Pre Calculus Please Explain

Question

What is the coefficient of the leading term of the polynomial function of minimal degree that has the partial solution set below?

Answer 1

If x = 4 with a multiplicity of 2 is a solution (or zero), it means that the factors of the polynomial associated with this would be:

(x - 4)(x - 4)

When x = ±√2 is a solution (or zero), it means that the factors of the polynomial associated with this would be:

(x + √2)(x - √2)

So that means the polynomial could be:

P(x) = (x - 4)(x - 4)(x + √2)(x - √2)

But, you could also have a multiplier in front of the factors and still get the same zeros:

P(x) = 2(x - 4)(x - 4)(x + √2)(x - √2)

P(x) = 3(x - 4)(x - 4)(x + √2)(x - √2)

P(x) = 4(x - 4)(x - 4)(x + √2)(x - √2)

etc

So the answer to the question is that the leading term could have ANY coefficient and still have the zeros x = {4 mult 2, ±√2}.

Is there more to the question that would narrow this down that you did not share?