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Long Division
© 2007 Thomas M. Kenyon
Use with permission: [email protected]
Review:
How many times does 3 go into 24?
24  3  8 times
How many times does 7 go into 35?
35  7  5 times
How many times does -4 go into 12?
12  4  3 times
How many times does 2x2 go into 8x5?
5
8x
3
8 x  2 x  2  4x times
2x
5
2
How many times does 3x go into 15x3?
15 x  3x  5 x times
3
2
Long division review:
68 5
74 50690
90
444
62
592
37
370
0
Steps:
How many times does 74 go into 370?
506?
629?
ans: 586
Multiply 5*74
6*74
8*74
ans: 370
444
592
Subtract
Bring down the next
DONE!
Repeat the steps
comment:
74 50690
Wouldn’t it be easier if we only had to worry
about the 7? The 4 makes it harder…
It’d be a lot easier if we just had to worry about
how many times 70 goes into 506 (7*70=490)
Oh well, we weren’t that lucky in 5th grade, but
we’re going to be that lucky when we do this with
polynomials!
A polynomial long division problem looks like this:
(12 x  7 x  10)  (3x  2)
2
or this:
3x  2 12 x  7 x  10
2
or this:
12 x  7 x  10
3x  2
2
All three notations mean the same thing: division.
Note: we are not dividing by a monomial this time!
We already learned division by a monomial. This time, the
divisor is a polynomial (we’re going to stick to binomials).
comment:
However, we’re going to learn a new way to divide
by a monomial. We will adapt this new method to
learning long division by a polynomial with 2 or more
terms in it.
3x 12 x  6 x  9 x
3
2
The steps for long division of polynomials are the same
as the steps for long division of numbers:
First, how many times does it go into the front part?
Second, multiply
Third, subtract
Fourth, bring down the next term (number)
Repeat these four steps until you’re done.
4x2
-2x
+3
3x 12 x  6x
6x  9x
3
2
12x3
0
-6x2
0
First, how many times 9x
does 3x go into 12x3? We already
32 and divide it by 3x.
saw atmany
the beginning
that3xwe
take
12x
How
times does
go
into
+9x?
-6x
?
0
2 times.
2 by
In this case,
it goes
into
it 4x
Divide
Positive
-6x
3 times.
3x to
find
out.
In this case, -2x times.
Second, we
Multiply
3 times
multiply
3x -2x
4x2 times 3x
3x.
Third, we subtract
Subtract
subtract.again.
Bring
Now,
we
down
bring
thedown
next term.
the next
Then
term.
repeat
Repeat
once the
more.
process.
Done!!!
Your turn:
Answer:
3x2
-2x + 4
5x 15x 10x
10 x  20
20xx
3
22
15x3
0
-10x2
0
20x
0
Note: in both of these problems, when we subtracted, we got
zero every time! That’s why long division with polynomials
is easier than long division with numbers.
2x2 -4x
+3
3
2
3x  5 6 x  2 x -11x
11x + 15
6x3 +10x2
0 (yikes!!)
-12x2
-12x2 -20x
0 + 9x 3
How many times does 3x go into 6x ??
9x
+
15
2 by 3x
Good
Now,
news!:
multiply
You
2x
don’t
have
+ 5.to worry
You have
about
to multiply
how
by the
Multiply
You
don’t need to worry about the 0
+ 5+yet…
Just,
Subtract.
0
Subtract
many
entire times
divisor,
3x+5
not goes
just the
into3x.
something…
That’s
the
You
only
only
thinghave
different
Multiply…
3
Bring
downtimes
the next
How many
does term.
3x go into 6x ?
to
do
“how
many
from
times
the
previous
does
3x
two
go9x?
into.
examples.
. .” 2
How
Notice:
many
we
still
times
always
does
get
3x
go
a
into
Multiply
-4x
by
the
whole
divisor
Now,
how
many
times
does 3x go into -12x ?
Be
careful!
When
you
subtract
Bring
down
the
next
term
Again,
be
careful
with
signs.
Multiply
thetimes
3 byterm
the
entire
divisor…
zero
for
the
first
when
we
3about
That
is,
-4x
3x
+
5
Again,
don’t
worry
the
+of5 out.
As
usual,
divide
6x
by
3x
to
find
aYou’re
polynomial,
you
subtract
all
subtracting
negative
20x.
is thethe
same!
distributive property)
3Everything
times
3x +else
5. (Use
are
subtracting.
2 divided
-12x
by
3x
is
-4x
times.
Subtract
the
terms. All
the signs
change.
Subtracting
changes
the sign,
so it’s the
2
2)
(again,
distributive
property)
This
time,
it’s
2x
(the
second
term
becomes
–
10x
same
as adding
20x. property)
(use the
distributive
Done!
example 2:
5x2 -6x
+7
2 x  4 10 x  32 x +38x
38x -28
28
3
2
10x3 -20x2
0 (yikes!!)
-12x2
-12x2 + 24x
0 + 14x
14x - 28
+ how
0
You don’t have to worry 0
about
Multiply
Good
news!:
Multiply…
Subtract.
Subtract
many
times
2x45x
goes
2 byinto
something…
3??You
only have
How
Notice:
Now,
many
multiply
we
still
times
always
does
get
2x
go
a
4.
into
You
10x
have
to
multiply
by the
How
many
times
does
2x
go
into
14x?
2?
to
Multiply
do
“how
-6x
many
by
the
times
whole
does
divisor
2x
go
into.
.
.”
Now,
how
many
times
does
2x
go
into
-12x
Multiply
the
7
by
the
entire
divisor…
zero
entire
for
divisor,
the
first
not
term
just
when
the
2x.
we
That’s
the
only
thing different
Bring
down
the next
Be careful!
Again,
be careful
When
with
youterm
signs.
subtract
That
is,down
-6x-3times
2x
-find
4 the
Again,
don’t
worry
about
-of
4
7are
times
2x
4.
Divide
subtracting.
10x
from
by
the
2xnext
previous
to
out.
examples
with monomials.
Bring
the
term.
a polynomial,
You’re
subtracting
you
subtract
24x.
all
2 divided
Subtract
Everything
else
is2x
the
-12x
bythe
is same!
-6xchange.
times.
the terms. All
signs
(again,
distributive
2 property)
(use
the
distributive
property)
This
time,
it’s
5x
(Use
the distributive
(the second
term
becomes
+20x2 ) property)
Done!
Your turn!!
3x2 -2x
-4
+12
5x  3 15x  19 x -14x
14 x 
12
3
2
15x3 -9x2
0 (yikes!!)
-10x2
-10x2 + 6x
0 - 20x
- 20x + 12
0 + 0