Analysis of a polynomial (induction)

[entropic11]

Homework Statement

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.

Homework Equations

Hint: Consider a polynomial $$q(y) = p(k_0 + y)$$

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...)

 

[mighty2000]

Hello!

Your reasoning for part (a) looks good! Just a small clarification, when you say "a_0,...,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 ?", it actually means that a_0, a_1, a_2, ..., a_n are all integers, but some of them may be non-positive (less than or equal to 0). So for example, a_0 = -1, a_1 = 2, a_2 = 0, a_3 = -3, and so on. The important thing is that a_n, the coefficient of the highest degree term, is greater than or equal to 1.

For part (b), the hint given is to consider a polynomial q(y) = p(k_0 + y). This means that we are shifting the polynomial p(x) to the right by k_0 units. So instead of looking at p(x), we are looking at q(y) = p(k_0 + y), where y is a variable. We know from part (a) that p(x) is always greater than or equal to 2 for any x greater than or equal to k_0. Can you use this information to show that there must be some value of y for which q(y) is not a prime number?

Also, to do a new line in LaTeX, you can use a double backslash (\\) or just press enter twice.