Once you've learned the basics of polynomials, the logical next step is learning how to manipulate them, just as you manipulated constants when you first learned arithmetic. Dividing polynomials might seem like the most intimidating of the operations to master, but as long as you remember the basic rules about adding and subtracting fractions and simplifying them, it's a surprisingly simple process.

#### TL;DR (Too Long; Didn't Read)

Write the division out as a fraction, with the polynomial as the numerator and the monomial as the denominator. Then break the polynomial apart into individual terms (each over the denominator/divisor) and simplify each term.

## Dividing a Polynomial by a Monomial

Consider the following example: Divide the polynomial 4x^{3} – 6_x_^{2} + 3_x_ – 9 by the monomial 6_x_ using the following steps:

## Write as a Fraction

Write the division out as a fraction, with the polynomial as the numerator and the monomial as the denominator:

(4x^{3} – 6_x_^{2} + 3_x_ – 9)/6_x_

## Break out Individual Terms

Rewrite the fraction as a series of individual terms, each over the denominator:

(4_x_^{3}/6_x_) – (6_x_^{2}/6_x_) + (3_x_/6_x_) – (9/6_x_)

## Simplify Each Term

Simplify each of the terms as much as possible. Continuing the example, this gives you:

(2_x_^{2}/3) – (*x*) + (1/2) – (3/2_x_)

#### TL;DR (Too Long; Didn't Read)

You can check your work by multiplying the result by the original divisor. Concluding this example, you'd have:

[(2_x_^{2}/3) – (*x*) + (1/2) – (3/2_x_)] × 6_x_ = 4x^{3} – 6_x_^{2} + 3_x_ – 9

Because multiplying gives you the same polynomial you started with, your answer is correct.