3.1.3—Factor and remainder theorems

For polynomial P, P(a)=0 exactly when x−a is a factor. Dividing by x−a leaves remainder P(a), so a quick substitution tests a candidate root.

After finding one factor, divide it out and solve the quotient. Repeated roots can be detected by testing the quotient or by checking derivatives when appropriate.

If P(−2)=0, x+2 divides P; a cubic then becomes a quadratic quotient that can be solved exactly.

The sign is tied to the factor: root a gives x−a, while root −a gives x+a.