I have to demonstrate that " is divisible byif and only if."

My work : If is divisible by , then I can write .

If , then , which is true. So ) is proved.

Now I have to show "if is divisible by , then ." I don't know how to continue this.

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- Jul 8th 2008, 01:45 PMarbolis[SOLVED] Help for a proof : polynomial divisibility
I have to demonstrate that " is divisible by

**if and only if**."

My work : If is divisible by , then I can write .

If , then , which is true. So ) is proved.

Now I have to show "if is divisible by , then ." I don't know how to continue this. - Jul 8th 2008, 01:56 PMgalactus
I am going to use (x-c) instead of alpha. OK?

Let's use the remainder theorem.

Then,

For some quotient q(x).

If P(c)=0, the P(x)=(x-c)q(x); that is, x-c is a factor of P(x).

Also, if x-c is a factor of P(x), then the remainder when we divide P(x) by

x-c must be 0. Therefore, by the remainder theorem P(c)=0.

Is that what you were getting at?. Unless you want to be a purist and prove the remainder theorem as well.(Nerd)(Clapping)

Is that want you wanna do?. - Jul 8th 2008, 02:05 PMarbolisQuote:

I am going to use (x-c) instead of alpha. OK?

I just understood your proof. I knew the remainder theorem, but not applied to polynomials! Thanks, I got it...

Quote:

Unless you want to be a purist and prove the remainder theorem as well.(Nerd)(Clapping)

Is that want you wanna do?.

**Z**. The proof I had to do now is an exercise I got in Calculus II, so I won't be a purist when it comes to algebra, ahaha.