Bézout's theorem. Factorisation

Bézout's theorem for polynomials states that if a polynomial $f(x)$ is divided by a linear polynomial $x - a$, the remainder is $f(a)$. This means $f(x)$ can be written as $f(x) = (x - a)q(x) + r$, where $r$ is a constant (the remainder), and $q(x)$ is the quotient polynomial. If $f(a) = 0$, then $x - a$ is a factor of $f(x)$. This is closely related to the Factor Theorem.

Important properties

• The remainder when dividing $f(x)$ by $x - a$ is $f(a)$.

• If $f(a) = 0$, then $x - a$ divides $f(x)$ exactly.

• This helps in factorising polynomials by finding roots.