### Learning Objectives

- Divide a binomial by a monomial
- Divide a trinomial by a monomial

The fourth arithmetic operation is division, the inverse of multiplication. Division of polynomials isn’t much different from division of numbers. In the exponential section, you were asked to simplify expressions such as: [latex]\displaystyle\frac{{{a}^{2}}{{({{a}^{5}})}^{3}}}{8{{a}^{8}}}[/latex]. This expression is the division of two monomials. To simplify it, we divided the coefficients and then divided the variables. In this section we will add another layer to this idea by dividing polynomials by monomials, and by binomials.

## Divide a polynomial by a monomial

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. In this example, you can add all the terms in the numerator, then divide by 2.

[latex]\frac{\text{dividend}\rightarrow}{\text{divisor}\rightarrow}\,\,\,\,\,\, \frac{8+4+10}{2}=\frac{22}{2}=11[/latex]

Or you can first divide each term by 2, then simplify the result.

[latex] \frac{8}{2}+\frac{4}{2}+\frac{10}{2}=4+2+5=11[/latex]

Either way gives you the same result. The second way is helpful when you can’t combine like terms in the numerator. Let’s try something similar with a binomial.

### Example

Divide. [latex]\frac{9a^3+6a}{3a^2}[/latex]

In the next example, you will see that the same ideas apply when you are dividing a trinomial by a monomial. You can distribute the divisor to each term in the trinomial and simplify using the rules for exponents. As we have throughout the course, simplifying with exponents includes rewriting negative exponents as positive. Pay attention to the signs of the terms in the next example, we will divide by a negative monomial.

### Example

Divide. [latex] \frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}[/latex]

## 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.