How can I prove that if α is a root of the polynomial S, then S (x) = Q (x) (x-α) where the highest exponent (Q) = n-1 is n the highest exponent of S.

This can be generalized as:

$$ KΠ_ {i = 1} ^ n (x-a_i) $$

I tried to prove this by generalizing Q as $ q_ {n-1} * x ^ {n + 1} + … + q_1x + q_o $

and decomposing it with α as $ (x-α) * (Q) = S $ so $ S = αq_ {n-1} * x ^ {n + 2} + … + αq_1x ^ {n + 1} + q_0 $

But now I'm more confused and I can not find a way to prove this correctly because I think this is not the right way to proceed. However, I have no idea of another way to start the test.