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Section 5.6Dividing Polynomials

Dividing a polynomial by a monomialDivide each term of the polynomial separately by

the monomial.

This process uses the quotient rule for exponents.

a

aa

3

153612 3 aa

a

a

a

3

15

3

36

3

12 3

aa

5124 2

Example

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

Divide 9 into 32. (What is the biggest multiple of nine contained in 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 6.

Divide 9 into 36.

Multiply 9 times 4.

Subtract 36 from 36.

This last subtraction gives zero, so the answer is 364, with no remainder.

Example 1: Long Division with integers1). 3276 ÷ 9

9 32763

27-___

-___5

3-___

54

6

7

6360

3276 ÷ 9 = 364

4

“Draw the line and change the sign.”

“Draw the line and change the sign.”

“Draw the line and change the sign.”

As you can see from the previous example, there is a pattern in the long division technique.• Divide• Multiply• Subtract• 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.)

“Draw the line and change the sign.”

Question:

How can you check your answer to this long division problem?

3276 ÷ 9 = 364

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)

Think of this as , then multiply both sid364 es 32

b :9

76y 9

Answer:

33 76 92 64

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

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.

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.

Example 2: Long Division with integers2). 3278 ÷ 9

9 32783

27-___

-___5

3-___

54

6

7

8362

Write answer as:

4

2364

9

Question: How can you check your answer to this long division problem?

3278 ÷ 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 productand see if you come up with the number you were dividing it into (3278). Check: 364 ∙ 9 + 2 = 3278 (do this multiplication by hand)

Think of this as , then multiply both 2

39

sides by 9:76

6432

9

Answer:

22

9 9 ( ) 9 9 93276

9

2364 364 364

9 9 /

/

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• 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.)

“Draw the line and change the signs.”

15232837 2 xxx

x4

xx 1228 2 35- x

5

1535 x

Divide 7x into 28x2.

Multiply 4x times 7x+3.

Subtract 28x2 + 12x from 28x2 – 23x.

Bring down -15.

Divide 7x into –35x.

Multiply -5 times 7x+3.

Subtract –35x–15 from –35x–15.

Nothing to bring down.

15-

So our answer is 4x – 5.

Example with polynomials:

Check: Multiply (7x + 3)(4x – 5) and see if you get 28x2 – 23x - 15.

“Draw the line and change the signs.”

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

86472 2 +-+ xxx

x2

xx 144 2+20- x

10-

7020 -- x78

Divide 2x into 4x2.

Multiply 2x times 2x+7.

Subtract 4x2 + 14x from 4x2 – 6x.

Bring down 8.

Divide 2x into –20x.

Multiply -10 times 2x+7.

Subtract –20x–70 from –20x+8.

Nothing to bring down.

8+

+)72(

78+x

x2 10-We write our final answer as

Example

“Draw the line and change the signs.”

86472 2 +-+ xxx

How do we check this answer?

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.

• Wednesdays– 10:10 am– 11:15 am

• Thursdays– 10:10 am– 11:15 pm

Gateway Quiz Retake Times(One new attempt allowed per week, beginning November 2)

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.

• Mondays– 1:25 pm– 2:30 pm

• Tuesdays– 1:25 pm– 3:35 pm

You may now OPEN

your LAPTOPSand begin working on the

homework assignment.