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.
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.
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.
Is that want you wanna do?.
O.K.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...
No! Thank you enough! I've already done that last year (in algebra), but for any number in 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.Unless you want to be a purist and prove the remainder theorem as well.
Is that want you wanna do?.