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Dividing Polynomials, Part 1

Section 5.6 Part 1
Dividing a polynomial by a
Divide each term of the polynomial separately by the monomial.
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

.
3y + y – 4y x x 2

Dividing a polynomial by a polynomial other than a monomial uses a “ long division
” technique that is similar to the process known as long division in dividing two numbers.
This process is reviewed in detail on the next slide, but first, try these two simpler examples in your notebook
(use long division, not your calculator):
1). 225 ÷ 9 (Answer: 25)
2). 232 ÷ 9 (Answer: 25 with a
..
remainder of 7, or 25 + 7/9)
Question: How can you check your answers on long division problems? (A: Multiply answer times divisor, e.g. 25 * 9 = 225)

1 6 8 Divide 43 into 72.
Multiply 1 times 43.
Subtract 43 from 72.
Bring down 5.
43
29 5
258
37 6
344
32
Divide 43 into 295.
Multiply 6 times 43.
Subtract 258 from 295.
Bring down 6.
Divide 43 into 376.
Multiply 8 times 43.
Subtract 344 from 376.
Nothing to bring down.
32 is smaller than 43 , so we are done.
We then write our result as 168
32
43

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.
We will incorporate this same repeated technique with dividing polynomials.

Now you try it
(And don’t forget to check your answer!)
Divide 3473 by 6 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: 578
5
6
(Can also be written as 578 + )
6

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 .
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.

Now you try it
(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.
ANSWER: 2x + 1
Check: Multiply (2x + 1)(3x – 2). What do you get?

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 .
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
2 x

10
4
4 x x
2
2 

6
14 x x

8

20 x


20 x

8
70
78
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.

Now you try it
(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.
Answer: 5x – 2 + 8
3x + 5
.

REMINDERS:
The assignment on t oday’s material ( HW 5.6A
) is due at the start of the next class session.
Homework Questions?
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We expect all students to stay in the classroom to work on your homework till the end of the 55minute class period. If you have already finished the homework assignment for today’s section, you should work ahead on the next one or work on the next practice quiz/test.