5.7 Dividing Polynomials by Polynomials; Synthetic Division
Objectives:
1. Divide a polynomial by a monomial
2. Divide a polynomial by a binomial
Dividing a Polynomial by a Monomial:
When the numerator has more than one term, use the distributive property
and divide each term by the denominator.
If
is any nonzero real number
abc a b c
  
x
x x x
x
EXAMPLE: Divide
14 x5  4 x 4  3x3 14 x5 4 x 4 3x3
a.)




6 x3
6 x3 6 x3 6 x3


5 xy 2  10 xy
b.)
25 xy
30a 2b 2  15a 2b  10ab 2
c.)
10ab
d.)
4 x 2 y 2  2  x 2 y 2  xy 
2 xy
a  b  a  b
2
e.)
2
2ab
1
Division of a polynomial by a polynomial like x 2  2x  15  x  5 is very similar to
the long division process we use to divide whole numbers. Let’s review this
procedure first:
 4356 is called the
dividend
4356  34 or 34 4356
 34 is called the divisor
 the answer is called the
quotient
Divide, Multiply, Subtract, pull
down.
We write the answer as:
remainder
4356  34  quotient 
divisor
We check the answer by: quotient  divisor  remainder  dividend
What happens if we get a zero remainder when dividing?
______________________
2
We follow a similar procedure for division of a polynomial by a binomial.
x
2
 2 x  15   x  5 or
x  5 x 2  2 x  15
x 2  2 x  15 is called the dividend x  5 is
called the divisor the answer is called the
quotient
PROCEDURE:
1.) Divide the first term in the dividend by
the first term in the divisor. This is the first
term in the quotient.
2.) Multiply this by each term in the
divisor.
3.) Subtract.
4.) Bring down
Repeat the process until the remainder
can no longer be divided.
We write the answer as:
remainder
2
x

2
x

15

x

5

quotient





divisor
We check the answer by making sure that:
quotient  divisor  remainder  dividend
3
GENERAL PROCEDURE:
1. Write the polynomials in standard form. (in order with descending
powers) If there is a missing term, fill it in with a zero times the missing
variable part. (i.e. x2  1  x 2  0 x  1)
2. Divide the first term in the dividend by the first term in the divisor. This is
the first term in the quotient.
3. Multiply each term in the divisor by the result of the previous step. Write
the answer beneath the dividend with similar terms under each other.
4. Subtract the product from the dividend.
5. Bring down the next terms and treat this as the new dividend. Start the
process over again. Repeat this until the remainder can no longer be
divided. This happens when the degree of the remainder is less than
the degree of the divisor.
8 y 3  y 4  16  32 y  24 y 2
EXAMPLE: Divide
y2
4
y4  y2  6 y
EXAMPLE: Divide:
y2  2
x4  1
EXAMPLE: Divide: 2
x 1
2 2
x  3x  2
3
EXAMPLE: Divide:
x 1
5
SYNTHETIC DIVISION: (optional)
We can use synthetic division to divide polynomials if the divisor is of the form
x  c . This method is quicker than long division.
Let’s see how the method works with x  3 x3  4 x 2  5 x  5 .
LONG DIVISION:
SYNTHETIC DIVISION:
3
x  3 x3  4 x 2  5 x  5
1 4 5 5
3
21 48
EXAMPLE: Use synthetic division to divide
a.) 8 y 3  y 4  16  32 y  24 y 2    y  2 
x3  7 x  6
b.)
x2
6