Synthetic Division

Let's review a very simple division problem for a few moments. 

If one is doing long division and wants to divide 7 by 2:

$2\overline{)7}$

1. Ask yourself what is the largest multiple of 2 that is smaller than or equal to 7. In this case that would be 3, so you put a three above the 7, and multiply 2 times 3 and put 6 below the 7.

$\begin{align} 3\\ 2\overline{)7}\\ 6 \end{align}$

1. Next you subtract the 6 from the 7 and get 1, which is the remainder because 1 is less than 2.
   $\begin{align} 3\\ 2\overline{)7}\\ -6\\ \overline{1} \end{align}$
2. To write the quotient using a mixed number, you place the remainder which is 1 over the divisor which is 2.
3. So, $7\div2=3\frac{1}{2}\:$which you  can discover using long division described above.

For this problem:

• 2 was the divisor,
• 7 was the dividend,
• 3 is the quotient, 
• 1 was the remainder

It will be useful to also remember that for any division problem, one can write an associated equation with multiplication and addition. In the case of the example above:

$7=2\cdot3+1$

More generally,

Dividend=(Divisor)(Quotient)+Remainder

It turns out that, in the case of long division, polynomials really don't behave differently. Which leads us to the following:

The Division Theorem

Let f(x) and g(x) be polynomials with g(x) of lesser degree than f(x) and g(x) of degree 1 or more. There exist unique polynomials q(x) and r(x) such that 

$f\left(x\right)=g\left(x\right)\cdot q\left(x\right)+r\left(x\right)$

where either r(x) = 0 or the degree of r(x) is less than the degree of g(x).

Synthetic Division

Synthetic division provides an efficient process for dividing a polynomial by a binomial of the form x − k.

The best way to learn synthetic division is to watch it do it's magic a few times, and then perform it a lot.

Here is a series of Khan videos showing a couple of simple examples of synthetic division and then another one explaining why it works:

So, to review, or just in case you could use one more example of synthetic division worked through. Here is another example: