Step by step solution to "Polynomial satisfying p(x)=3x for x∈N"

College
Algebra
Abstract algebra

haguemarineo6h

Answered question

2022-04-25

Polynomial satisfying

p (x) = 3^{x}

for

x \in N

Answer & Explanation

Dexter Conner

Beginner2022-04-26Added 15 answers

Step 1
$p (x + 1) = 3 p (x)$ on
$N \Rightarrow p (x + 1) = 3 p (x) \Rightarrow p (- 1) = \frac{1}{3}$ contra $p (x) \in Z [x]$
Or, continuing, $p (- n) = 3^{- n} :\Rightarrow p (x) : p (- x) = 1$ on $ℕ \Rightarrow p (- x) : p (x) = 1$ contra degree comparison. This proof works over much more general coefficient rings, e.g. $Q$