5.4 Dividing Polynomials and Synthetic Division
What You Will Learn
 Divide polynomials by monomials and write
in simplest form.
 Use long division to divide polynomials by
polynomials.
 Use synthetic division to divide and factor
polynomials.
2
Dividing a Polynomial by a Monomial
3
Dividing a Polynomial by a Monomial
4
Example 1 – Dividing a Polynomial by a Monomial
Perform the division and simplify.
Solution
5
Example 1 – Dividing a Polynomial by a Monomial
cont’d
6
Long Division
7
Example 2 – Long Division Algorithm for Positive Integers
Use the long division algorithm to divide 6584 by 28.
Solution
8
Example 2 – Long Division Algorithm for Positive Integers
cont’d
So, you have
In Example 2, 6584 is the dividend, 28 is the divisor, 235
is the quotient, and 4 is the remainder.
9
Long Division
10
Example 3 – Long Division Algorithm for Polynomials
The remainder is a fractional part of the divisor, so you can
write
11
Synthetic Division
12
Synthetic Division
13
Example 7 – Using Synthetic Division
Use synthetic division to divide x3 + 3x2 – 4x – 10 by x – 2.
Solution
The coefficients of the dividend form the top row of the
synthetic division array. Because you are dividing by x – 2,
write 2 at the top left of the array.
To begin the algorithm, bring down the first coefficient.
Then multiply this coefficient by 2, write the result in the
second row, and add the two numbers in the second
column.
14
Example 7 – Using Synthetic Division
cont’d
By continuing this pattern, you obtain the following.
15
Example 7 – Using Synthetic Division
cont’d
The bottom row shows the coefficients of the quotient.
So, the quotient is
and the remainder is 2.
So, the result of the division problem is
16
Synthetic Division
Synthetic division (or long division) can be used to factor
polynomials.
If the remainder in a synthetic division problem is zero, you
know that the divisor divides evenly into the dividend.
17
Example 8 – Factoring a Polynomial
Completely factor the polynomial x3 – 7x + 6 given that one
of its factors is x – 1.
Solution
The polynomial x3 – 7x + 6 can be factored completely
using synthetic division. Because x – 1 is a factor of the
polynomial, you can divide as follows.
18
Example 8 – Factoring a Polynomial
cont’d
Because the remainder is zero, the divisor divides evenly
into the dividend:
From this result, you can factor the original polynomial as
follows.
x3 – 7x + 6 = (x – 1)(x2 + x – 6)
= (x – 1)(x + 3)(x – 2)
19
Homework:
Page 254 #’s 1 – 12 all