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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Finding this Polynomial Subspace

Let $A = k[x^{\pm 1}, y^{\pm 1} ] $, considered as a $k$ - algebra. Can someone give me a nice description of the (vector) subspace: $$ A_0 = \lbrace (f,g) \in A^2 : \frac{ \partial f}{\partial y} = ...
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How prove such $f(x)g(x)$ coefficient is $1$ or $-1$

show that: there are exsit two polynomial $f(x),g(x)$ with the coefficient is integer,and every polynomial coefficient at least one more than $2014$,and $f(x)g(x)$ coefficient is $1$ or $-1$ and ...
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Is it possible to calculate the roots of the difference between successive terms of this polynomial series $\rm{P}_n (x)=x\rm{P}_{n-1}-r\rm{P}_{n-2}$

Consider the polynomial series defined by the following recursion formula: $$ \begin{align} &\mathrm{P}_0 = 1 \\ &\mathrm{P}_1 = x-r \\ &\mathrm{P}_n = x\mathrm{P}_{n-1} - ...
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Please help me to reduce this equation $6xy + 8 y^2 -12x-26y + 11 = 0$ to canonical one of a second-order curve

I have this polynom $$ 6xy + 8 y^2 -12x-26y + 11 = 0 $$ and I need to reduce it to a canonical equation of a second-order curve. The correct answer from the textbook is that it is a hyperbola $$ ...
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Solve the following equation: $\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$

Solve the following equation: $$\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$$ I know it's from a Math Olympiad but I don't know which and I couldn't find it on the internet. Expanding everything ...
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48 views

How to find a basis for a linear space of polynomials?

Questions such like, (1) Let $V = P^4$ be the vector space of all real valued polynomials of degree less than or equal to four. Let $W =\{p(x)\in P^3 |p(−2)=p(2)\}$. Find the basis for $W$ (2) Let ...
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How to prove that a polynomial with leading coefficient $1$ has no fractional solutions

I want to prove that the equation $$x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots+a_0=0$$ has no solutions in the form of $p/q$ when $p$ and $q$ and coprime and $q>1$. With this polynomial, $a_n=1$ and ...
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24 views

Change of basis from falling powers to powers for polynomials up to degree $n$

Notice that $$(1, x, x^{\underline{2}}, x^{\underline{3}}, \dots)$$ and $$(1, x, x^2, x^3, \dots) $$ both are bases of $\mathbb{R}[x]$ (where $x^{\underline{n}}$ is the falling power). Now suppose the ...
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43 views

Is the non-existence of a general quintic formula related to the impossibility of constructing the geometric median for five points?

In particular, in the Computation section of in the Wikipedia page for geometric median, there is this statement: ...but no such formula is known for the geometric median, and it has been shown ...
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Transformation matrix of a polynomial

I would really appretiate some help about the following transformation matrices. We have to write a tranformation matrix in basis $B = \{ 1 + x, x + x^2, x^2 \}$ with a polynomial $(Ap)(x) = (x^2 - ...
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Galois group of a splitting field

$f=x^4-5x^2+6 \in \operatorname Q[x]$ $f=(x^2-2)(x^2-3)$ $\operatorname F=\operatorname Q(\sqrt[2]{2},\sqrt[2]{3})$ $f$ is irreducible for Eisenstein's criterion, $\operatorname{char} \operatorname ...
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24 views

Galois group of the splitting field of $ x^6 - 5$

$f = x^6 - 5$ $\in \operatorname Q [x]$ I want to find a splitting field $\operatorname F$ of $f$ over $ \operatorname Q$ $\sqrt[6]{5}$ is a real root of $f$ $u$ is the 6-th root of unity then $ ...
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2answers
21 views

Taylor series and differentiation

The Taylor series about 0 for the function $$f(x) = \left(\frac14 + x\right) ^ {-\frac{3}{2}}$$ is $$f(x) = 8 - 48x+240x^2-1120x^3 + \dots$$ for $-\frac{1}4 < x <\frac14$ Use differentiation ...
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Finding the underlying model from data

Let's say I have "k" data points in "n" dimensions. I also have "k" results of an unknown function of this data. Is it possible, simply given the data and the results from the function, to guess the ...
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31 views

Determining the intersection of an arbitrary number of polynomials.

