Dividing by a Monomial

Learning Objective(s)

·         Divide a monomial by a monomial.

·         Divide a polynomial by a monomial.

Introduction

The fourth arithmetic operation is division, the inverse of multiplication. Division of polynomials isn’t much different from division of numbers. Let’s start with dividing a monomial by another monomial, which is the basis for dividing a polynomial by a monomial.

Dividing Monomials by Monomials

When you multiply two monomials, you multiply the coefficients together and then you multiply the variables together. Similarly, when dividing monomials, you divide the coefficients and then divide variables. When there are exponents with the same base, the law of exponents says you divide by subtracting the exponents. Consider this example:

 Example Problem Group the monomial into numerical and variable factors. Divide the coefficients, and divide the variables by subtracting the exponents of each y term. Answer =

Here’s another example:

 Example Problem A rectangle has an area of 8x2 and a length of 4x. Find the width of the rectangle using the formula: . Substitute known values. Divide coefficients, and divide the variables by subtracting the exponents of each x term. Answer width =  units

Sometimes division requires simplification.

 Example Problem Divide. Group the monomial into numerical and variable factors. Simplify to . Divide the variables by subtracting the exponents of r. Note that the variable has a negative exponent. Simplify  by rewriting it as the inverse of r. Multiply. Answer =

Remember that a term is not considered simplified if it contains a negative exponent; this is why  was rewritten as .

 Divide:   A) 11x4   B) 22x3   C) 11x3   D) 22x4   Show/Hide Answer

Dividing Polynomials by Monomials

The distributive property states that you can distribute a factor that is being multiplied by a sum or difference, and likewise you can distribute a divisor that is being divided into a sum or difference (as division can be changed to multiplication.)

Or you can distribute the 2, and divide each term by 2.

Let’s try something similar with a polynomial.

 Example Problem Divide. Distribute 2x over the polynomial by dividing each term by 2x. Divide each term, a monomial divided by another monomial. Answer =

Let’s try one more example, watch the signs.

 Example Problem Divide. Divide each term in the polynomial by the monomial. Simplify. Remember that 18 can be written as 18y0. So the exponents are  0 – 1 = −1. Write the final answer without any negative exponents. Answer =

 Divide:   A)   B)   C)   D)     Show/Hide Answer

Summary

To divide a monomial by a monomial, divide the coefficients (or simplify them as you would a fraction) and divide the variables with like bases by subtracting their exponents. To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. Be sure to watch the signs! Final answers should be written without any negative exponents.