Math 101 Intermediate Algebra
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Polynomial Division
Chapter 5, Sections 4

Dividend, Divisor, Quotient, and Remainder

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

Problem and answer will look like dividend
divisor		 = quotient + remainder
divisor

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

Answer: x + 3 + 3
x + 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 + 9
3xy		 =   3x2y
3xy		   +   5xy3
3xy		   +   9
3xy
  =   x   +   5y2
3		   +   3
xy
Answer: 3x   +   5y2   +   3
xy

Example

a2b2c - 6abc2 + 5a3b5
2abc2		 =   a2b2c
2abc2		   -   6abc2
2abc2		   +   5a3b5
2abc2
  =   ab
2c		   -   3   +   5a2b4
2c2
Answer: ab
2c		   -   3   +   5a2b4
2c2

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