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.
Review Questions
How do you express a polynomial division using the Division Algorithm?
What must be true about the degree of the remainder in comparison to the divisor in polynomial division?
Explain how you would use synthetic division to divide a polynomial by a binomial.