If p (x) = x^3-3x^2-x+3 and p (3) = 0, what is a factor of p (x)

Answer: x-3

Since p (3) = 0, this means x = 3 plugs into p (x) to get 0

We can write p (x) as p (x) = (x-3) q (x) where q (x) is some other polynomial that multiplies with (x-3) to lead to x^3-3x^2-x+3

Let's plug in x = 3 and see what happens

p (x) = (x-3) q (x)

p (3) = (3-3) q (3)

p (3) = 0*q (3)

p (3) = 0

No matter what the result of q (3) was, it doesn't matter because it multiplies with 0 to get 0.

The general rule is: if p (k) = 0, then x-k is a factor of p (x). This is a special case of the remainder theorem.