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POLYNOMIALS

-1
112
2
+280

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

Mar 21, 2020

#1
+2277
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!

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.

Mar 23, 2020