Give one reason that a 4th degree polynomial function might not have four factors.
(This was a test question)
The Attempt at a Solution
I wrote that a 4th degree polynomial function might contain an irreducible quadratic, such as
(x-4)(x-2)(x^2+1). Because of the factor (x^2+1), the polynomial cannot be factored over
the set of real numbers into four linear factors.
But my teacher wanted me to say that the order of the zeros may be higher than 1, so an expression such as (x-2)^2(x-4)^2 would only have two factors because there are two double roots.
My question is, doesn't the expression (x-2)^2(x-4)^2 still have four factors? The expression
(x-2)^2(x-4)^2 can be written as (x-2)(x-2)(x-4)(x-4) so it still have four factors. Just because the factors are repeated doesn't mean that it only counts as one factor, right?
The question asks about factors, not zeros. In any case, wouldn't I be right to assume that "factor" refers to linear factors?