Question

Find the quotient and remainder using synthetic division for

\frac(x^3 + 3 x^2 + 4 x + 8)(x+2)

Answer 1

Any ideas on how to get started?

Answer 2

I'm pretty sure it's divison. that's why it says \frac in front.

Answer 3

I know, the \frac means "fraction"

but do you have any ideas on how to start the synthetic division?

Answer 4

I have no idea what synthetic division even means.

Answer 5

Have you learned polynomial long division?

Answer 6

Yes, I'm doing it right now on a different problem but I don't know what synthetic divison is.

Answer 7

synthetic division is like polynomial division, but you only focus on the coefficients and it's slightly faster and easier to remember (in my opinion)

Answer 8

however, synthetic division can't be used all the time

Answer 9

does it end up having a remainder at the end? If so, then I'm doing this current problem wrong and I've been doing this problem exactly right.

Answer 10

to get started, you need to identify the coefficients

x^3 + 3 x^2 + 4 x + 8

is the same as

1x^3 + 3 x^2 + 4 x + 8

The coefficients are 1, 3, 4, 8

Because we're dividing by x+2, this means we're using -2 as a test zero (since -2 makes x+2 equal to 0)

So set up the synthetic division table to get

-2 | 1 3 4 8
|
----------------------------------------


Then drop the first coefficient (1) to get this

-2 | 1 3 4 8
|
----------------------------------------
1


Then multiply the outer test zero (-2) by that number in the very bottom to get -2*1 = -2. Write this under the 3 like so

-2 | 1 3 4 8
| -2
----------------------------------------
1


Then add 3 and -2 to get 3+(-2) = 1. This goes right under the -2 to get this

-2 | 1 3 4 8
| -2
----------------------------------------
1 1

Then you repeat the last two steps

-2 | 1 3 4 8
| -2 -2
----------------------------------------
1 1


-2 | 1 3 4 8
| -2 -2
----------------------------------------
1 1 2


Keep going until you fill up the second and third rows with numbers.

Answer 11

So you need to do 3 more steps (because you'll stop when the last row has 4 values in it.

Answer 12

I am so confused by this

Answer 13

Do you have any notes about synthetic division? If not, then you may need to review it.

Answer 14

No notes, nothing. My teacher just kind of shows things on the board and says "do this"

Answer 15

Do my steps help at all?

Answer 16

not really /:

Answer 17

do you see how I got the coefficients 1 3 4 8

Answer 18

yes I understand that. the numbers that are part of the problem are the coefficients, but I don't understand is how to exactly get the answer that I'm looking for

Answer 19

do you see how I got the -2 (that's on the very left)

Answer 20

basically plugging it in for x to make 0 or something like that

Answer 21

yes, solving x+2 = 0 gives you x = -2

Answer 22

so

x+2 = 0

-2+2 = 0

0 = 0

Answer 23

This gives us the starting table


-2 | 1 3 4 8
|
----------------------------------------

Answer 24

The next step after this is to write the first coefficient 1 under the line to get this


-2 | 1 3 4 8
|
----------------------------------------
1

Answer 25

Step 3) Multiply -2 by 1 (the 1 that's in the bottom row) to get -2. Then write this under the 3 in the first row to get this


-2 | 1 3 4 8
| -2
----------------------------------------
1


Step 4) Add 3 and -2 to get 1. Write this under the -2


-2 | 1 3 4 8
| -2
----------------------------------------
1 1

Answer 26

You keep repeating these two steps until you reach the end of the row

Answer 27

-2 | 1 3 4 8
| -2 -4
--------------------------
1 1 2 4

is that right?

Answer 28

close, the last row is correct

Answer 29

the second row is not correct

Answer 30

but you put that down yourself....

Answer 31

I put this down


-2 | 1 3 4 8
| -2 -2
----------------------------------------
1 1

but I wasn't finished (since I wanted to see if you could finish or not)

Answer 32

The full and completed synthetic division table should be this


-2 | 1 3 4 8
| -2 -2 -4
----------------------------------------
1 1 2 4

Answer 33

that is basically what I put down. I just filled in the blanks...

Answer 34

you were missing a -2 in the second row though

Answer 35

okay. I get the point. Making this table still doesn't really give me the answer I'm looking for

Answer 36

it does, if you know how to read it

the bottom row is what is what we take away from the table

the bottom row is 1, 1, 2, 4

The first three values form the coefficients of the quotient and the last value is the remainder

Answer 37

got it

Answer 38

so the coefficients of the quotient is 1, 1, 2

So the quotient is \[\Large x^2+x+2\]

Answer 39

x^2+x+2

remainder is 4

Answer 40

you nailed it

Answer 41

thanks

Answer 42

yw