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 Expansion Proof Estimation

1-(1-x)^n where x is a value between 0 and 1 and n is a large value. This estimates to around x*n. I am having trouble with the polynomial expansion. According to Pascal's triangle, the first few ...
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25 views

Building a cubic polynomial with certain parameters

I am trying to build a cubic polynomial $y = ax^3 + bx^2 + cx + d$ with the following conditions: $y(0) = 3$ $y'(0) = 2$ $y'(1) = 0$ $y(1) = 4$ Unfortunately, solving the required system yields ...
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Partial sum formula of a polynomial series?

I am trying to find the partial sum formula of the following series: $$ \sum_{y=1}^{\infty} \frac{4y^2-12y+9}{(y+3)(y+2)(y+1)y} $$ I have tried using Faulhaber's formula without success. I have also ...
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A necessary and sufficient condition for a polynomial to be univalent in the unit disk

I want to prove: Show that a polynomial $p(z) = z + a_2z^2+\cdots+a_nz^n$ is univalent in $\Bbb D$ if and only if its associated polynomials ...
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1answer
18 views

Proof in Apostol Polynomial Zeros

If $f(a)=0$ and $f$ is a polynomial of degree $n\geq1$, show that $f(x)=(x-a)h(x)$, where $h$ is a polynomial of degree $n-1$. I can't figure this proof out. Will someone help?
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1answer
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Strategy for problems like deciding whether $f(x,y) = x^2-y^2+1$ is irreducible in $F[x,y]$

As title says, let $f(x,y) = x^2-y^2+1$ be a polynomial in 2 variables. Suppose $F = \mathbb{Z}/7\mathbb{Z}$, what can we say about the irreducibility of $f(x,y)$ in $F[x,y]$? My thoughts are: ...
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3answers
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Why do polynomial terms with an even exponent bounce off the x-axis?

Like if it's $f(x) = (x-5)^2(x+6)$ Why, at $x=5$, does the graph reflect off the x-axis?
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When do Leibniz-like rules lead to unique linear operators

Background Usually one defines differentiation in terms of limits, and then shows that differentiation satisfies the Leibniz (product) rule, $$\frac{d}{dx}(f \cdot g) = f\frac{dg}{dx} + g ...
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1answer
15 views

Bounds on the roots of complex polynomial

How does the positive root of the polynomial $|a_n|x^n - \sum\limits_{i=0}^{n-1} |a_i|x^i = 0$ an upper bound on the absolute value of the roots of the polynomial $\sum\limits_{i=0}^n a_ix^i$ ?
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Irreducibility in Rings [on hold]

$f(x,y)=x^{2}-y^{2}+1$. Is it irreducible in $\mathbb{C}$? In $\mathbb{Z}_{7}$? Please include the reason and the result. I have read Eisenstein Criterion and many other theorems. None of them ...
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Divisibility !?? wihout mathematical induction if possible PLZ …

Hello I need to prove if $n\, |\, (x-a)$ and $n \, | \, f(x)$ then $n \, | \, f(a)$. It is true if $\operatorname{deg}(f)=1$ or $2$, but what for greater degree of $f(x)$? I don't know how to ...
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51 views

To make a polynomial with coefficients in a finite field uniform at random

We define the polynomial ring $R[x]$ consist of all polynomial with coefficients from $\mathbb Z_p$. Let $P_1$ be a polynomial such that $P_1 \in R[x]$. The aim is to compute $P_2=P_1 . r$, where ...
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2answers
443 views

What does a complex root signify?

What does it tell me when I find that a polynomial has complex roots, except for the obvious fact that it crosses zero for these values? To me it seems that the existance of complex roots must have ...
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1answer
20 views

Increasing Function or Polynomial with Prescribed Values

Consider $n$ points $(a_1,b_1), (a_2,b_2),\cdots, (a_n,b_n)$ in Euclidean plane with $a_1<a_2<\cdots < a_n$ and $b_1<b_2<\cdots < b_n$. It is easy to construct a polynomial of degree ...
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15 views

Roots of the Lagrange polynomials

This question follows my previous one Coefficients of Lagrange polynomials. Notations : $ n\in\mathbb{N}^*$ $[|1,n|]=\{1,2,\dots,n\}$ $A=(a_1,\dots,a_n)\in\mathbb{K}[X]^n$ all different numbers ...
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Polynomial Irreducibility Test [on hold]

Is there any polynomial with constant c=8 that make this polynomial reducible over field Q but Eisensten Irreducibility Test does not apply?
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1answer
38 views

