# Thread: Notate Coefficients of a Polynomial

1. ## Notate Coefficients of a Polynomial

Hi,

Not sure If this thread is in the right part of the forum.

Anyway, quite simply if one has a polynomial as below,

Can the coefficents of an arbitrary polynomial be notated as a series or set?

$\displaystyle (x+1)(x^4 + 2x^3 + 2x^2 + 2x + 1) \:=\:x^5 + 3x^4 + 4x^3 + 4x^2 + 3x + 1$

i.e. from this example i would need...

$\displaystyle s = [1, 3, 4, 4, 3, 1]$

If so how?

Thanks Alex

2. I dont think this would fall under the catagory of discrete mathematics.

I am not sure exactly what you want, but the first thing that came to my mind was this...

let $\displaystyle S=\{1,x,x^{2},...,x^{n-1},x^{n}\}$. This set spans $\displaystyle \mathbb{R}^{n}$ because any polynomial fucntion $\displaystyle p(x)=a_{0}+a_{1}x+...+a_{n-1}x^{n-1}+a_{n}x^{n}$ in $\displaystyle \mathbb{R}^{n}$ can be written as,

$\displaystyle p(x)=a_{0}(1)+a_{1}(x)+...+a_{n-1}(x^{n-1})+a_{n}(x^{n})$

So, in your example, your basis is the set $\displaystyle S=\{1,x,x^{2},x^{3},x^{4},x^{5}\}$ in $\displaystyle \mathbb{R}^{5}$. So any polynimial of degree 5 can be created by these vectors. Since you have coeficients $\displaystyle \{1,3,4,4,3,1\}$ we can write $\displaystyle p(x)=1(1)+3(x)+4(x^{2})+4(x^{3})+3(x^{4})+1(x^{5})$ which is your polynomial. Thus any polynomial or the form $\displaystyle p(x)=a_{0}+a_{1}x+...+a_{5}x^{5}$ can be created from the set S. The only thing that changes are your coeficients $\displaystyle (a_{0},a_{1},...,a_{5})$, so I guess you could just list them in a set.

Thats my stab at it.