How does the remainder theorem allow for zeros of a polynomial to be identified?

If substituting a given number a into the polynomial results in a value of zero, then the remainder theorem states that dividing the polynomial by (x - a) gives a remainder of zero. This implies that x = a is one of the zeros of the polynomial. Having identified (x - a) as a factor of the polynomial (since it leaves a remainder of zero), we can then divide the polynomial to reduce its degree and make it easier to find any further zeros.