I'm studying AS maths, and am trying to connect my thoughts around polynomials and the factor and remainder theorems.
I understand the factor theorem and its application: it helps me find roots of a polynomial.
The remainder theorem is also easy to understand, until I ask myself what the remainder is useful for. Why do I need it?
Here's a concrete example:
When $x^3 + 3x + k$ is divided by $x - 1$ the remainder is 6. Find the value of $k$.
So if I evaluate $f(1)$ for the polynomial, I'm given a constant, thus:
$f(1) = 6$ therefore $6 = 1^3 + 3(1) + k$ which when rearranged equals $2 = k$. So in this example it's easy to understand why I'd use it.
What else is the remainder useful for? Or is it just to find constants?