# 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.

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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 ...
23 views

### 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 ...
1 vote
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55 views

### 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 $A^{2}$ ...
24 views

### Can a compositional radical function be a polynomial?

Is the following compound function a polynomial? If not, why?
36 views

### 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 ...
1 vote
55 views

### 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 ...
43 views

### 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 ...
24 views

### 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 ...
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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) ...
1 vote
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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 ...
43 views

### 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 ...
130 views

### 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 vote
92 views

### 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: $$P_0 = x^4-4x^2-8x+12$$ $$P_1 = 4x^3-8x-8$$ $$P_2 = 3x^3+2x^2-18$$ $$P_3 = 8x^2/3+8x-16$$ $$P_4 = 60-39x$$...
1 vote
140 views

### 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. ...
1 vote
40 views

### 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 $a = -2 \pm 2i\sqrt{3}$, ...