Questions tagged [polynomials]
For both basic and advanced questions on polynomials in any number of variables, including, but not limited to solving for roots, factoring, and checking for irreducibility.
26,178
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Polynomial degrees
I have this very trivial question, but I think I might have interpreted it incorrectly.
The expression shown below is a polynomial of what degree?
$$x^3 {(x+\frac{1} {x})} {(1 + \frac {1} {x+1} + \...
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Flatness of $\mathbb{C}[x_1,\ldots,x_n]$ over $\mathbb{C}[f]$, $f \in \mathbb{C}[x_1,\ldots,x_n]$, $n \geq 1$
Let $f \in \mathbb{C}[x_1,\ldots,x_n]$, $n \geq 1$.
Call $f$ 'good' if $\mathbb{C}[x_1,\ldots,x_n]$ is flat over $\mathbb{C}[f]$.
Is it true that every $f \in \mathbb{C}[x_1,\ldots,x_n]$ is good?
$f ...
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Is the set of polynomials of odd degree dense in $\mathcal{C}^{0}([a,b])$?
Let $$E=\{\sum\limits_{k=0}^{n}a_{k}\,x^{2k+1}\, | \, n\in\mathbb{N}\cup\{0\}\}$$ be the set of polynomials of odd degree in each term defined on $[1,2]$.
(a) Show that $E$ is not closed in $\mathcal{...
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How to prove $\sum_{n=1}^{\infty}(-1)^na_n$ is conditionally convergent?
Let $f_n(x)=x^n+nx-1$, let $a_n$ denote its unique positive root. Then prove $\sum_{n=1}^{\infty}(-1)^na_n$ is conditionally convergent. Below is my solution.
First, $$a_n(a_n^{n-1}+n)=1,$$
because $...
- 417
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2
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Finding roots of cubic equation by factorisation
I want to find solutions of a cubic equation which is in $x$. I know a cubic equation will have three roots. Consider this cubic equation
$(a-xb)[(c-xd)(e-fx)-g]=0$
Here $a,b,c,d,e,f,g$ are constants. ...
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How do I write a polynomial function with given characteristics?
Write the equation of a polynomial function with the given characteristics. Use a leading coefficient of 1 or -1 and make the degree of the function as small as possible.
Crosses the x-axis at -4, 0, ...
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simpler functional form for polynomial sequence related to the Bessel polynomials
Does this set of polynomials have a name, or how can we find a closed-form in terms of special functions? It's seemingly related to the Bessel polynomials
$$B(k,x)=\overset{k}{\underset{n=0}{\sum}}\! \...
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Is $x^6 + bx^3 + b^2$ irreducible?
Let $b\in \mathbb{Q}^*$ be rational number. We factorise $x^9-b^3\in \mathbb{Q}[x]$ and obtain $$x^9-b^3=(x^3-b)(x^6+bx^3+b^2).$$
Is the polynomial $x^6+bx^3+b^2$ irreducible?
If $b=1$ we get a ...
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Product between matrix-polynomial and vector
I was wondering if it is possible to optimize the evalutaion of the product of a matrix polynomial and a vector.
$$ \vec{y} = \left( \sum_{i=0}^{n}a_iM^i \right)\vec{x}$$
Matrix size is maybe ...
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Solving $f'(x)g(x)-f(x)g'(x)=c \in \mathbb{C}-\{0\}$
Let $c \in \mathbb{C}$.
How to find a general form of $f=f(x),g=g(x) \in \mathbb{C}[x]$ that satisfy $f'(x)g(x)-f(x)g'(x)=c$?
I think I can solve this algebraically by writing $f=a_nx^n+\cdots+a_1x+...
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A doubt regarding the extension of Weierstrass factorization theorem
According to the Weierstrass factorization theorem, the sine function can be represented as a product of its factors:
$$\sin(x)-0=(x) \left(1 - \frac{x}{\pi} \right) \left(1 + \frac{x}{\pi} \right) \...
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Generalization of Eisenstein's Criterion [duplicate]
Let $f(X)=a_{2n+1}X^{2n+1}+\ldots+a_0\in \mathbb{Z}[X]$ with
$$\begin{align*}
a_{2n+1}&\not \equiv 0 \pmod p\\
a_{2n},\ldots,a_{n+1} &\equiv 0 \pmod p\\
a_n,\ldots,a_0&\equiv 0 \pmod{p^2} ...
