The Division Algorithm for polynomials states that given any two polynomials, a dividend and a non-zero divisor, there exist unique quotient and remainder polynomials. The degree of the remainder polynomial is less than the degree of the divisor.
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The Division Algorithm can be written as $f(x) = d(x)q(x) + r(x)$ where $f(x)$ is the dividend, $d(x)$ is the divisor, $q(x)$ is the quotient, and $r(x)$ is the remainder.
The degree of the remainder polynomial $r(x)$ must be less than the degree of the divisor polynomial $d(x)$. If $r(x) = 0$, then $d(x)$ divides $f(x)$ exactly.
Polynomial long division and synthetic division are common methods used to apply the Division Algorithm.
The Remainder Theorem states that if a polynomial $f(x)$ is divided by $(x - c)$, then the remainder is $f(c)$.
Finding zeros of polynomial functions often involves using the Division Algorithm to factorize polynomials.