Calculus - please help?

Question

The 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?

I got this wrong on a test and I really don't know the answer.

Answer 1

well necessary and sufficient are logical terms - so the type of mathematical problem you are looking at doesnt matter.

suppose you want to say that P is necessary for Q
if Q then P ( Q --> P)

suppose you want to say that P is sufficient for Q
that would be ( P --> Q)

which leads to "P is necessary and sufficient for Q" is the same as P if and only if Q

so above lets let P be "x-a is a factor of f(x)"
Q be "f(a) = 0"

so the factor theorem then looks like P if and only if Q
so P is 'necessary and sufficient' for Q

also recall iff is a biconditional

so one can say Q if and only if P also
so Q is 'necessary and sufficient' for P

that should be enough help :)

Answer 2

f(a) = 0 is a necessary, but not sufficient condition.

The sufficient condition is that f(x) = (x - a) g(x); where g(x) is a polynomial subset of f(x). Here we can see that f(a) = (a - a) g(a) = 0.

I don't know where your math is, but to prove the theorem, you need to do a Taylor expansion of the polynomial f(x). [See source.]