# Analysis of a polynomial (induction)

1. Feb 26, 2009

### entropic11

1. The problem statement, all variables and given/known data
Consider the polynomial

$$p(x) = a_0 + a_1x + \cdots + a_nx^n$$

where each of $$a_0,...,a_n$$ is an integer (some of them may be non-positive), and $$a_n \geq 1$$

a) Show that there is $$k_0 \in \aleph$$ such that $$p(k) \geq 2 for each k \geq k_0$$

b) Show that there is an integer $$k \geq k_0$$ such that p(k) is not a prime number.

2. Relevant equations
Hint: Consider a polynomial $$q(y) = p(k_0 + y)$$

3. The attempt at a solution
I've attempted (a)... but (b) I'm clueless. I tried (a) by induction...
Want to show $$p(k_0) \geq 2$$
Base: x=2
Since $$a_n \geq 1$$, take the lowest possible value it could be, ie, $$p(k) = a_0 + a_1(x) = 0 +1(x)$$
Then $$p(2) = 0 + 1(2) = 2$$
So the base case holds, $$p(k_0) \geq 2$$

Induction
Assume $$p(k_0)$$ true.
$$p(k_0 + 1) = a_0 + a_1(k_0+1)$$
$$=[a_0 + a_1(k_0)]+a_1$$
$$=[ 2 ] + 1$$ by induction hypothesis, and since $$a_n \geq 1$$
$$=3 \geq 2 \$$
So $$p(k_0+1) \geq 2 \forall k \geq k_0$$

First, is this legitimate reasoning? I'm still trying to get the hang of induction/analysis/etc. For example, $$a_0,\cdots,a_n$$ - does this mean that if a1 =1, then a2 =2, a3=3 and so on, just so long as $$a_n \geq 1$$ ?

Second, I'm not even sure where to start on (b). I'm assuming the hint will be useful... but where should I take it?

Thank you! (ps, I also just recently learned basic LaTeX... super useful that this forum has it! Also, takes some damn time getting used to writing things like that, haha! How do I do a new line? Just a backslash ( \ ) won't work...)

Last edited: Feb 26, 2009