All Questions
12
questions
3
votes
5
answers
218
views
Coefficient of $x^{21}$ in $(1+x+x^2+\dots+x^{10})^4$
Find the coefficient of $x^{21}$ in $(1+x+x^2+\dots+x^{10})^4$
I tried splitting the terms inside the bracket into two parts $1+x+\dots+x^9$ and $x^{10}$, and then tried binomial theorem, but that ...
- 565
2
votes
2
answers
108
views
A general formula for $\mathcal{F}_{n} = \prod_{i=1}^n (a_ix + b_iy)$
So I am trying to simplify following product, $$\mathcal{F}_{n} = \prod_{i=1}^n \left(a_ix + b_iy\right)$$ in terms of products and summation. This is what I have come up with so far.
We see that for ...
- 141
7
votes
3
answers
275
views
Proof of a neat pattern in polynomials
Let $f_1:\mathbb{R}\to\mathbb{R}$ such that $$f(x) = ax + b\space\space \forall\space x \in \mathbb{R}$$
It can be easily verified that $$f(x)-2f(x-1)+f(x-2)=0 \space \forall \space x \in
\mathbb{R}--...
- 271
-1
votes
1
answer
77
views
What is the coefficient of $x^{11}$ in $(x+x^2+x^3+x^4+x^5)^4(1+x+x^2+...)^4$?
I got
$$(x+x^2+x^3+x^4+x^5)^4=x^4((1-x^5)/1-x)^4$$
and
$$(1+x+x^2+...)^4=1/(1-x)^4,$$
what I should do next? Please help me with it, thank you.
- 1
2
votes
2
answers
781
views
For what values of $n$ give a constant term when $\left(\frac{1}{x^2}+x\right)^n$ is expanded? Also, what is this constant term(in terms of $n$)?
Here's my question:
What values/restrictions of $n$ give a constant term in the expansion of $$\left(\frac{1}{x^2}+x\right)^n$$? Also, for the expansions that do have a constant term, what is this ...
- 2,846
0
votes
1
answer
111
views
Coefficients of $(1+x+x^2)^{2018}$ [closed]
The question is
How many of the coefficients of $(1+x+x^2)^{2018}$ are not divisible by 3?
Somebody asked me the question, and I have no idea how to solve it. I am not sure if the coefficients are ...
- 9,961
15
votes
4
answers
790
views
Find coefficient of $x^{20}$
Find the coefficient of $x^{70}$ in the expansion
$$(x-1)(x^2-2)(x^3-3)(x^4-4)\cdots (x^{12}-12)$$
$\mathcal {\text {Now I have solved this question}}$. What I did was I noticed that the highest ...
- 9,803
1
vote
0
answers
307
views
An Olympiad problem ( need explanation to the given answer)
$P(x)$ is a polynomial of degree $3n$ such that
\begin{eqnarray*} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-...
- 9,803
5
votes
3
answers
208
views
Is there a quick way of finding the coefficients in an expression like $(ax^3+bx^2+cx+d)^3$?
We can raise a sum to the power of $n$ quickly and easily using Pascal's triangle, due to the binomial theorem:
$$(a+b)^n = \sum_{i=0}^n {n \choose i} a^i b^i$$
For sums of more than one term, we ...
- 68.1k
2
votes
1
answer
2k
views
Number of terms in multivariate polynomial
We know that the number of terms in a univariate polynomial of degree n is n+1.
But what about if there are multiple variables:
for eg: for variables $x,y$ polynomial of degree 2 will have:
$1+x+y+xy+...
0
votes
3
answers
592
views
Binomial expansion to find a specific term (coefficient)
for this question I tried to use binomial theorem to find a specific term. However, I eventually cannot find a valid value of n and r and p. My working is shown in the picture and please tell me my ...
- 1
5
votes
1
answer
150
views
How to calculate this efficiently?
If in the expansion of $(1 + x)^m \cdot (1 – x)^n $, the coefficients of $ x $ and $ x^2 $are 3 and -6 respectively, then m is ?
I solved it in the following way :
Expanding we get, the coefficient ...
Tretwick Marain