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How to calculate the min and max points of an ellipse
The question is inspired by: Solve the equation $3x^2+2y^2-4xy+x-2=0$ in the set of integer numbers..
The first way to solve the equation is by solving the quadratic equation for $x$ - $x_{1,2}=\frac{...
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Finding the polynomial with roots equal to the square of the roots of another polynomial
Let's consider the polynomial:
$$a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+...a_{n-1}x+a_{n}=0 ...(i)$$
if $A$ solves $(i)$ i.e.
$$a_{0}A^{n}+a_{1}A^{n-1}+a_{2}A^{n-2}+...a_{n-1}A+a_{n}=0$$
then $A^{2}$ ...
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Can a compositional radical function be a polynomial?
Is the following compound function a polynomial? If not, why?
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Generators of $I$ in $\mathbb{Z}[X,Y]$
Let $p$ be prime and $(a,b)\in \mathbb{Z}^2$. Prove $I=\{f(X,Y)| f(a,b)\equiv 0 \mod p\}$ can be generated by three explicit elements.
Hilbert's Theorem tells me that $I$ must really be finitely ...
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Show that $a(\gcd (n,k))$ is generated from roots of generating function of $\sum _{h=0}^{\infty } \left(\sum _{k=1}^n x^{h n+k} a(\gcd (n,k))\right)$
Let $a(n)$ be the Dirichlet inverse of the Euler totient function:
$$a(n) = \sum\limits_{d|n} d \cdot \mu(d) \tag{1}$$
And let the matrix $T$ be:
$$T(n,k)=a(\gcd(n,k)) \tag{2}$$
Compute the ordinary ...
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Determine x,y if they are integers number [closed]
Determine x,y who belongs to Z(integers number) if xy+x+y=3 and (x+1)(y+1)=4
I tried to make an equation system ,to equalise them , but I couldn't bring them to a ' easier form".
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Solve the equation $3x^2+2y^2-4xy+x-2=0$ in the set of integer numbers.
QUESTION
Solve the equation $3x^2+2y^2-4xy+x-2=0$ in the set of integer numbers.
MY IDEA
I decided to write it as:
$2x^2+2y^2-4xy+x^2+x-2=0={(x\sqrt{2}+y\sqrt{2})}^2+x^2+x-2$
I was thinking of ...
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Show that the below polynomial has a root in [0,1] without using Rolle's theorem [duplicate]
Show that the polynomial $p(x)=a_0+a_1 x+a_2x^2+\cdots+ a_nx^n$
has a zero in $[0,1]$ when it is given that,
$$\frac{a_1}{2}+\frac{a_2}{6}+\cdots+ \frac{a_n}{(n+1)(n+2)}=0.$$
I know there exists an ...
2
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If $S$ is a unital commutative ring, and $R$ is a subring with unity $1_R = 1_S$, is it the case that $R[x]$ is an $S$-algebra?
If $S$ is a commutative ring, and $R$ is a subring with unity $1_R = 1_S$, is it the case that $R[x]$ is an $S$-algebra?
I don´t see why this would be true; I asked GPT-$3.5$ and it gave me $f:S \to R[...
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If $R$ is a unital commutative ring, then the polynomial ring $R[x]$ is an $R$-algebra.
Assume we have a unital commutative ring $R$ and a polynomial ring $R[x]$. How is the ring-homomorphism $$f:R \to R[x]$$ explicitly defined, such that $$f(R) \subset Z(R[x])?$$ That is, how can we ...
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Multiplicities of roots of a smooth family of polynomials
Let $p_x \left( z \right)$ be a smooth family of complex polynomials in $z$, smoothly parametrized by $x$ in a connected open subset $U \subseteq \mathbb R^n$. Suppose that the roots of $p_x$ do not ...
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Reed-solomon coding: question on proof of minimum distance
In the "coefficient view" of Reed-Solomon encoding, the message is interpreted to be coefficients of a polynomial m(x). The code word is $c(x) = m(x)*g(x)$ where $g(x)$ is a generator ...
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How to find the remainder when a large degree polynomial is divided by a cubic polynomial with repeated roots.
I was solving a question in which we had to find, as a step, the remainder when $x^{32}$ was divided by $x^3-x^2-x+1$. at first I tried to find it by using the remainder theorem in parts, once for ...
