Section 5.4
Dividing Polynomials
Review of Long Division
672  21
What terms do we use to describe 672 and 21?
Because the reminder is zero, we know that 21
is a factor of 672.
How could you write 672 using the divisor and
quotient?
• Writing the ratio of:
P( X )
D( X )
P( X )
REMAINDER
 Q( X ) 
D( X )
D( X )
• Writing P(x) in terms of
the divisor, quotient,
and remainder
P( X )  D( X )  Q( X )  Re mainder
Using long division to Divide
polynomials
672  21
2
6
x
  7 x  2    2x  1
 4x
2
 23x  16    x  5 
Is x
2
1 a
factor of
3x  4x  12x  5
4
3
2
Synthetic Division
• Synthetic division simplifies the long division
by using only the zero of the divisor and the
coefficients of the dividend
• Example 3:  x 3  14x 2  51x  54    x  2 
x
3
 57 x  56    x  7 
3
2
x

7
x
 38x  240
The polynomial
expresses the volume,
in cubic inches, of a shadow box. What are the
dimensions of the box if one side is x + 5?
Remainder Theorem
If you divide a polynomial P(x) of degree n>1 by
x – a, then the remainder is P(a)
Example:
5
3
2
P
(
x
)

x

2x

x
2
What is the remainder when
Is divided by x – 3?
How many ways could you find the remainder?
Which is most efficient?