Factor Theorem

Remainder Theorem

When \(f(x)\) is divided by \((x - a)\) then the remainder is \(f(a)\). The remainder will always have a degree less than the degree of the divisor.

Factor Theorem

Factor theorem is a special case of the remainder theorem where \(f(a) = 0\). If the remainder is equal to zero, then we know that \((x - a)\) is a factor of \(f(x)\).

Factor and Remainder Theorem Extended

If \((ax - b)\) is a factor of \(f(x)\) then \(f(\frac{b}{a}) = 0\). When \(f(x) = a_0x^n + a_1x^{n-1} + ... + a_{n-1}x + a_n\) then any possible factors of \(f(x)\) will be a factor of \(a_n\) divided by a factor of \(a_0\)