Uniformly at random polynomial

We have a polynomial of degree $d$, and multiply it by a polynomial whose coefficients are chosen uniformly at random and its degree is equal to or less than $d$. My question is whether the result is ...
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62 views

Prove that the two polynomials intersect each other only at a single point

Here are the polynomials: $$D^K_1(\theta)=\sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i(1-\theta)^{K-i}$$ and $$D^K_2(\theta)=\frac{1}{2}\sum_{i=\lceil{K/2}\rceil}^K ...
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3answers
64 views

Play with a,b,c polynomial [on hold]

I want to get $a^4-b^4-c^4 +2(ab)^2 + 2(ac)^2 +2(bc)^2 = (...)*(...)$ Thank you in Advance !
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1answer
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dimension of quotient space

Let $f(x)=x^4+3x^3-x^2-4x-3$ and $g(x)=3x^3+10x^2+2x-3$ and $U = \{u(x)f(x)+v(x)g(x) | u(x),v(x) \in \mathbb{F}[x]\}$, find the dimension of quotient space $\mathbb{F}[x]/U$ If $V$ is a finite ...
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1answer
37 views

Using the pseudoinverse to find the linear combination of functions?

I'm working out this problem with a friend of mine on a group project and we are both stuck Our professor insists that we do all of our work in Maple. I like Maple, but it's not as great as ...
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Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question: Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order ...
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Location of Roots Symmetric Polynomial

I'm trying to prove (or disprove) that the roots of an even degree real symmetric coefficient polynomial are all on the unit circle. If it is not true, I will then try to find the conditions such that ...
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2answers
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Roots of $x^{4} -28 x^{2}+49$ with Horner

I am studying Horner's algorithm and I got a problem I can't solve. The polynomal is $x^{4} -28 x^{2}+49$. After trying $\pm 1, \pm 7, \pm49$ with Horner I couldn't find any solution. Wolfram alpha ...
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1answer
60 views

Number of (distinct) roots of derivative of polynomial

Let $f(x) = (x-a)(x-b)^3(x-c)^5(x-d)^7$, where $a,b,c,d \in \mathbb{R}$ and $a<b<c<d$. Thus $f(x)$ has 16 real roots counting multiplicities and among them 4 are distinct from each other. ...
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proportion of primes in a polynomial sequence

It is conjectured (Bunyakovsky) that when $P(x)$ is a polynomial from $\mathbb{Z}[X]$, irreducible, with positive leading coefficient and so that the integers $P(n)$ , $n\gt0$ do not share a common ...
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1answer
18 views

Find the size of squares cut from a box.?

This has been taking me days to do and I really want to do it for test practice. I actually have absolutely no idea how to even start this, so if I can get a hint, advice, or something to start me ...
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Solution to Quartic, Pentic, Hexic and Sietic Polynomials? [on hold]

Is there a mistake in this article: Solution to Quartic, Pentic, Hexic and Sietic Polynomials and isn't it in contradiction with Galois theory?
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Basis-free and noncommutative versions of the two-polynomials-over-ring problem (McCoy theorem etc.)

There is a rather canonical bunch of exercises in commutative algebra which tend to come up time and again on math.stackexchange: recently in #948010 and #83121, formerly in #227787 and #413788, and ...
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37 views

Factorizing Cubic Equations.

Factorization of Cubic Equations has always obstructed my way to the solution to a problem. Is there any simple technique to factorize them?
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38 views

If $\deg(f) > p^k$ then $f$ as an irreducible divisor of degree $> k$

Let $p$ be prime and let $f \in \mathbb{F}_p[X]$ with no repeated roots. Let $k \in \mathbb{N}^*$ such that $\deg(f) > p^k$. Show that $f$ has an irreducible divisor of degree $> k$. My ...
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How can I simplify the polynomial $x^4+1$ into quadratic factors? [on hold]

The teacher gave us a hint that this polynomial expression can be written as the multiplication or sum of quadratic factors at the most. How can I do this?
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Show that there exists no integer b such that f(b) is 1993.

We are given a polynomial $f$ with integer coefficients such that for 4 distinct integers $a_1,a_2,a_3$ and $ a_4$, $f(a_1)=f(a_2)=f(a_3)=f(a_4)=1991$. Show that there exists no integer $b$ such that ...
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A Polynomial that Passes through the following four points?

I'm trying to do this for practice but I'm just going nowhere with it, I'd love to see some work and answers on it. Thanks :) Find a polynomial that passes through the points (-2,-1), (-1,7), ...
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Minimal polynomial of f restricted to its image

Let $f:V\to V$ be a $F$-linear map, $V$ an $n$-dimensional vector space over $F$, $\operatorname{rank} E=r$, $W=\operatorname{Im} f$, $\tilde f:=f|_W:W\to W$. Let $\mu$ be the minimal polynomial of ...
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1answer
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Secure Computation and polynomial evaluation

Consider we evaluate a polynomial P of degree d on some points (say 2d+1 points) to obtain Y's. My questions are: A) Given 2d+1 (or more) Y's can anybody 1) recover the original polynomial 2) ...
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1answer
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Finding roots of characteristic polynomial of 3x3 matrix

I have never learned how to solve cubic equations and unfortunately need to do it in an upcoming exam for finding eigenvalues. I have been searching on the web for good resources, but whenever I find ...
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Polynomial satisfying $ P (P (x))=P (x)+ P(x*x)$

If $P(x)$ is a polynomial with integer coefficients such that for all integer $x$, $$P (P (x)) = P (x)+P (x*x).$$ I've tried solving it putting it as a function instead. Not much though. How do you ...
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The min degree of polynomials of two variables with a special form

Let $f(x)$, $u(x,y)$ and $v(x,y)$ be non-constant polynomials over complex number field $\mathbb{C}$. Assume that $u(x,y)$ is not a polynomial only on $y$, and $v(x,y)$ is not a polynomials of only ...
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How do generating function created from solution of system of polynomials

What are the examples of generating function derived from solution of system of polynomials? how to count the number of points in varieties which are solution of system of polynomials? From Wiki, ...
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Question about multiple solutions to a polynomial

Assume that $f(X,Y,Z,V,W)\in \mathbb{Z}[X,Y,Z,V,W]$ is some polynomial and assume that $f(x,y,z,v,w)=0$. I would like to know if there is some way to figure out if there are non-trivial constants in ...
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Special representation of polynomial

How to prove that for natural $n$ the polynomial $(x^4-6x^2+1)^n$ can't be represented in such a way $$ (x^4-6x^2+1)^n=f(x)^2+1, (x^4-6x^2+1)^n=g(x)^3-1, $$ where$f(x), g(x)$ are polynomials. ...
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1answer
200 views

Coefficients of Lagrange polynomials

Let $n\in\mathbb{N}^*,A=(a_1,\dots,a_n)\in\mathbb{K}[X]^n$ all different numbers and $B=(b_1,...,b_n)\in\mathbb{K}[X]^n$ all different numbers. Let $L_{A,B}$ be the polynomial of degree $n-1$ ...
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1answer
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0-1 roots in a free algebra

Let $\mathbb{F}$ be a field and consider the free algebra $\mathbb{F}\langle x_1,\ldots, x_n \rangle $, that is, the algebra of non-commutative polynomials with coefficients from $\mathbb{F}$. Let ...
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LCM of two polynmials when they are represented as point-value.

I`m wondering if we can obtain least common multiple of two polynomial when each polynomial represented as point-value. To be more clear, can we do any computation on these point-values and obtain ...
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Irrational roots of unity?

Is it possible to take irrational roots of unity? For example, say I wanted to solve $f(x)=(x+1)^{\sqrt{2}}=1$. I found that one solution is the obvious $x=0$, and another one can be written nicely as ...
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Degree of the zero polynomial

In “Linear Algebra Done Right” by Axler, while defining the degree of a polynomial, it is stated that the zero polynomial is said to have degree $- \infty$. Why is this so?
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Solving equations: reasoning doesn't work backwards?

In doing my (high school) math homework, I came to an issue that doesn't make sense to me. Given an equation $0 = a_1 + a_2x + a_3x^2 + \dots$, we can multiply both sides by $x$ to obtain $0 = a_1x + ...
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Evaluate each algebraic expression at the indicated values of the variables. Answer problem below showing work [closed]

Evaluate each algebraic expression: $4x-x^2=$ Simplify by combining like terms for the problem below: $(16pq-7p^2) + (5pq+5p^2)$
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1answer
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A positive polynomial is the sum of two squares in $\mathbb{R}[X]$ [duplicate]

Let $P\in\mathbb{R}[X]$ be a positive polynomial. I want to show that there exists $A,B\in\mathbb{R}[X]$ so that $P=A^2+B^2$ $\displaystyle ...