2.1.2—Polynomial division

Divide a polynomial by x−a using long division or synthetic division. The remainder theorem gives remainder P(a), and P(a)=0 means x−a is a factor.

Write descending powers, include zero coefficients, and verify by divisor×quotient+remainder. Use a known factor to reduce the degree before solving the remaining polynomial.

For P(x)=x³−2x²−x+2, P(1)=0, so x−1 is a factor; division gives x²−x−2=(x−2)(x+1).

A zero remainder proves divisibility by the chosen factor only; it does not make every nearby value a root.