Say $f_1,f_2,\dots,f_k\in k[x_1,x_2,\dots,x_n]$, where $k$ is a field. Clearly, every $f_i$ is a polynomial. How does one represent the intersection of all these polynomials? In other words, I want ...
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Finding the leading exponent of a binary number

Let's say that the binary representation of a number $k$ is $2^{X_n} + 2^{X_{n-1}} + \dots + 2^{X_0}$ with each term in this polynomial having a $1$ or $0$ multiplied to it (I just haven't showed them ...
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Representing multivariable polynomials as matrices

Is there a nice way to represent polynomials in $x$ and $y$ of degree say $n$ as matrices, so that multiplication works out in a nice way? Maybe a ring homomorphism or something? I'm sorry that this ...
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29 views

Is the intersection of polynomials in $n$ degrees a polynomial? [on hold]

I have been told that the intersection of two polynomials is a polynomial. Given two polynomials in $n$ variables, how do I represent their intersection? I initially thought it would be $f-g$. ...
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70 views

Always “one double root” between “no root” and “at least one root” ? (Second version)

Let $a<b$ be two real numbers. Let $f(x,y)$ be a bivariate polynomial. Suppose that $f(x,.)$ has no real roots in the interval $[a,b]$ when $x<0$, but has at least one real root in the interval ...
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Always a double root between “no roots” and “at least one root”?

Let $f(x,y)$ be a real bivariate polynomial. Suppose that $f(x,.)$ has no real roots when $x<0$, but has at least one real root when $x>0$. Does it automatically follow that $f(0,.)$ has a ...
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Explicit expression for polynomial recurrence relation.

I am (as here) working on Machin formula and with some works I'd like to explore the following polynomial recurrence relation : $$P_0 = 1, P_1 =b,\quad P_{n+2} = XP_{n+1} - P_n$$ It seems relation ...
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Euclidean division - For what values of a, does the polynomial g(t) get divided by f(t) in the complex ring

They want to find the values of a where g(t) can be divided by f(t). $f(t) = t^2 + it − ai$ $g(t) = t^4 + (1 − i)t^3 + (1 − 2i)t^2 − 3at − (4 + 2i)a$ Euclidean algorithm: $g(t) = f(t)q(t) + ...
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183 views

Roots of polynomial of degree 6

I'm struggling to find the complex roots of $x^6-9x^3+8 = 0$. I've managed to find the real roots (1 and 2) by letting a variable, say $α = x^3$ and substituting where relevant, leading to a ...
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Polynomial of degree 5 divisible by other polynomials.

I need some help with a problem. Find a polynomial $f(x)$ of degree $5$ such that both of these properties hold: $f(x)-1$ is divisible by $(x-1)^3$. $f(x)$ is divisible by $x^3$. I can't seem to ...
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219 views

Polynomial $f(x)$ degree problem.

Suppose the polynomial $f(x)$ is of degree $3$ and satisfies $f(3)=2$, $f(4)=4$, $f(5)=-3$, and $f(6)=8$. Determine the value of $f(0)$. How would I solve this problem? It seems quite complicated... ...
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How to solve the inequality: $\prod_{k=1}^N\left(x^k-k^2\right)\gt0$

Given the inequality: $$\displaystyle\prod_{k=1}^N\left(x^k-k^2\right)\gt0$$ how can I solve it? I suppose there is a difference if $N=2n$ or $N=2n+1$ with $n\in\mathbb{N}$, but I'm unable to find a ...
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179 views

A question about algebraically closed fields

A field $\mathbb{K}$ is said to be algebraically closed in practice if every polynomial over $\mathbb{K}$ of positive degree less than or equal to $10^{10}$ has zero belonging $\mathbb{K}$. The ...
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57 views

prove $\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$

Show that if $a,b,c,d \geq 0$ and $ab+bc+cd+da=1$ :$$\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$$ yet again it should be solved with Cauchy inequality. thing i have done so far: ...
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How to prove $\sum_{i=1}^k(\frac{1}{\alpha_i}\prod_{j\neq i}^k\frac{\alpha_j}{\alpha_j-\alpha_i})=\sum_{i=1}^k\frac{1}{\alpha_i}$?

How to prove $\sum_{i=1}^k(\frac{1}{\alpha_i}\prod_{j\neq i}^k\frac{\alpha_j}{\alpha_j-\alpha_i})=\sum_{i=1}^k\frac{1}{\alpha_i}$? Where $\alpha_1, \alpha_2,\ldots, \alpha_k$ are $k$ distinct ...
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Polynomial Division - “Define the largest natural number…” [closed]

Would someone mind helping me with this question? The more detailed possible so I can have 100% of understanding. Thanks. Question: Define the largest natural number m such that the polynomial ...
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27 views

Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
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51 views

Please help me with this problem on binomial expansion [closed]

Please simplify this expression $\frac{n-r+1!}{n-r-1!}$ I have read the textbook and seen examples but I still do not understand.
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67 views

poles of a polynomial

What are the poles of a polynomial? Are they the same as the roots?
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25 views

polynomial factorization and equation solving

Given a polynomial equation: $$x^4+Ax^3+(B+C+D)x^2+(AB+AC)x^2+BD=0$$ where $A$, $B$, $C$, $D$ are known. Numerically I know it has complex solutions. However, I tried but failed to analytically ...
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Polynomial representation

Why is the polynomial $P(x)$ represented as $$ P(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots+ a_2 x^2 + a_1 x + a_0 \text{ ?}$$ A polynomial can be $5x^4 + 3x^3 + 7x^2 + 10x -2$ and it ...
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How do you solve part (b) to this polynomial interpolation question?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the polynomial interpolation method. (b) Assuming $f(x)\in C^3$, evaluate ...
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$\forall x \,\exists k$ s.t. $f^{(k)}(x)=0$, then $f$ is a polynomial

My friend sent me the following problem: Suppose that $f$ is real analytic on $(a,b)$, and that for all $x$ in $(a,b)$ there exists a non-negative integer $k$ such that $f^{(k)}(x)=0$. Show ...
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Product of Chebyshev polynomials of the second kind?

