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the Section 5.4 homework?
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Section 5.6
Dividing Polynomials
Dividing a polynomial by a monomial
Divide each term of the polynomial separately by
the monomial.
This process uses the quotient rule for exponents.
Example
 12a 3  36a  15
 12a 3 36a 15



3a
3a
3a 3a
5
2
 4a  12 
a
Problem from today’s homework:
Dividing a polynomial by a polynomial other
than a monomial (i.e. one with two or more
terms) uses a “long division” technique that is
similar to the process known as long division
in dividing two numbers.
An example of this type of problem
with polynomials would be dividing
6x2 + 17 x + 5 by 3x+1.
Because long division (without a calculator) is
kind of a lost art these days, we’ll work these
two examples with numbers before we move
on to dividing polynomials:
1). 3276 ÷ 9
2). 3278 ÷ 9
Example 1: Long Division with integers
3 64
9 3276
-___
27
57
-___
54
36
-___
36
0
1). 3276 ÷ 9
Divide 9 into 32. (What is the biggest multiple of nine contained in 32?
Multiply 3 times 9.
Subtract 27 from 32. “Draw the line and change the sign.”
Bring down 7.
Divide 9 into 57.
Multiply 9 times 6.
Subtract 54 from 57.
Bring down 6.
Divide 9 into 36.
Multiply 9 times 4.
Subtract 36 from 36.
“Draw the line and change the sign.”
“Draw the line and change the sign.”
This last subtraction gives zero,
so the answer is 364, with no remainder. 3276 ÷ 9 = 364
As you can see from the previous example,
there is a pattern in the long division
technique.
•
•
•
•
•
Divide
Multiply
Subtract “Draw the line and change the sign.”
Bring down
Then repeat these steps until you can’t bring
down or divide any longer.
• Last step: Check your answer by multiplying
the answer by the divisor (see next slide for steps.)
(Always check your answer – you’ll need to know
how to do this for the last quiz and test.)
Question:
3276 ÷ 9 = 364
How can you check your answer to
this long division problem?
Answer:
3276
Think of this as
 364, then multiply both sides by 9:
9
3276  364  9
So we check by multiplying the answer (364)
by the number you divided by (9),
and see if you come up with the
number you were dividing it into (3276).
Check: 364 ∙ 9 = 3276 (do this multiplication by hand)
Now you try it in your notebook:
Use long division
(NOT YOUR CALCULATOR)
to divide 771 by 3.
3 771
Show all of your steps,
and ask for help if you get stuck on any step.
Answer: 257
Example 2: Long Division with integers
3 64
9 3278
-___
27
57
-___
54
38
-___
36
2
2). 3278 ÷ 9
Divide 9 into 32.
Multiply 3 times 9.
Subtract 27 from 32.
Bring down 7.
Divide 9 into 57.
Multiply 9 times 6.
Subtract 54 from 57.
Bring down 8.
Divide 9 into 38.
Multiply 9 times 4.
Subtract 36 from 38.
Write answer as:
This last subtraction leaves us with the
2
number two, and nothing else to bring down,
364 
so the answer is 364, with a remainder of 2.
9
Question:
How can you check your answer to this long division problem?
Answer:
3278 ÷ 9 = 364 +2/9
3276
2
Think of this as
 364  , then multiply both sides by 9:
9
9
3276
2
2
9
 9  (364  )  9  364  9/   9  364  2
9
9
9/
So we check by multiplying the answer (364)
by the number you divided by (9),
then add the remainder (2) to this product
and see if you come up with the
number you were dividing it into (3278).
Check: 364 ∙ 9 + 2 = 3278 (do this multiplication by hand)
Now you try it
(And don’t forget to check your answer!)
Divide 1639 by 7 using long division.
Then check your answer.
Do this in your notebook now, and make sure
you ask if you have questions about any step.
This will be crucial to your understanding of
long division of polynomials.
(ANSWER: 234 + 1/7)
Now we’ll apply this long division pattern to
dividing a polynomial by another polynomial
with two or more terms:
•
•
•
•
•
Divide
Multiply
Subtract “Draw the line and change the signs.”
Bring down
Then repeat these steps until you can’t bring down
or divide any longer.
• Last step: Check your answer by multiplying the
answer by the divisor and then adding the
remainder, if there is one.
(Always check your answer – you’ll need to know how to
do this for the last quiz and test.)
Example with polynomials:
4x  5
7 x  3 28x  23x  15
2
28 x  12 x
 35 x  15
 35x 15
2
So our answer is 4x – 5.
Divide 7x into 28x2.
Multiply 4x times 7x+3.
Subtract 28x2 + 12x from 28x2 – 23x.
“Draw the line and change the signs.”
Bring down -15.
Divide 7x into –35x.
Multiply -5 times 7x+3.
Subtract –35x–15 from –35x–15.
Nothing to bring down.
Check: Multiply (7x + 3)(4x – 5)
and see if you get 28x2 – 23x - 15.
Now you try it
(And don’t forget to check your answer!)
Divide 6x2 – x – 2 by 3x – 2
using long division.
Then check your answer.
Do this in your notebook now, and make sure
you ask if you have questions about any step.
ANSWER: 2x + 1
Example
2x  10
2
2 x  7 4x  6 x  8
2
4 x  14 x
20x  8
20x  70
78
Divide 2x into 4x2.
Multiply 2x times 2x+7.
Subtract 4x2 + 14x from 4x2 – 6x.
“Draw the line and change the signs.”
Bring down 8.
Divide 2x into –20x.
Multiply -10 times 2x+7.
Subtract –20x–70 from –20x+8.
Nothing to bring down.
We write our final answer as 2 x  10 
78
( 2 x  7)
How do we check this answer?
2 x  7 4x  6 x  8
2
Final answer:
2x – 10 + 78 .
2x - 7
How to check: Calculate (2x + 7)(2x – 10) + 78.
If it comes out to 4x2 – 6x + 8,
then the answer is correct.
Now you try it
(And don’t forget to check your answer!)
Divide 15x2 + 19x – 2 by 3x + 5
using long division.
Then check your answer.
Do this in your notebook now, and make sure
you ask if you have questions about any step.
Answer: 5x – 2 +
8
3x + 5
.
Problem from today’s homework:
Reminders:
1.
This homework assignment on section 5.6 is due
at the start of next class period.
2.
You should also start looking at Practice Quiz 3
before the next class period, when we’ll be
reviewing for Quiz 3 on sections 5.1-5.4 & 5.6.
3.
If you have yet to pass the Gateway Quiz and
haven’t taken this week’s version yet, come in
and take it during the scheduled hours this
week. There are only two more weeks to go in
the semester after this week.
Gateway Quiz Retake Times
(One new attempt allowed per week, beginning November 2)
• Mondays
• Wednesdays
– 1:25 pm
– 2:30 pm
– 10:10 am
– 11:15 am
• Tuesdays
• Thursdays
– 1:25 pm
– 3:35 pm
– 10:10 am
– 11:15 pm
SIGN UP IN THE MATH TLC OPEN LAB!
If NONE of the above times work for you…
email Krystle Mayer, Math TLC Coordinator (JHSW 201),
to set up a date and time.
You may now
OPEN
your LAPTOPS
and begin working on the
homework assignment.