2.1.3—Factor and remainder theorems

The factor theorem says x−a is a factor of P(x) exactly when P(a)=0. The remainder theorem says division by x−a leaves remainder P(a).

Test simple candidates first, divide out a confirmed factor, then solve the lower-degree quotient. A repeated root also requires P′(a)=0.

For P(x)=x³−4x²+x+6, P(2)=0, so x−2 is a factor; division reduces the cubic before finding the remaining roots.

P(a)=0 identifies a factor x−a, not x+a; substitute the sign carefully.