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)).
1. The sufficient condition of the factor theorem is that if (x-a) is a factor of f(x), then f(a) = 0. This means that if you have a polynomial f(x) and you divide it by (x-a) and the remainder is 0, then (x-a) is a factor of f(x).
2. The necessary condition of the factor theorem is that if f(a) = 0, then (x-a) is a factor of f(x). This means that if you have a polynomial f(x) and you know that (x-a) is a factor of f(x), then when you substitute a into f(x) (i.e., evaluate f(a)), the result should be 0.
To understand why your answer was incorrect, let's break down the if and only if statement in the factor theorem:
- The statement "x-a is a factor of f(x)" is the sufficient condition.
- The statement "f(a) = 0" is the necessary condition.
The if and only if statement means that both conditions are true. In other words, if (x-a) is a factor of f(x), then f(a) = 0, and if f(a) = 0, then (x-a) is a factor of f(x).
So your correct answer should be:
1. The sufficient condition of the factor theorem is f(a) = 0.
2. The necessary condition of the factor theorem is (x-a) is a factor of f(x).
Remember that in an if and only if statement, both conditions are necessary and sufficient for each other.