Dividing by a Monomial

In a previous chapter we learned about the properties of exponents. In particular, we learned that to divide two terms with the same base, we subtract the exponents and keep the common base:

$\frac{x^m}{x^n}=x^{m-n}$

We will now use this quotient property of exponents to divide two monomials.

Technically, any divisor cannot equal zero, as division by zero is undefined. Also, if we get an answer of $x^0$, the answer will always be $1$, provided that $x\ne0$.  $0^0$ is undefined.  Notice that dividing two monomials does not always result in a monomial.  For example, $\frac{z^4}{z^7}=\frac{1}{z^3}$ does not result in a monomial. Remember that monomials cannot have negative exponents; the exponent must be a whole number.

When there are coefficients attached to the variables, we divide the coefficients and divide the variables.

Dividing a Polynomial by a Monomial

The distributive property states that we can distribute a factor that is being multiplied by a sum or difference: $a(b+c)=ab+ac$.  If the term being multiplied is a fraction, $\frac{1}{a}$, the distributive property tells us:

$\begin{equation}\begin{aligned}&\;\;\;\; \\ \frac{1}{a}(b+c)&=\frac{1}{a}\cdot b+\frac{1}{a}\cdot c\end{aligned}\end{equation}$

But, $\frac{1}{a}(b+c)=\frac{b+c}{a}$ and $\frac{1}{a}\cdot b+\frac{1}{a}\cdot c=\frac{b}{a}+\frac{c}{a}$ by multiplication of fractions.

So, $\frac{b+c}{a}=\frac{b}{a}+\frac{c}{a}$

In other words, we can distribute a divisor that is being divided into a sum or difference.

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

$\frac{\text{dividend}\rightarrow}{\text{divisor}\rightarrow}\,\,\,\,\,\, \frac{8+4+10}{2}=\frac{22}{2}=11$

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

$\frac{8}{2}+\frac{4}{2}+\frac{10}{2}=4+2+5=11$

Either way gives the same result. The second way is helpful when we can’t combine like terms in the numerator, as in a polynomial divided by a monomial.

The distributive property can be extended to any number of terms, so the next example applies the same ideas to divide a trinomial by a monomial. We can distribute the divisor to each term in the trinomial and simplify using the rules for exponents. Remember that simplifying with exponents includes rewriting negative exponents in the numerator as positive exponents in the denominator. Also remember to pay attention to the signs of the terms.

No matter the number of terms in the polynomial, we can use the distributive property to divide by a monomial.