Remainder and factor theorems
Dividing a polynomial takes a page of work. What can one evaluation tell you instead, and how many roots must a polynomial have?
Where you are 3 inside
What this area is about
These are results about polynomials and their roots. They link three things: the value p(c), the remainder when p(x) is divided by x - c, and whether x - c is a factor. Together they say where to look for roots, and how many there are.
How these fit together
All three children are theorems, and each is used as a tool.
The remainder and factor theorems: dividing p(x) by x - c leaves the remainder p(c), so x - c is a factor exactly when p(c) = 0.
The rational root theorem: for a polynomial with integer coefficients and a_0 \ne 0, any rational root in lowest terms has a numerator dividing the constant term a_0 and a denominator dividing the leading coefficient a_n. That gives a short list of candidates to test.
The fundamental theorem of algebra: every polynomial of degree n \ge 1 with complex coefficients has a complex root, so it has exactly n roots when repeats are counted.
A path through it
The remainder and factor theorems come first; the rational root theorem and the fundamental theorem both build on them.
Next: whether a graph crosses or touches at a zero, and later irreducibility tests in abstract algebra.