Is x squared minus x minus twelve a prime?

Answer 1

Just like numbers, polynomials can be prime in some way, although this is usually referred to as being irreducible. Your polynomial is not in fact irreducible since it can be decomposed into (x-4)(x+3). This also means that plugging in any integer value for x gives a factorisation for the number x^2-x-12, which is (x-4)(x+3), so the value of x^2-x-12 for any integer x can only be prime if either x-4 = 1,-1 or x+3 = 1,-1 so for x=5,3,-2,-4. Now for x=5 we get 25-5-12=8, which is not prime. For x=3 we get 9-3-12=-6, which is not prime either. For x=-2 we have 4+2-12=6, which again is not prime. And for x=-4 we have 16+4-12=8 again, which of course is still not prime. So not only is your polynomial reducible, also any value it takes when plugging in an integer x will be composite. An alternative way of establishing this fact would have been to note that this polynomial only takes even values for any integer x and that the value 2 is not in fact attained for any integer x.