Download Lesson 9-4b: Polynomial Division – Synthetic Division

Lesson 9-4b: Polynomial Division – Synthetic Division
Yesterday we learned how to divide polynomials using long division. There is another
method that we will learn today. It has a strange name: synthetic division.
Synthetic division? What does synthetic mean?
I honestly don’t know where the name “synthetic” division comes from. I assume it is
because something that is synthetic isn’t natural. This method of division is not a
“natural” way of dividing. Long division is a “normal” way to approach division because it
is the accepted way to divide big numbers. Synthetic division is…well…not something
you’d naturally think of.
Synthetic division is a short cut way to divide polynomials. It has some advantages:
1. We only need to keep track of the coefficients (the numbers in front of each
variable or term).
2. It is only 3 rows tall…it doesn’t cascade all the way down your paper!
Synthetic division has one key limitation:
1. It only works with a x ± k type divisor.

Examples of divisors that work are x + 2, x – 3, etc.

Examples of divisors that don’t work are x 2  3 , 3x  5 , 2 x2  x  5 , etc.
So it only works when you are dividing by an x  # type of polynomial.
I’m not even going to try to describe it to you. I’ll show you an example and explain as we go.
Synthetic division
Let’s divide 3x5  8x3  2 x2  x  1by x  2 using synthetic division.
First, what are the coefficients of the dividend? In order they are 3,  8, 2,  1and 1. The 3
comes from the 3x5 …where did that 0 come from? If you notice, the dividend is missing
the x 4 term. We need to account for all of the terms in order.
To do synthetic division, lay out the coefficients of the dividend like this. Notice there is
what looks like an up-side-down long division bar. This is how you set up the synthetic
division problem:
2
3 8
2 1 1
Okay, I can see where the numbers inside the bar come from…they are the dividend
coefficients. Where did that -2 on the outside come from? It comes from the divisor x  2 .
The number on the outside is the negative of the constant term of the divisor.
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Lesson 9-4b: Polynomial Division – Synthetic Division
To do the synthetic you just follow a pattern over and over. Start from the left inside and
work to the right. Pull down the first number to under the bar at the bottom:
3 0  8 2 1 1
2
3
Now take the number you pulled down (3) and multiply it by the number on the outside (2)… we get -6. Put this new number inside just under the next number...under the 0:
3 0  8 2 1 1
2
6
3
Now add the 0 and the -6. Write the answer (-6) under the bar right below the -6:
3 0  8 2 1 1
2
6
3 6
Now just start repeating the pattern: take the number outside (-2) and multiply it by the
new number (-6) underneath. Put the answer (+12) in the 2nd row inside below the next
number.
3 0  8 2 1 1
2
 6 12
3 6
What next? Add the -8 and the 12. Write the answer (4) below the 12:
3 0  8 2 1 1
2
 6 12
3 6
4
Multiply -2 by 4, write answer (-8) inside at end of row 2. Add, write below. Repeat until
have used up all the numbers on the top row inside. Here is the final picture:
3 0  8 2 1
1
2
 6 12  8 12  22
3 6
4  6 11  21
The numbers in the row under the bar are the coefficients of the answer. The last number
is the remainder. All you need to do is plug in the variables. The degree of the answer
will always be one less than the degree of the dividend. Our dividend here is degree 5.
Therefore the answer will be degree 4. The 1st term will be 3x 4 . The full answer is:
3x4  6 x3  4 x2  6 x  11 R  21
Take a second and compare the answer with the end of the synthetic division process.
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Lesson 9-4b: Polynomial Division – Synthetic Division
More examples
Divide x4  2 x3  31x  4 by x  4 using synthetic division.
Don’t forget there is a missing term! The x 2 term is missing. You must keep it’s
place with a 0!!!
4
1 2
1
0  31  4
4
8
32
4
2
8
1
0
The dividend is degree 4…the answer will be degree 3. Fill in the answer:
x3  2 x 2  8 x  1
The remainder is zero so we don’t list it.
Divide 2 x3  11x2  18x  9 by x  3 using synthetic division.
3
2 11 18 9
 6  15  9
2
5
3
0
The answer is 2 x2  5x  3 with no remainder.
The Remainder Theorem
One very cool thing about synthetic division is found in what is called the Remainder
Theorem. The Remainder Theorem basically gives us a shortcut way to plug a number
into a big ugly polynomial functions and find out what the answer is. You just use
synthetic division and the remainder is the answer!
For instance, if I asked you what the value of the function f ( x)  3x5  8x3  2 x 2  x  1 is
when x  2 , you could answer the question by plugging -2 into the function for x and
crunching things out. Or you could use synthetic division with -2 on the outside and the
remainder would be the answer:
2
3
0 8
2 1
1
 6 12  8 12  22
3 6
4  6 11  21
The remainder is -21. Thus we can now say that f ( x)  3x5  8x3  2 x 2  x  1 at x = -2:
f (2)  21 . The fancy way of saying what we just did is we “evaluated the function at
the given value of x using the Remainder Theorem. 
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