Math 101 Intermediate Algebra    Polynomial Division
Chapter 5, Sections 4

Dividend, Divisor, Quotient, and Remainder

 Problems will look like dividenddivisor or dividend ÷ divisor,
where the dividend and divisor are polynomials.

 Problem and answer will look like dividenddivisor = quotient + remainderdivisor

Check that the division has been performed correctly (i.e., the correct quotient and remainder have been found) by

quotient · divisor + remainder = dividend

That is, multiplying the quotient by the divisor and adding in the remainder has to result in the dividend.

Example

 Problem: x2 + 5x + 9x + 2 Dividend: x2 + 5x + 9 Divisor:  x + 2

 Answer: x + 3 + 3x + 2 Quotient: x + 3 Remainder: 3

Note that (x + 3)(x + 2) + 3 = x2 + 5x + 9,
that is, quotient · divisor + remainder = dividend.

Dividing a Polynomial by a Monomial

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

That means, apply the distibutive property and simplify.

Example

 3x2y + 5xy3 + 93xy = 3x2y3xy + 5xy33xy + 93xy = x + 5y23 + 3xy
 Answer: 3x + 5y2 + 3xy

Example

 a2b2c - 6abc2 + 5a3b52abc2 = a2b2c2abc2 - 6abc22abc2 + 5a3b52abc2 = ab2c - 3 + 5a2b42c2
 Answer: ab2c - 3 + 5a2b42c2

Long Polynomial Division

Long polynomial division may always be used when the divisor has more than one term. That is, the divisor is a binomial or trinomial or etc.

Long polynomial division is a technique for finding the quotient and remainder given the dividend and divisor.

Long polynomial division is performed much like long division of numbers.

The following examples will demonstrate how to do long polynomial division:

Example 1
Example 2
Example 3