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Dividing
Polynomials:
Synthetic Division
Essential Question
How do I use synthetic division to
determine if something is a factor of a
polynomial?
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Synthetic Division divide a polynomial by a polynomial
To use synthetic division:
•There must be a coefficient for
every possible power of the variable.
•The divisor must have a leading
coefficient of 1.
Ex6 :
5x
4

 4 x  x  6  ( x  3)
2
5x
4

 4 x  x  6  ( x  3)
2
Step #1: Write the terms of the
polynomial so the degrees are in
descending order.
4
3
2
5x  0x  4x  x  6
Since the numerator does not
contain all the powers of x,
3
you must include a 0 for the x .
5x
4

 4 x  x  6  ( x  3)
2
Step #2: Write the constant r of the
divisor x-r to the left and write
down the coefficients.
4
3
2
5x  0x  4x  x  6
3
5
0
-4
1
Since the divisor is x-3, r=3
6
5x
4

 4 x  x  6  ( x  3)
2
Step #3: Bring down the first
coefficient, 5.
3
5
5
0
-4
1
6
5x
4

2
 4 x  x  6  ( x  3)
Step #4: Multiply the first
coefficient by r, so 3  5  15
and place under the second
coefficient then add.
3
5
0
15
5
15
-4
1
6
5x
4

2
 4 x  x  6  ( x  3)
Step #5: Repeat process multiplying
the sum, 15, by r; 15  3  45
and place this number under the
next coefficient, then add.
3
5
5
0
-4
15
45
15
41
1
6
 5x
4
2
 4x  x  6
  ( x  3)
Step #5 cont.: Repeat the same procedure.
Where did 123 and 372 come from?
3
5
5
0
-4
1
6
15
45
123 372
15
41 124 378
 5x
4
2
 4x  x  6
  ( x  3)
Step #6: Write the quotient.
The numbers along the bottom are coefficients
of the power of x in descending order, starting
with the power that is one less than that of the
dividend.
3
5
5
0
-4
1
6
15
45
123 372
15
41 124 378
 5x
4
2
 4x  x  6
  ( x  3)
The quotient is:
378
5x  15x  41x  124 
x3
3
2
Remember to place the
remainder over the divisor.
Ex 7:
5x
5
 21x  3x  4x  2x  2  x  4
4
3
2
Step#1: Powers are all accounted
for and in descending order.
Step#2: Identify r in the divisor.
Since the divisor is x+4, r=-4 .
4
5
 21 3
4
2 2
 5 x  21x  3 x  4 x  2 x  2    x  4 
Step#3: Bring down the 1st coefficient.
Step#4: Multiply and add.
Step#5: Repeat.
5
4
4
5
3
2
 21 3
20
-1
4
1
4
-4
0
2 2
0 8
-2 10
-5
10
4
3
2
5 x  x  x  2 
x4
Assignment
Complete the worksheet.