2. 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?
How many times does 3x go into 15x3?
times
8
3
24
times
5
7
35
times
3
4
12
times
4x
2
8
2
8 3
2
5
2
5
x
x
x
x
times
5
3
15 2
3
x
x
x
Review:
3. Long division review:
74 50690
Steps:
How many times does 74 go into 506?
ans: 6
6
Multiply 6*74
ans: 444
444
Subtract
Bring down the next
Repeat the steps
62
9 How many times does 74 go into 629?
ans: 8
Multiply 8*74
ans: 592
Subtract
Bring down the next
Repeat the steps
8
592
37
0 How many times does 74 go into 370?
ans: 5
5
Multiply 5*74
ans: 370
370
Subtract
0
DONE!
4. 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!
comment:
5. A polynomial long division problem looks like this:
)
2
3
(
)
10
7
12
( 2
x
x
x
or this:
10
7
12
2
3 2
x
x
x
2
3
10
7
12 2
x
x
x
or this:
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
by a monomial. We will adapt this new method to
learning long division by a polynomial with 2 or more
terms in it.
x
x
x
x 9
6
12
3 2
3
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.
comment:
7. x
x
x
x 9
6
12
3 2
3
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.
4x2
Second, we multiply 4x2 times 3x.
12x3
0
Now, we bring down the next term. Repeat the process.
2
6x
Third, we subtract.
How many times does 3x go into -6x2?
Divide -6x2 by 3x to find out. In this case, -2x times.
-2x
Second, we multiply -2x times 3x
-6x2
Third, we subtract again.
0
x
9
Bring down the next term. Then repeat once more.
How many times does 3x go into +9x?
Positive 3 times.
+ 3
Multiply 3 times 3x
9x
Subtract
0
Done!!!
9. 15
11
2
6
5
3 2
3
x
x
x
x
(yikes!!)
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!
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
2x2
Now, multiply 2x2 by 3x + 5. 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)
6x3 +10x2
Subtract.
Be careful! When you subtract
a polynomial, you subtract all of
the terms. All the signs change.
(the second term becomes – 10x2 )
0 -12x2
-11x
Bring down the next term.
Now, how many times does 3x go into -12x2?
Again, don’t worry about the + 5
-12x2 divided by 3x is -4x times.
-4x
Multiply
Multiply -4x by the whole divisor
That is, -4x times 3x + 5
(again, distributive property)
-12x2 -20x
Subtract
Again, be careful with signs.
You’re subtracting negative 20x.
Subtracting changes the sign, so it’s the
same as adding 20x.
0 + 9x
Bring down the next term
+ 15
How many times does 3x go into 9x?
+ 3
Multiply…
Multiply the 3 by the entire divisor…
3 times 3x + 5.
(use the distributive property)
9x + 15
Subtract
0 + 0
Done!
Notice: we still always get a
zero for the first term when we
are subtracting.
10. 28
38
32
10
4
2 2
3
x
x
x
x
(yikes!!)
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!
How many times does 2x go into 10x3??
Divide 10x3 by 2x to find out.
This time, it’s 5x2
5x2
Now, multiply 5x2 by 2x - 4. 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)
10x3 -20x2
Subtract.
Be careful! When you subtract
a polynomial, you subtract all of
the terms. All the signs change.
(the second term becomes +20x2 )
0 -12x2
+38x
Bring down the next term.
Now, how many times does 2x go into -12x2?
Again, don’t worry about the - 4
-12x2 divided by 2x is -6x times.
-6x
Multiply
Multiply -6x by the whole divisor
That is, -6x times 2x - 4
(again, distributive property)
-12x2 + 24x
Subtract
Again, be careful with signs.
You’re subtracting 24x.
0 + 14x
Bring down the next term
-28
How many times does 2x go into 14x?
+ 7
Multiply…
Multiply the 7 by the entire divisor…
7 times 2x - 4.
(use the distributive property)
14x - 28
Subtract
0 + 0
Done!
Notice: we still always get a
zero for the first term when we
are subtracting.
example 2:
