Division Algorithm

Definition

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.

5 Must Know Facts For Your Next Test

1. 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.
2. 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.
3. Polynomial long division and synthetic division are common methods used to apply the Division Algorithm.
4. The Remainder Theorem states that if a polynomial $f(x)$ is divided by $(x - c)$, then the remainder is $f(c)$.
5. 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.

Related terms

Polynomial Long Division: A method used to divide polynomials similar to arithmetic long division, producing a quotient and a remainder.

Synthetic Division: A simplified form of polynomial division, particularly useful when dividing by linear factors.

Remainder Theorem: $\text{If } f(c) \text{ equals zero when } f(x) \text{ is divided by } (x - c), \text{ then } (x - c) \text{ is a factor of } f(x).$