Another repost... pls just answer the question here or at this question chat: https://web2.0calc.com/questions/polynomials_23

madyl Mar 21, 2020

#1**0 **

Here is a hint to help you out:

We know that a composite polynomial p(x) can be decomposed into two or more polynomials only if there are polynomials w(x) and c(x) for which p(x) = w(c(x)).

If we know that p(x) is nonnegative, what must w(c(x)) be?

Hope this helped!

CalTheGreat Mar 21, 2020

#2**+1 **

We know that P(a) - P(b) is divisible by a - b. This means if a - b is not a prime number, then P(a) - P(b) is not a prime number. So by modular arithmetic, we can find a positive integer n such that P(n) is composite.

If P(a_1) and P(a_2) are composite, then P(a_1) - P(a_2) will be divisible by some prime number. Similarly, if P(a_1) - P(a_2) is divisible by some prime number, and P(a_1) or P(a_2) is composite, then the other one is also composite.

We can then take a_1 = n. So there exists a positive integer a_2 such that P(a_2) is composite. But since P(a_2) is composite, we can apply the result above, to find a positive integer a_3 such that P(a_3) is composite. In this way, we can generate an infinite number of positive integers n such that P(n) is composite.

Guest Mar 23, 2020