Synthetic Division
What does this mean?
What am I solving for?
When we have a polynomial1 that needs to be
divided by a binomial2, we can use a special
format of division, called synthetic division, for
easier calculation.
Through synthetic division, we can do a
sequence of operations that is much faster than
traditional long division with polynomials.
This sequence of operations allows us to use
only the coefficients3 of each polynomial.
What’s the benefit? We don’t have to use those
“pesky” x-variables.
Steps of Synthetic Division
Detailed Examples
1. Descending Order: Ensure that the
dividend is in descending order and that all
place holders are accounted for.
Example 1 (Step by Step)
(
))
- 5 + 10
2. Proper Form: Make sure that the divisor is
a binomial in the form of x-c.
3. Divisor: Write down c as your new divisor
and the coefficients of the dividend.
4. Bring down the first value of the dividend
to the quotient6.
2
3
-6
0
-5
10
3
3
6
0
-6
0
-5
10
3
2 3
6
0
-6
0
0
0
+
2
+
The 0 is a place holder
Bring down the 3
Multiply 2 and 3 to get 6
Add -6 and 6 to get 0
Multiply 2 and 0 to get 0
Add 0 and 0 to get 0
-5
10
But, the catch is that the divisor4 you use must 5. Multiply the first value of the quotient by
+
6
0
0
Multiply 2 and 0 to get 0
be in the form of x-c, but we can convert an x+c the divisor and write the product7 below the
3
0
0
-5
Add -5 and 0 to get -5
into x-(-c). (See how these are actually the
second value of the dividend.
2 3 -6
0
- 5 10
same?? It’s the same concept of (2) = -(-2)!!)
6. Add the second column and write that
+
6
0
0
-10 Multiply 2 and -5 to get -10
value as the second value for the quotient.
How is this different from long division with
3
0
0
-5
0 Add 10 and -10 to get 0,
polynomials?
which means that we have no remainder!
7. Repeat steps 5 and 6 until necessary.
+
+ 0 -5 Degree is one less than the dividend
Long division with polynomials is very similar
-5
The 0 coefficient place holders aren’t needed
to the long division that we do with numbers.
8. Remainder: If the last number of your
any longer
quotient
is
a
non-zero
number,
then
this
is
Example 2 (Condensed)
We take the leading term in the divisor and ask
the numerator in your remainder. If this
“how many times does this go into the leading
( +1) ) +
-10
number is zero, then you have no remainder.
term of the dividend5?”
Put divisor in the form of x-c
Then, we take that new term, multiply it by our
entire divisor and subtract that new
polynomial by our entire dividend.
However, synthetic division only uses
multiplication and addition!
9. Final Answer: Don’t forget to put the
variables back into the answer. Note: Always
start with one degree less than your original
dividend.
-1
1 0 5 0 0 -10
by making it x-(-1), so c is -1!
-1 1 -6 6 -6
Add place holders!!
1 -1 6 -6 6 -16 Follow the steps: Multiply, Add!
Put the variables back in
-6 +6 Start with one degree less!!
Use Remainder Theorem
1. Polynomial: an algebraic expression involving terms of x to varying degrees. 2. Binomial: an algebraic expression with only two terms. 3. Coefficient: the number in front of the x.
4. Divisor: the binomial you are dividing by. 5. Dividend: the polynomial you are dividing into. 6. Quotient: the answer from a division problem. 7. Product: the answer from a multiplication problem.