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prove that if $\lambda$ is a root of the cubic equation $x^3 + ax^2 + bx + c = 0$ (real or complex), then $| \lambda | \leq 1.$
Let $a,b,c$ be three real numbers such that $1 \geq a \geq b \geq c \geq 0$. prove that if $\lambda$ is a root of the cubic equation $x^3 + ax^2 + bx + c = 0$ (real or complex), then $| \lambda | \leq ...
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Proof the Presence of a Constant (Either $1$ or $-1$) in Chebyshev Polynomials of the First Kind ($T_n$) for Even Values of $n$
The Chebyshev Polynomials of the first kind are given as follows:
\begin{align*}
T_0(x) & = \mathbf{1} \\
T_1(x) & = x \\
T_2(x) & = 2x^2 - \mathbf{1} \\
T_3(x) & = 4x^3 - 3x \\
T_4(x) ...
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Can a rational function have infinitely many vertical asymptotes?
My first approach to this problem is to make a rational function that has infinitely many distinct real roots or zeros at distinct values of $a_1,\dots,a_i$ for $n$ approaching infinity
$$\frac{p(x)}{...
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Parity of the output of a polynomial
$n \epsilon Z+$
For $n=1,2,3...$
$f(n)=(1/2)(n^2-n)$ produces an output of 2 odd integers followed by 2 even integers, $(2O2E)$
$g(n)=(1/6)(-n^3+6n^2+1)$ produces an output of 3 odd integers ...
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Reduce Polynomial Over Real Numbers
I was given the question $x^8 + 16$ and told to reduce it as much as able over the real numbers.
Here is what I tried
$x^8 + 16$
$(x^4+4)^2-8x^4$
$(x^4+4-2^{3/2}x^2)(x^4+4+2^{3/2}x^2)$
I can not ...
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functional equation polynomial with real roots
if $f(f(x))=0$ has atleast one real root,and there be a real root $a$ of $f(x)$,is it necessary for the equation $f(x)=a$ to have atleast one real solution/root.if so why?
my contention is if $f(a)=0$,...
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Is $\cap_{c \in \mathbb{C}} \langle x_1-c \rangle=0$ in $\mathbb{C}[x_1,\ldots,x_n]$, $n \geq 2$?
It is well-known that the Jacobson radical, the intersection of all maximal ideals, in $\mathbb{C}[x_1,\ldots,x_n]$ is zero, $n \geq 1$, see this.
In particular, in $\mathbb{C}[x_1]$ we have $\cap_{a \...
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Corollaries of algebraic independence
The numbers
$$\pi,e^{\pi},\Gamma (1/4)$$
are algebraically independent over $\mathbb{Q}$. It is a corollary of the fact that
$$\frac{3}{\pi},3\frac{\Gamma (1/4)^8}{(2\pi)^6},e^{-2\pi}$$
are ...
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How to prove that the real solution to this equation is greater than 1 without solving it? $x^3 - x^2 +2x - 4 =0$
I was attempting a problem online and reached a point where I needed to prove that the real solution to this equation is greater than 1 but couldn't seem to find a way to do it. Please advise!
If you ...
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Inequality for a complex polynomial
Let $p(z), q(z)$, and $r(z)$ be polynomials with complex coefficients in the complex plane. Suppose that $|p(z)|+|q(z)| \leq|r(z)|$ for every $z$. Show that there exist two complex numbers $a, b$ such ...
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$f(x,y,z) := y^2z + yz^2-x^3+xz^2$ is irrreducible in $\mathbb{Z}[x,y,z]$?
Let $f(x,y,z) := y^2z + yz^2-x^3+xz^2.$ Then $f$ is irreducible in $\mathbb{Z}[x,y,z]$ so that $I:=(f)$ is a prime ideal of $\mathbb{Z}[x,y,z]$?
I think that this is my first time of seeing a problem-...
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How to prove $V$ is a linear space and find $\dim V$?
Let $f(x)$ be a cubic real polynomial with a leading coefficient of $1$.
Let $\alpha$ be all the complex roots of $f(x)$.
$$V=\{y\,|\,y=g(\alpha),g(x) \text{ is a real polynomial}\}$$
Prove that $V$ ...
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Does sequence of monomials form a Frame in $L^2[0,1]$?
A sequence $(x_n)_{n \in \mathbb{N}}$ is said to be Frame, if there exist constants $A,B > 0$
such that
$$ A \lVert x \rVert^2 \leq \sum_{n=0}^{\infty} \lvert (x,x_n) \rvert^2 \leq B \lVert x \...
