>> Let's try it through another division problem using synthetic division. Here we are taking this polynomial and dividing by x minus 6. Now, to divide by x minus 6 synthetically, we put 6 in this position and then along this row we enter the coefficients and the constant involved in this polynomial. So, we'll have 3 and then leave some space then minus 16. And notice there is no first degree term in this polynomial, so we need a zero for a place holder for that term and then minus 72. Alright, and then we go through the synthetic division process. We'd bring down the 3 and then multiply, add, multiply, add, multiply, add, 6 times 3, 18 adding here negative 16 with 18 that's 2, 6 times 2, 12 and then adding 12, 6 times 12, 72. Adding, collecting you see we get zero. So, zero then is our remainder that means the division went evenly. Now, our answer polynomial is right here, but the fact that our division went evenly implies that we can show a factored form that is x minus 6 went into this rascal evenly and so we can factor that as, x minus 6 times whatever our answer was. To emphasize that it is this idea that 3x cubed minus 16x squared minus 72 can be rewritten as, let's see, our divisor was x minus 6 and our answer in this synthetic division process was, well let's see, it's 3x 2, an exponent one smaller than the exponent on the term implied by this entry. Now, remember this came from the 3x cubed, it came from this business. So, what we have here is 3x squared and then plus 2x and then plus 12 as the factored form. And we may or may not be able to continue the factorization, but the point is that when the synthetic division goes evenly it implies a factored form. Now, this is cleverly called the Factor Theorem, but it is then a way to test to see if we have a factor of a polynomial that is if we're testing to see if x minus 6 is a factor of this, then we would go through the synthetic division process. And if that remainder is zero, then yes it is a factor and if the remainder is not zero, then no it's not a factor, you see. So, we can make a determination about factorability or whether or not we have a factor by using this process.