1. Long Division
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Thomas M. Kenyon
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© 2007 Thomas M. Kenyon
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2. How many times does 3 go into 24?
Review:
How many times does 3 go into 24?
How many times does 7 go into 35?
How many times does -4 go into 12?
How many times does 2x2 go into 8x5?
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How many times does 3x go into 15x3?

3. 6 8 5 74 50690 9 444 62 592 37 370 Long division review: Steps: DONE! 9
How many times does 74 go into 370?
ans: 5
How many times does 74 go into 506?
ans: 6
How many times does 74 go into 629?
ans: 8
444 62
Multiply 5*74
ans: 370
Multiply 8*74
ans: 592
Multiply 6*74
ans: 444
592 37
Subtract
Subtract
Subtract
370
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Thomas M. Kenyon
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Bring down the next
Bring down the next
DONE!
Repeat the steps
Repeat the steps

4. 74 50690 Wouldn’t it be easier if we only had to worry
comment:
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 (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!
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5. A polynomial long division problem looks like this:
or this:
or this:
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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).

6. However, we’re going to learn a new way to divide
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.
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.
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7. 4x2
-2x
+ 3
12x3
-6x2 9x
First, how many times does 3x go into 12x3? We already
saw at the beginning that we take 12x3 and divide it by 3x.
In this case, it goes into it 4x2 times.
How many times does 3x go into +9x?
Positive 3 times.
How many times does 3x go into -6x2?
Divide -6x2 by 3x to find out. In this case, -2x times.
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Multiply 3 times 3x
Second, we multiply 4x2 times 3x.
Second, we multiply -2x times 3x
Subtract
Third, we subtract.
Third, we subtract again.
Bring down the next term. Then repeat once more.
Now, we bring down the next term. Repeat the process.
Done!!!

8. 3x2 -2x + 4 15x3 -10x2 20x Your turn: Answer:
-10x2
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20x
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.

9. Done! 2x2 -4x + 3 -11x + 15 6x3 +10x2 0 -12x2 -12x2 -20x 0 + 9x
(yikes!!)
-12x2 -20x
0 + 9x
How many times does 3x go into 6x3??
You don’t need to worry about the + 5 yet… Just,
How many times does 3x go into 6x3?
As usual, divide 6x3 by 3x to find out.
This time, it’s 2x2
9x + 15
Good news!: You don’t have to worry about how
many times 3x+5 goes into something… You only have
to do “how many times does 3x go into. . .”
Everything else is the same!
Now, multiply 2x2 by 3x You have to multiply by the
entire divisor, not just the 3x. That’s the only thing different
from the previous two examples.
(Use the distributive property)
Multiply
Multiply -4x by the whole divisor
That is, -4x times 3x + 5
(again, distributive property)
0 + 0
Subtract.
Be careful! When you subtract
a polynomial, you subtract all of
the terms. All the signs change.
(the second term becomes – 10x2 )
Subtract
Again, be careful with signs.
You’re subtracting negative 20x.
Subtracting changes the sign, so it’s the
same as adding 20x.
Multiply…
Multiply the 3 by the entire divisor…
3 times 3x + 5.
(use the distributive property)
Bring down the next term.
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Thomas M. Kenyon
6455 Habgood Road
Black Creek, NY 14714
Notice: we still always get a
zero for the first term when we
are subtracting.
How many times does 3x go into 9x?
Now, how many times does 3x go into -12x2?
Again, don’t worry about the + 5
-12x2 divided by 3x is -4x times.
Bring down the next term
Done!
Subtract

10. Done! +38x -28 5x2 -6x + 7 10x3 -20x2 0 -12x2 -12x2 + 24x 0 + 14x
example 2:
5x2
-6x
+ 7
+38x
-28
10x3 -20x2 x2
(yikes!!)
-12x2 + 24x x
14x - 28
Multiply
Multiply -6x by the whole divisor
That is, -6x times 2x - 4
(again, distributive property)
Good news!: You don’t have to worry about how
many times 2x- 4 goes into something… You only have
to do “how many times does 2x go into. . .”
Everything else is the same!
Multiply…
Multiply the 7 by the entire divisor…
7 times 2x - 4.
(use the distributive property)
Subtract.
Be careful! When you subtract
a polynomial, you subtract all of
the terms. All the signs change.
(the second term becomes +20x2 )
Subtract
Again, be careful with signs.
You’re subtracting 24x.
How many times does 2x go into 10x3??
Divide 10x3 by 2x to find out.
This time, it’s 5x2
Notice: we still always get a
zero for the first term when we
are subtracting.
Now, multiply 5x2 by 2x You have to multiply by the
entire divisor, not just the 2x. That’s the only thing different
from the previous examples with monomials.
(Use the distributive property)
How many times does 2x go into 14x?
Now, how many times does 2x go into -12x2?
Again, don’t worry about the - 4
-12x2 divided by 2x is -6x times.
Bring down the next term
Done!
Bring down the next term.
Subtract

11. -14x 3x2 -2x - 4 +12 15x3 -9x2 0 -10x2 -10x2 + 6x 0 - 20x - 20x + 12
Your turn!!
3x2
-2x
- 4
-14x
+12
15x3 -9x2 x2
(yikes!!)
-10x2 + 6x x
- 20x + 12

12. This presentation may be used by k-12 public education teachers.
Contact me for permission to use this presentation in whole or in part
for any other purpose. Feel free to send suggestions or to comment
on any errors/omissions. or
Thomas M. Kenyon
6455 Habgood Road
Black Creek, NY 14714
This presentation may be used by k-12 public education teachers.
Contact me for permission to use this presentation in whole or in part
for any other purpose. Feel free to send suggestions or to comment
on any errors/omissions. or
Thomas M. Kenyon
6455 Habgood Road
Black Creek, NY 14714