concept of synthetic division?

Synthetic division is similar in principle to long division. Suppose we want to divide 9756 by 35. We want to find a number x such that 35x + remainder = 9756, where the remainder is less than 35. We find x one piece at a time. We first find the highest power of 10 for x. In this case it would be 100, which multiply by 2 to get 200. 35(200 + X2) = 9756. We put the 2 on top, multiply 35 by 200, giving 7000 and subtract from 9756, giving 35X2 = 2756. We then continue in the same way.

Now suppose we want to divide p(x)= x^3+7x+3x+4 by x+6. We want to find a polynomial.q(x) such that p(x) = (x+6)q(x) + r(x), where r(x) is of degree less than (x+6), which means that it will be a constant. Again we find q(x) one step at a time. We look for the highest power of x in q(x). We know this must be x^2. This gives (x^2 + q2(x))(x+6) = p(x), where the highest power of q2(x) is less than 3.. Analogously to what was done for long division, we subtract x^2(x+6) from p(x), giving q2(x) and then continue in the same way, now searching for q2(x) satisfying (x+6)q2(x) = p(x) - x^2(x+6).