All Questions
Tagged with binomial-coefficients polynomials
188
questions
2
votes
2
answers
781
views
For what values of $n$ give a constant term when $\left(\frac{1}{x^2}+x\right)^n$ is expanded? Also, what is this constant term(in terms of $n$)?
Here's my question:
What values/restrictions of $n$ give a constant term in the expansion of $$\left(\frac{1}{x^2}+x\right)^n$$? Also, for the expansions that do have a constant term, what is this ...
- 2,846
1
vote
1
answer
148
views
Finding pattern in a sequence of polynomials
So I have a set of polynomials, with variables $n_0, n_1, n_2$. I would like to figure out the general formula for these polynomials given the number.
$$f(2) = n_{0} + 2 n_{1} + n_{2}$$
$$f(3) = n_{...
- 1,799
0
votes
1
answer
92
views
Eisenbud 1.21a $G(n):= F(n)-F(n-1)$, given $G(n)$ is integer-valued, then so is $F(n)$.
This question is based on a subproblem from 1.21 (a) from Eisenbud's commutative algebra
Given $F: \mathbb Z \to \mathbb Z$, we then define $G(n):= F(n)-F(n-1)$. Is it then true that:
$G(n)$ is ...
user459879
1
vote
1
answer
40
views
Is there a sub-quadratic method of calculating the expanded polynomial from its roots?
If I have the roots of a polynomial, $r_1, r_2,\ldots, r_n$ is there a fast way (hopefully sub-quadratic) to find the expanded polynomial (or equivalently its coefficients)?
I know of Vieta's ...
- 31
4
votes
3
answers
225
views
Prove that $x^2+px+p^2$ is a factor $(x+p)^n-x^n-p^n$, if $n$ be odd and not divisible by $3$.
Question:
Prove that $x^2+px+p^2$ is a factor of $(x+p)^n-x^n-p^n$, if $n$ is odd and is not divisible by $3$.
My approach:
$$(x+p)^n-x^n-p^n=\sum_{r=0}^n\limits {n\choose r} x^{n-r}p^r-x^n-p^n$$ What ...
- 730
9
votes
3
answers
415
views
An interesting problem of polynomials
In the polynomial
$$
(x-1)(x^2-2)(x^3-3) \ldots (x^{11}-11)
$$
what is the coefficient of $x^{60}$?
I've been trying to solve this question since a long time but I couldn't. I don't know whether ...
- 1,135
4
votes
2
answers
462
views
Show that a polynomial vanishes
When doing the expansion of the following polynomial expression, the result is zero.
\begin{align*}
&- 1 (-1 - 2 x) (0 - 2 x) (1 - 2 x) (2 - 2 x) \\
&+5 (-1 - 1 x) (0 - 1 x) (1 - 1 x) (2 - 1 x)...
- 3,716
1
vote
2
answers
551
views
How to find the coefficient of $x^3y^5$ in the expression $(1+xy+y^2)^n$
We are given the expression $(1+xy+y^2)^n$ where $n$ is positive integer. We are required to find the coefficient of $x^3y^5$ in the expansion of the given expression.
I know how to expand a binom to ...
- 1,409
1
vote
1
answer
145
views
could someone please give some geometric explanation about $(a+b)^4$?
This is Visualisation of binomial expansion up to the 4th power
could someone please give some geometric explanation about $(a+b)^4$, does that move the cube a length of distance? and then compute ...
- 1,454
2
votes
0
answers
168
views
Polynomial Division of a "Special" Polynomial
Let $$f(m)=(2n+1)((2n+1)^2-1^2)((2n+1)^2-3^2)\ldots((2n+1)^2-(2m-3)^2)/(2m-1)!$$
for some Positive integers $n,m$
we have to find the coefficients of $t^{1-k}$ quotient on polynomial Division of
$...
- 39
10
votes
0
answers
331
views
Expanding a product of linear combinations with coefficients $1$ and $-1$
For any odd natural number $n$, denote $t \equiv \frac{n-1}{2}$. Let $K$ be a field such that $\operatorname{char} K \neq 2$. Working over the polynomial ring $K\left[x_1,x_2,...,x_{n} \right]$, ...
- 124
-1
votes
1
answer
39
views
Coefficient of an expansion
Find the coefficient of $x^k$ in $(x+a)(x+b)...(x+n)$ where $a$, $b$ and $n$ are integers.
I am not able to approach this problem.
- 524
0
votes
0
answers
91
views
Is there a generalization of Faulhaber's formula is sense of summation bounds?
Well known Faulhaber's formula is stated as
$$\sum _{k=1}^{n}k^{p}=\frac{1}{p+1}\sum _{j=0}^{p}{p+1 \choose j}B_{j}n^{p+1-j}$$
But is there any formula which holds for power sum of the form
$$\sum_{...
- 558
1
vote
3
answers
450
views
Coefficients of polynomial $(x+1)(x+2)...(x+n)$
I was trying to calculate the integral
$$
I(m,n)=\int_0^\infty\frac{x^me^x}{(1-e^x)^n}\mathrm{d}x
$$
It's possible to evaluate this in closed form by using the zeta function, for example:
$$
I(m,4)=m!(...
- 2,148
1
vote
3
answers
165
views
Prove that $\sum_{X=0}^N u(X) {N \choose X} p^X (1-p)^{N-X}=0 \iff u(X)=0, \space \forall X\in\{ 0,1,...,N \}$
I am trying to prove that $Bin(N,p)$ where $N$ is fixed is a complete distribution.
Thus my goal is to show
$$E[u(X)]=0 \iff u(X)=0$$
While I was attempting to prove this I have noticed that
$$\...
- 5,175