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 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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1answer
60 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
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
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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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1answer
48 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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2answers
47 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 $x$?
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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 ...
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1answer
20 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 ...
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0answers
28 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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4answers
350 views

How to prove $f\equiv 0$ without Weierstrass theorem?

Let $\,f:[0,1] \to \mathbb{R}$ continuous. Show that: If $$\int_0 ^1 x^k f(x)\, dx=0,$$ for all $k\in\mathbb N$, then $f\equiv 0$. I know that it can be proved using Weierstrass Theorem, ...
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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 ...
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3answers
75 views

Characterizing kernel of ring morphism

Let $K$ be a field and define a ring morphism $$\psi: K[x_1,x_2, \dots , x_n, y_1, y_2, \dots , y_n] \rightarrow K(x_1,x_2, \dots , x_n)$$ by $\psi(x_i) =x_i$ and $\psi(y_i) =\frac{1}{x_i}$. I ...
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0answers
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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 ...
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0answers
54 views

Solving a system of polynomial equations in three variables ($x^2-yz=18$, $y^2-zx=8$, $z^2-xy=-7$)

I am looking for a way to solve the following system of polynomial equations in three variables: $\begin{align*} x^2-yz&=18\\ y^2-zx&=8\\ z^2-xy&=-7 \end{align*} $ I've tried ...
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1answer
23 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
38 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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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 ...
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0answers
49 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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0answers
52 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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4answers
75 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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0answers
21 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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1answer
85 views

A polynomial agreeing with a function and its derivatives

If we want $$p(x_i)=a_i, \qquad x_1 < \dotsb < x_{n+1},$$ then there is a unique polynomial of degree $\leq n$ that accomplishes this (Lagrange interpolation). If we want $$p(x_i)=a_i, \qquad ...
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2answers
80 views

Modified Hermite interpolation

I asked similar questions here and here, but I tried to formulate this one in a sharper way. Is anyone aware of some literature on polynomial interpolation where we sample the function and its ...
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0answers
14 views

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
31 views

Finding polynomial optimal in terms of least squares approximation

Find polynomial $w$ of degree at most $2$ optimal in terms least squares approximation for a function $f(x)=x^3$ in the norm $\|g\|=\sqrt{(g,g)}$, given that: $$ (f,g) = \int\limits^1_0 f(x)g(x)dx. ...
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1answer
265 views

Families of Polynomials Irreducible in $\mathbb{Z}$ but reducible in $\mathbb{Z}/p\mathbb{Z}$ for all primes $p$.

I am wondering if there exist classification of polynomials that are irreducible in $ \mathbb{Z}$ but reducible $\pmod p$ for all primes $p$. I am aware that $\Phi_n$ has this property if ...
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0answers
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0
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1answer
47 views

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
23 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 ...
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4answers
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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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2answers
350 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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1answer
594 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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2answers
30 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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1answer
28 views

Estimate the difference between $f$ and $p$ interpolating $f$

Suppose $p$ is the unique polynomial of degree $\leq 2$ that agrees with a function $f$ at points $a_1 < a_2 < a_3$. If the third derivative $f^{(3)}$ exists, and $x\in (a_1,a_3)$, then we can ...
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1answer
34 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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0answers
36 views

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
19 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 ...
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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!
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1answer
295 views

Rank of a rectangular Vandermonde Matrix to which weighted columns are added

A Vandermonde matrix: $\left(\begin{array}{ccc} 1 & \alpha_{0} & \dots & \alpha_{0}^{n} \\ 1 & \alpha_{1} & \dots & \alpha_{1}^{n} \\ \vdots & \vdots & \ddots & ...
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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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1answer
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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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1answer
54 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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3answers
47 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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3answers
61 views

complex roots calulation question

How can we find the roots of an equation such as:$z^2 +z +1=0 ,z \in \mathbb{C} $ ?
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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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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) ...
2
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2answers
85 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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3answers
39 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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Conjecture: Tract version of Gauss--Lucas Theorem for higher derivatives.

The Gauss--Lucas Theorem states that all zeros of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros of ...