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C3 Remainder Theorem

Using the identity

F(x) ≡ Q(x) x divisor + remainder

How do you know when the remainder is quadratic or linear?

Im doing myself so i need a bit of help.
It depends on what order polynomial you start with, and what you're dividing by.
eg. start with x^4+1/x^2 then you get a quad and a integer remainder.
I've never seen this being asked though in an exam:s-smilie: don't quote me though.
Reply 2
Avatar for Bengaltiger
Bengaltiger
OP
i still dont understand why the remainder is an interger in this case. Can someone explain when to use a linear remainder or an interger or a quadratic remainder...!!
Well if you were to do algebraic division, you can see this.
I've just tried a couple of examples to see. I remember thinking exactly the same some time back, so i figured out that if you take the numerators index away from the denominators ie. 4-2. you know that it's going to be a binomial. ie x^2 is going to be the Q(x) therefore you must allow yourself Ax+B at least. then the remainder, you need to look at the eq again, and in x^4+1, you see that 1 is not divisible by x^2. so it's an integer remainder so only add D

If however, it was x^4+x^2+x./x^2 then do the same procedure 4-2. so (Ax^2+Bx+C)(x^2) + Dx+E as you can see x is not divisible by x^2.

Even though some letters equal 0, that's not a problem. it's correct. check by division.
hope that clears it up a bit for you.
Reply 4
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Bengaltiger
OP
Hmm....thanks for that little help there....Ill do some examples and see if this works.

Thanks again.
Reply 5
Avatar for RichE
RichE
15
Bengaltiger
Using the identity

F(x) ≡ Q(x) x divisor + remainder

How do you know when the remainder is quadratic or linear?

Im doing myself so i need a bit of help.


In general the degree of the remainder will be strictly less than that of the divisor.

So if you divide by a quadratic the remainder will be at most degree one; it may also be a constant or even zero if the quadratic divides exactly

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