Factor theorem states that "x-a is a factor of f(x) if and only if f(a)=0"
1. What is the sufficient condition of the factor theorem?
2. What is the necessary condition of the factor theorem?
My answer was x-a is a factor of f(x) for (1.) and f(a)=0 for (2.) but it is wrong.
You have an if and only if statement, what does that tell you?
I don't know..
I did it wrong on a test and I don't know why.
From my notes I know
sufficient condition (if)
necessary condition (only if)
Ok, you might've missed something about if and only if. We abbreviate 'if and only if' by iff, so P if and only if Q is the same as P iff Q.
We might also use symbols like P<=>Q to denote P iff Q.
In any event, P iff Q is the same as P =>Q 'and' Q=>P. Both implications must be true. The statement P iff Q is another way of saying the two statements are logically equivalent. In this type of statement 'both' parts of the statement are sufficient and neccessary conditions for the other part. Any time you see an iff theorem think "logical equivalence". This is a very common statement type in mathematics.
Your answer would be correct if we had P=>Q. In that case P is sufficient and Q is necessary.
You are incorrect however in associating 'only if' with necessary. P only if Q is another way of saying Q=>P
Just a word about the factor theorem. That thm is a statement of equivalence between the statements: a is a root of p(x) and (x-a)|p(x). (a is a root of some poly. p and (x-a) divides p(x)).