What is the remainder theorem and how does it save time in polynomials?

The remainder theorem says that if you divide a polynomial p(x) by a linear factor (x − a), the remainder is simply p(a). So instead of doing full long division, you just substitute x = a into the polynomial. For example, to find the remainder when p(x) = x³ − 2x² + 5x − 1 is divided by (x − 2), compute p(2) = 8 − 8 + 10 − 1 = 9. The remainder is 9. If the divisor is (x + 3), write it as (x − (−3)) and substitute x = −3. This only works for linear divisors. It is a huge time saver in exams and is also the basis of the factor theorem, where remainder 0 means the divisor is a factor.