“Why is b a factor of 24 in this example? It is not immediately obvious to me.”

This isn’t part of the factor theorem but “common sense”, if the polynomial factors at at then:$

$(x – b)(x-c)….(x -d) = x^3 -3x^2 – 10x + 24$ then all the b, c, …d multiplied together will be 24. So b is a factor of 24.

if it were 70 you’d look for factors of 70.

“Does it only work for degree 3 and more”

No, it works for degrees 2 and 1. But the quadratic equation and linear solutions make those cases easier.

Proof:

Well, okay… For any polynomial can rewrite P(x) = Q(x)Z(x) + R(x) where the degree of P is equal to the degree of Q plus the degree of Z and the degree of R(x) {the remainder polynomial; what is left when your force the division} is less than the degree of Z or Q. Basically this is dividing P(x) by Z(x) to get Q(x) plus a remainder polynomial R(x). You can always do this. It just might not do you any good.

So we can write: P(x) = Q(x)(x – b) + R(x) where degree of R(x) is less then the degree of (x – b) which is 1. So the degree of R(x) = 0 so R(x) = r. For some real number r.

So P(x) = Q(x)(x – b) + r

but if P(b) = 0 the P(b) = Q(b)(b – b) + r = Q(b)0 + r = r. So r = 0.

So P(x) = Q(x)(x -b) for some polynomial Q(x) so (x -b) divides P(x).