
35 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 ...
0
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0answers
20 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*} ...