3.6 Synthetic Division

Boy will you be mad that you haven’t learned this before.

What is Synthetic Division?

Synthetic Division is long division in another form.

1. It involves only coefficients and roots. 2. It is short hand (but can’t be used

with quadratic divisors). 3. It saves time and paper, and is a great

way to factor a polynomial with high power > 2. (More on that in section 3.7)

Lets try a problem

Please divide by long division.

Keep track of the answer.

3 2x 3x x 6x 1

This is Synthetic Division

This is the equivalent problem in synthetic division form:

The Pattern is “drop, multiply , add, multiply, add, etc.

1 1 3 -1 -6

zero of divisor

Coefficients of Numerator

This is synthetic Division

Try synthetic Division and see what you get:

1 4 3 1 4 3 -3

1 1 3 -1 -6

That is the beauty of synthetic division! You get the coefficients of the quotient plus any remainder. It takes less paper and is much quicker and easier!!

Try another problem:

3x 3x 2x 4

Here’s another problem with a bit of a twist.

If your last name begins with A-M, do this problem by long division.

If your last name begins with N-Z, do this problem by synthetic division.

3 22x x 3x 72x 1

What did you notice?

Therefore, what is the rule when the denominator has a coefficient in front of x?