So Wikipedia has this formula for a product of two Chebyshev polynomials of the second kind evaluated at a fixed $x$ with different indices: $$ U_n(x)U_m(x)=\sum_{k=o}^{n}U_{m-n+2k}(x) $$ Which would ...
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A Question about the Proof of Eisenstein's Irreducibity Criterion

Statement: Let $f(x) = a_n x^n + a_{n-1} x^{n-1}+ \cdots + a_0 \in \mathbb Z[x]$. If there is a prime $p$ such that $p \nmid a_n, p \mid a_{n-1}, \dots,p \mid a_0$ and $p^2 \nmid a_0 $, then $f(x)$ ...
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primitivity of a polynomial over a field

Suppose that we have a polynomial $f(x)=ax^3-bx^2+cx-d\in\mathbb{Z}[x]$ then $f(x)$ will be called primitive if $(a,b,c,d)=1$ I have been told that over a field $F$ there is no notion of ...
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2answers
44 views

How prove that: $\left | a \right |+\left | b \right |\leq 5$ for $P(x)=x^{3}+ax^{2}+bx+1$?

Let $P(x)=x^{3}+ax^{2}+bx+1$ and $\left | P(x) \right |\leq 1$ for all x such that $\left | x \right |\leq 1$. How prove that: $\left | a \right |+\left | b \right |\leq 5$?
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Find scaling factor that minimizes f(x) - round(f(x))?

Let's say I have a function $f(x)$, which has a fractional component $\{ f(x) \} = f(x) - \lfloor f(x) \rfloor$. I would like to add a scaling factor $h(x)$, where $h(x)$ is a polynomial, such that ...
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Whether $\sum_{i=1}^k\frac{\prod_{j\neq i}(\alpha_j-\beta)}{\prod_{j\neq i}(\alpha_j-\alpha_i)}=1$ is true

Suppose we have k positive numbers: $\alpha_1, \alpha_2, ..., \alpha_k$, for any number $\beta>0$, is $$\sum_{i=1}^k\frac{\prod_{j\neq i}(\alpha_j-\beta)}{\prod_{j\neq i}(\alpha_j-\alpha_i)}=1$$ ...
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Differentiability of polynomials

Trivial question but I am confused with the notation If $p_{n-1}$ is a polynomial of degree $n-1$, is it $\in$ the differentiability class C^n$? Obviously if $p_n$ is a polynomial of degree $n$, ...
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31 views

A question from the mod p irreducibility test's proof

Let $p$ be a prime an suppose that $f(x) \in \mathbb Z[x]$ with $\deg f(x) \geq 1$. Let $f_1(x)$ be the polynomial in $\mathbb Z_p[x]$ obtained from $f(x)$ by reducing all the coefficients of $f(x)$ ...
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1answer
14 views

Multivariate Polynomials Sage

Sorry if I'm in the wrong Stackexchange (but sage is a math program...) I'm computing something on multivariate polynomials: I have a primary variable $x$ and several other variables $a, b, c, ...
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52 views

If $a^2=b^2+c^2$ and $0<n<2$ prove $a^n<b^n+c^n$

If $a^2=b^2+c^2$ and $a,b,c$ are positive real numbers, prove (a) if $n>2$ then $a^n>b^n+c^n$, (b) if $0<n<2$ then $a^n<b^n+c^n$. Part (a) was easy to prove: $a^2=b^2+c^2$ and ...
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21 views

Find a basis and state its dimension of a $C$-vector space polynomial.

The $C$ vector space $V$ of polynomials $P(t) \in C[t]$ of degree at most $n$ and such that $P(a) = P'(a) = 0$ for $a \in C$ fixed. Indication : prove that $P(t) \in V \Leftrightarrow (t − a)^2$ ...
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Changing the order of the elements of the divided difference Polynomial Interpolation

Apparently this is rather trivial but I don't understand why what I've highlighted in green is correct.
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115 views

roots of a polynomial inside a circle

I am asked to show that for $n$ larger or equal to $2,$ the roots of $1 + z + z^{n}$ lie inside the circle $\|z\| = 1 + \frac{1}{n-1}$ Attempt1: Induction for the case $n = 2,$ the roots of $1 + z + ...