Remainder & Factor Theorems

The Remainder Theorem: dividing P(x) by (x − c) leaves a remainder of exactly P(c).

So to find a remainder, you can just substitute. No division required.

The Factor Theorem is the special case where that remainder is zero: if P(c) = 0, then (x − c) is a factor, and c is a root.

These three statements describe the same fact:

• c is a root of P
• (x − c) is a factor of P
• (c, 0) is an x-intercept of the graph

That equivalence is the reason factoring finds roots, and finding roots helps you factor.

From the division algorithm P(x) = (x − c)Q(x) + R with R constant, substituting x = c gives P(c) = R — the Remainder Theorem. The Factor Theorem is the case R = 0.

The practical procedure for a higher-degree polynomial is: find one root (by inspection, the rational root theorem, or a graph), divide it out synthetically to depress the degree, and repeat on the quotient until it is quadratic and can be solved directly.

The equivalence of root, factor and x-intercept underlies the connection between algebra and graphing throughout the course. It also extends to multiplicity: (x − c)^m divides P exactly when c is a root of multiplicity m.

Note that the theorems hold over any field, including ℂ, so a complex root c yields a complex linear factor. Combined with the Fundamental Theorem of Algebra, this gives the complete factorisation of any polynomial into linear factors over ℂ.