Final answer:
The answer to whether there is only one polynomial with the factors (x-2), (x+1), and (x-3) is False. There is an infinite number of polynomials with these factors, as any nonzero constant multiple of the product of these factors will also produce a polynomial with these factors.
Step-by-step explanation:
The question is asking whether there is only one polynomial that has the factors (x-2), (x+1), and (x-3). The answer is False. There is not just one polynomial with these factors, but rather an infinite number of polynomials. Any polynomial that is the product of these three factors multiplied by any nonzero constant will also have these factors.
To see why this is the case, consider the polynomial P(x) = k*(x-2)*(x+1)*(x-3), where k is any nonzero constant. Since k does not change the root structure of the polynomial, it means that each of these polynomials will still have x-values of 2, -1, and 3 as roots, regardless of what nonzero value k takes. Therefore, there are infinite polynomials with these factors, differing by a multiplicative constant.