Question

In a polynomial division problem, the dividend does not have an x^3-term. When the problem is written in long division form, what should be the coefficient of the x^3-term in the dividend?

A. 1
B. 0
C. -1
D. 3

Answer 1

Do we get to see the problem?

Answer 2

That is the problem @fibonaccichick666

Answer 3

ok, so let's think about this then

Answer 4

first, can you tell or show me what a dividend is?

Answer 5

yes, its the certain amount that you want to divide up
dividend/divisor=quotient

Answer 6

ok, cool, now just one quick note before we continue, division is just fractions, so keep fraction rules and division rules in mind. However, this problem is much simpler than we think. Are you ok with me using a generic form for a polynomial or would you prefer a specific example?

Answer 7

What ever works best.. Just be prepared to explain if i get confused. :P

Answer 8

ok, why don't I do a specific example, if you are not in theoretical math, it will be much easier to understand.

Answer 9

yes that works!!

Answer 10

So let us try doing \(x^4+2x^2+3\) divided by \(x^3+x^2+1\)

Answer 11

Can you just try setting it up for me?

Answer 12

Try setting it up using our old long division format

Answer 13

let me know what you get or if you are confused

Answer 14

Just set it up, don't try to solve it

Answer 15

ok sorry my internet went down.. so let me read what you said real quick

Answer 16

it's cool, I just thought you were trying to solve it

Answer 17

it would be \[\left( x^4+2x^2+3 \right) \div \left( x^3+x^2+1 \right)\]

Answer 18

You mean like that?

Answer 19

not quite, so long division style means (assume the square root is just a division thingy, I don't actuall know what it is called...)
\[(x^3+x^2+1)\sqrt{x^4+2x^2+3}\]

Answer 20

poop I was close.

Answer 21

you were, and this way of writing these is only used when doing poly div.

Answer 22

so anyway, that is the first part

Answer 23

so all this wants to know is how else we can write \(x4+2x2+3\) so that it "contains" an \(x^3\) term

Answer 24

whoopsie \(x^4+2x^2+3\)

Answer 25

so what does that mean? It means we sort of add a ghost term in, one that is there but the things are still equal with or without it.

Answer 26

so we start by adding in the x^3 term, with an unknown coeficient

Answer 27

\(x^4+ax^3+2x^2+3\)

Answer 28

So now, we have hit the meat and potatoes of your question, what value of a makes it the exact same as \(x^4+2x^2+3\)

Answer 29

1?

Answer 30

What do you think a is?

Answer 31

well, if a=1, we have \[x^4+ax^3+2x^2+3=x^4+1x^3+2x^2+3\] is that the same as \(x^4+2x^2+3\)?

Answer 32

no?

Answer 33

correct, no they are not the same

Answer 34

the x^3 term will not go away

Answer 35

we need an a value where the x^3 term goes away or vanishes

Answer 36

i think we can eliminate 1 and -1 right?

Answer 37

nope, because we only can put 1 variable in

Answer 38

forget the multiple choice

Answer 39

just be logical

Answer 40

3?

Answer 41

5+?=5
or x+?=x think about what the question mark would be

Answer 42

0

Answer 43

what does the x^3 term have to equal in order for the polynomial to remain the same

Answer 44

you are guessing

Answer 45

provide me with why

Answer 46

Ok I think its 0 because. 5+?=5 if you take 5+3 = 8 it equals 8 as 5+0=5

Answer 47

ok, so, yes the ? will =0. It is due to the additive identity if we want to be technical

Answer 48

so how does that help us?

Answer 49

any ideas?

Answer 50

uhm we want something that'll will get rid or keep it the same?

Answer 51

ok, so whatever we add must equal what?

Answer 52

what can we add to any number and it will remain the same?

Answer 53

x^3?

Answer 54

no, we cannot add x^3 to any number and have it be the same

Answer 55

oh oh oh 0

Answer 56

brainfart moment

Answer 57

ok, there we go(and it happens to all of us)

Answer 58

so we need to add zero somehow, no

Answer 59

we have not arrived at your answer yet

Answer 60

oh xD ok

Answer 61

now, we need to add zero, but we also need to add an x^3 term, so we need to find a value of a such that 0=ax^3

Answer 62

Once you figure out what value of a works, you will have your answer. As I do not think you fully understand, please check this reference. An example is located about halfway down the page. http://www.purplemath.com/modules/polydiv3.htm