How to Divide Polynomials By Monomials

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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 4x3 – 6_x_2 + 3_x_ – 9 by the monomial 6_x_ using the following steps:

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

    (4x3 – 6_x_2 + 3_x_ – 9)/6_x_

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

    (2_x_2/3) – (x) + (1/2) – (3/2_x_)


    • 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_ = 4x3 – 6_x_2 + 3_x_ – 9

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

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