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General formula for $\sum_{k=0}^n k^a \binom{n}{k}$ is somehow hypergeometric? [duplicate]
Original question is a duplicate of this question. Please refer to after the edit
Original question:
I was investigating formulas of the form
$\sum_{k=0}^n k^a \binom{n}{k}$ after noticing on ...
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1
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Uniqueness of interpolation for distinct positive real numbers by non-negative coefficients $x_i$ and $\sum_{i=1}^n x_i =1$
Let $a_1,a_2,\dots,a_n>0$ be distinct positive real numbers and let $x_1,x_2,\dots,x_n \ge 0$ be non-negative real coefficients such that $\sum_{i=1}^n x_i = 1$.
Is it possible to find another set ...
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Prove that $m^2+n^2+1\geq m+n+mn$ where $m,n\in\mathbb {R}$ [duplicate]
I have an interesting task, I need to prove following inequality by creating brackets in which the value of the variables will not play a role, here is the task: $m^2 + n^2 + 1 \geq m + n + mn$
I got ...
1
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3
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92
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How many roots of $x^4-4x^2-8x+12$ lie in the range $[-2,2]$
I tried to use the Sturm Theorem and computed the Sturm sequence as the following:
$$
P_0 = x^4-4x^2-8x+12
$$
$$
P_1 = 4x^3-8x-8
$$
$$
P_2 = 3x^3+2x^2-18
$$
$$
P_3 = 8x^2/3+8x-16
$$
$$
P_4 = 60-39x
$$...
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A polynomial $a_0x^n+a_1x^{n-1}+…+a_{n-1}x+a_n$,$a_i\in\{-1,1\}$ has all its roots real, find maximum degree possible and number of such polynomials.
For polynomials of the form $$a_0x^n+a_1x^{n-1}+…+a_{n-1}x+a_n$$ where $a_i\in\{-1,1\}$ for $i=0,1…n$ having all its roots real, find the maximum degree possible as well as number of such polynomials.
...
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Solving a polynomial equation using substitution
The equation is:
$x^4 + 4x^2 + 16 = 0$
I tried solving it by substitution and then using the quadratic formula:
$x^2 = a$
$a^2 + 4a + 16 = 0$
using the quadratic formula I got $a = -2 \pm 2i\sqrt{3}$,
...
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0
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Finding regular and singular roots of a cubic perturbed polynomial using rescaling
Question: Find the rescalings for the roots of
$$\epsilon^5 x^3 - (3 - 2\epsilon^2 + 10\epsilon^5 - \epsilon^6)x^2 + (30 - 3\epsilon -20 \epsilon^2 + 2\epsilon^3 + 24\epsilon^5 - 2\epsilon^6 - 2\...
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2
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Finding the matrix representation of a transform of a polynomial space
I am having trouble with this problem from Sergei Triel's Linear Algebra Done Wrong, problem, problem 3.3c:
3.3. For each linear transformation below find [its] matrix
...
c) $T : \mathbb{P}_n → \...
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0
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Develop perturbation solutions of a cubic polynomial
Question: Develop perturbation solutions to
$$x^3 + (3+4\epsilon + \epsilon^2)x^2 + (3 + 9\epsilon + 7\epsilon^2 + 2\epsilon^3)x + 1 + 5\epsilon + 8\epsilon^2 + 5\epsilon^3 + \epsilon^4 = 0$$
finding ...
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1
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Others abbreviated calculation formulas...
I started sloving an algebric problem and i wonder if we can write $x^2+y^2+z^2$ or $a^2+b^2+c^2+2a+2b+2c+3$ as a product of terms.
By product of terms i think of writing does terms as a product:
Ex: $...
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0
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Interpolation, advantages and disadvantages
So during our numerics course we learned a few interpolation methods Aitken/Neville, divided differences,Lagrange and the Vandermonde matrix. How these work is clear to me for the most part, I'm just ...
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1
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Approximating $\frac{1}{(1+x)^2}$ in $L^2$ norm with polynomial of degree $1$
I'm currently looking at an exercise where I'm supposed to get the best approximation of $ \frac{1}{1+x^2}$ regarding $ ||f||= \sqrt{\int_{-1}^{1}f(x)^2dx}$ with a polynomial $p(x)=ax+b$. So my idea ...
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