# recursive def

• Jul 30th 2007, 10:29 PM
TheRekz
recursive def
Give a recursive definition of the set of polynomials with integer coefficients.

What I don't understand here is what is the set of polynomials with integer coefficients? Can someone give me an example?
• Jul 30th 2007, 10:47 PM
tukeywilliams
I believe the following would be a recursive definition of the set of polynomials with integer coefficients:

$\displaystyle 0 \in S$(some arbitrary set)
If $\displaystyle p(x) \in S$, then $\displaystyle p(x) + cx^n \in S$, $\displaystyle c \in \mathbb{Z}, \ n \in \mathbb{Z}, \ n \geq 0$.

The set of polynomials with integer coefficients is for example $\displaystyle \{x, x+1, x^2,2x^2+6x+7, x^3, \ldots \}$.
• Jul 31st 2007, 01:50 AM
CaptainBlack
Quote:

Originally Posted by TheRekz
Give a recursive definition of the set of polynomials with integer coefficients.

What I don't understand here is what is the set of polynomials with integer coefficients? Can someone give me an example?

It is the set of all polynomials with integer coefficients, like $\displaystyle 7x^4+2x+3$, and can be recursivly defined:

$\displaystyle 1 \in P$

$\displaystyle \forall p \in P, \forall a,b \in \mathbb{Z},\ axp+b \in P$

RonL
• Jul 31st 2007, 07:32 AM
TheRekz
which answer's right? captain black or tukeywilliams
• Jul 31st 2007, 08:01 AM
CaptainBlack
Quote:

Originally Posted by TheRekz
which answer's right? captain black or tukeywilliams

They could both be. So long as the both generate all polynomials with integer coefficients they are doing the same job. Now I think one is a betters answer than the other, but then I would wouldn't I.

RonL