Example: determine the zeroes of following polynomials.

P ( x)= 5x⁵ - 20x⁴+5x3 + 50x2 - 20x - 40 = 5 (x + 1)² ( x - 2)³

Solution

In this the factoring has been done for us.  Do not worries regarding factoring anything like this?  You won't be asked to perform any factoring of this type anywhere in this material. There are only here to make the point that the zero factor property works here as well. 

In this case we do have a product of 3 terms however the first is a constant and will not make the polynomial zero. So, from the last two terms it looks like the polynomial will be zero for x = -1 and x =2 .Thus, the zeroes of this polynomial are, x = -1 & x = 2