Example 1.
9
x
5
+6
x
4

18
x
3

24
x
2
3
x
2
Divide each term in the numerator by
3
x
2
9
x
5
3
x
2
+
6
x
4
3
x
2

18
x
3
3
x
2

24
x
2
3
x
2
Reduce each fraction
,
subtracting exponents
3
x
3
+2
x
2

6
x

8
Our Solution
Example 2.
8
x
3
+4
x
2

2
x
+6
4
x
2
Divide each term in the numerator by
4
x
2
8
x
3
4
x
2
+
4
x
2
4
x
2

2
x
4
x
2
+
6
4
x
2
Reduce each fraction
,
subtracting exponents
Remember negative exponents are moved to denominator
2
x
+1

1
2
x
+
3
2
x
2
Our Solution
The previous example illustrates that sometimes we will have fractions in our
solution, as long as they are reduced this will be correct for our solution. Also
interesting
in
this
problem
is
the
second
term
4
x
2
4
x
2
divided
out
completely.
Remember that this means the reduced answer is 1 not 0.
Long division is required when we divide by more than just a monomial. Long
division
with
polynomials
works
very
similarly
to
long
division
with
whole
numbers. An example is given to review the (general) steps that are used with
whole numbers that we will also use with polynomials
Page 99
Example 3.
4

631
Divide front numbers
:
6
4
=1
...
1
4

631
Multiply this number by divisor
:1
·
4=4

4
Change the sign of this number
(
make it subtract
)
and combine
23
Bring down next number
1
5
Repeat
,
divide front numbers
:
23
4
=5
...
4

631

4
23
Multiply this number by divisor
:5
·
4=
20

20
Change the sign of this number
(
make it subtract
)
and combine
31
Bring down next number
15
7
Repeat
,
divide front numbers
:
31
4
=7
...
4

631

4
23

20
31
Multiply this number by divisor
:7
·
4=
28

28
Change the sign of this number
(
make it subtract
)
and combine
3
We will write our remainder as
a
fraction
,
over the divisor
,
added to the end
157
3
4
Our Solution
This same process will be used to divide polynomials. The only difference is we
will replace the word “number” with the word “term”.
Dividing Polynomials
1. Divide front terms
2. Multiply this term by the divisor
3. Change the sign of the terms and combine
Page 100
4. Bring down the next term
5. Repeat
Step number 3 tends to be the one that students skip, not changing the signs of
the terms would be equivalent to adding instead of subtracting on long division
with whole numbers. Be sure not to miss this step! This process is illustrated in
the following two examples.
Example 4.
3
x
3

5
x
2

32
x
+7
x

4
Rewrite problem as long division
x

4

3
x
3

5
x
2

32
x
+7
Divide front terms
:
3
x
3
x
=3
x
2
3
x
2
x

4

3
x
3

5
x
2

32
x
+7
Multiply this term by divisor
:3
x
2
(
x

4)=3
x
3

12
x
2

3
x
3
+
12
x
2
Change the signs and combine
7
x
2

32
x
Bring down the next term
3
x
2
+
7
x
Repeat
,
divide front terms
:
7
x
2
x
=7
x
x

4

3
x
3

5
x
2

32
x
+7

3
x
3
+
12
x
2
7
x
2

32
x
Multiply this term by divisor
:7
x
(
x

4)=7
x
2

28
x

7
x
2
+
28
x
Change the signs and combine

4
x
+7
Bring down the next term
3
x
2
+7
x

4
Repeat
,
divide front terms
:

4
x
x
=

4
x

4

3
x
3

5
x
2

32
x
+7

3
x
3
+
12
x
2
7
x
2

32
x

7
x
2
+
28
x

4
x
+7
Multiply this term by divisor
:

4(
x

4)=

4
x
+
16
+4
x

16
Change the signs and combine

9
Remainderputoverdivisorandsubtracted
(
duetonegative
)
3
x
2
+7
x

4

9
x

4
Our Solution
Page 101
Example 5.
You've reached the end of your free preview.
Want to read all 200 pages?
 Summer '19
 Addition, Subtraction, Elementary arithmetic, Negative and nonnegative numbers