Suppose you divide a polynomial by a binomial. how do you know if the binomial is a factor of the polynomial? create a sample problem that has a binomial which is a factor of the polynomial being divided, and another problem that has a binomial which is not a factor of the polynomial being divided.

The divisor is a factor if you get the remainder  zero after division.

Just as you know that 4 is a factor of 16 because 16/4 has zero remainder. 

For example, 

(2x-1)(x+2)(3x+7) / (x+2) 
= (6x^3+23x^2+15x-14)/(x+2) 

we know that the remainder is zero, because we constructed it that way. To make that a fraction where the divisor is NOT a factor, just change any of the coefficients. Then the divisor will no longer go in evenly.