and turn off and put away your cell phones,

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

12 a
3 
36 a

15
3 a


12 a
3
3 a

36 a
3 a

15
3 a
 
4 a
2 
12

5 a

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
6x 2 + 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

1). 3276 ÷ 9
3 6 4
9 3276
5 7
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.
3 6
0
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:
Think of this as
32 76
9

364 , then multiply both sid es b y 9 :

3 64

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)

3 771
Show all of your steps, and ask for help if you get stuck on any step.
Answer: 257

2). 3278 ÷ 9
3 6 4
9 3278 Divide 9 into 32.
Multiply 3 times 9.
5 7
Subtract 27 from 32.
Bring down 7.
3 8
2
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.
This last subtraction leaves us with the number two, and nothing else to bring down, so the answer is 364, with a remainder of 2.
Write answer as:
2
364

9

Question:
How can you check your answer to this long division problem?
Answer:
3278
÷ 9
= 364 +2/9
Think of this as
32 76
9

3 64

2
, then multiply both sides by 9:
9
9

3276
9
 
364

2
)

9
9

364 9 /
2

9

364

2
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)

(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 sign s
.”
•
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:
7 x

3
4 x

5
28
28 x
2 
23 x
2 
12 x x

15

35 x


35 x

15
15
Divide 7 x into 28 x 2 .
Multiply 4 x times 7 x +3.
Subtract 28 x 2 + 12 x from 28 x 2 – 23 x .
“Draw the line and change the signs.”
Bring down -15.
Divide 7 x into –35 x .
Multiply -5 times 7 x +3.
Subtract –35 x –15 from –35 x –15.
Nothing to bring down .
So our answer is 4x – 5.
Check: Multiply (7x + 3)(4x – 5) and see if you get 28x 2 – 23x - 15.

(And don’t forget to check your answer!)
Divide 6x 2 – 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
2 x

7
2 x

10
4
4 x x
2
2 

6
14 x x

8

20 x


20 x

8
70
78
Divide 2 x into 4 x 2 .
Multiply 2 x times 2 x +7.
Subtract 4 x 2 + 14 x from 4 x 2 – 6 x .
“Draw the line and change the signs.”
Bring down 8.
Divide 2 x into –20 x .
Multiply -10 times 2 x +7.
Subtract –20 x –70 from –20 x +8.
Nothing to bring down .
We write our final answer as 2 x

10

( 2
78 x

7 )

2 x

7 4 x
2

6 x

8 Final answer:
2x – 10 + 78 .
2x - 7
How to check: Calculate (2x + 7)(2x – 10) + 78.
If it comes out to 4x 2 – 6x + 8, then the answer is correct.

(And don’t forget to check your answer!)
Divide 15x 2 + 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
.

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 March 7)
• Mondays
– 1:25 pm
– 2:30 pm
• Tuesdays
– 10:10 am
– 11:15 am
• Wednesdays
– 10:10 am
– 11:15 am
• Thursdays
– 1:25 pm
– 2:30 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) or
Dr. Laura Schmidt, to set up a date and time.

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