## Monomial Division

By now we have seen how to add and subtract polynomials by collecting like terms
and multiplying polynomials using the distributive property. Now, we will learn how to
divde a polynomial by a monomial. Remember, a **monomial** is a polynomial with only one term. That means no addition or subtraction. Dividing by
polynomials is a little more complicated so we will save that for later.

To divide by a monomial, we will divide each term in the numerator by the monomial.

\( \cfrac{15x^5 - 10x^4 + 5x^3}{\textcolor{blue}{5x^2}} \)

\( =\cfrac{15x^5}{\textcolor{blue}{5x^2}} + \cfrac{- 10x^4}{\textcolor{blue}{5x^2}} + \cfrac{5x^3}{\textcolor{blue}{5x^2}}\)

You can think of this like the distributive property. We are distributing \(\frac{1}{5x^2}\) into the numerator.

Next, we will simlify each term. First, divide the coefficients:

\( \cfrac{15x^5 - 10x^4 + 5x^3}{\textcolor{blue}{5x^2}} \)

\( =\cfrac{15x^5}{\textcolor{blue}{5x^2}} + \cfrac{- 10x^4}{\textcolor{blue}{5x^2}} + \cfrac{5x^3}{\textcolor{blue}{5x^2}}\)

\( =\cfrac{\cancel{15}x^5}{\textcolor{blue}{\cancel{5}x^2}} + \cfrac{\cancel{-10}x^4}{\textcolor{blue}{\cancel{5}x^2}} + \cfrac{\cancel{5}x^3}{\textcolor{blue}{\cancel{5}x^2}}\)

Next, deal with the variables! Remember the
exponent rules for dividing exponents with the same base:

\( =\cfrac{3x^5}{\textcolor{blue}{x^2}} + \cfrac{-2x^4}{\textcolor{blue}{x^2}} + \cfrac{x^3}{\textcolor{blue}{x^2}}\)

\( =\cfrac{3\cancel{x^5}}{\textcolor{blue}{\cancel{x^2}}} + \cfrac{-2\cancel{x^4}}{\textcolor{blue}{\cancel{x^2}}} + \cfrac{\cancel{x^3}}{\textcolor{blue}{\cancel{x^2}}}\)

\( =3x^{5-2} + -2x^{4-2} + x^{3-2}\)

\( =3x^3 + -2x^2 + x^1\)

\( \boxed{=3x^3 + -2x^2 + x}\)

Simplify the following:

\( \cfrac{4x^2y + 12xy -8xy^2}{2xy} \)

Show Answer
First, divide each term in the numerator by the denominator:

\( =\cfrac{4x^2y}{\textcolor{blue}{2xy}} + \cfrac{12xy}{\textcolor{blue}{2xy}} + \cfrac{-8xy^2}{\textcolor{blue}{2xy}}\)

Next, simplify the coefficients:

\( =\cfrac{2x^2y}{\textcolor{blue}{xy}} + \cfrac{6xy}{\textcolor{blue}{xy}} + \cfrac{-4xy^2}{\textcolor{blue}{xy}}\)

Next, simplify the exponents on the \(x\):

\( =\cfrac{2xy}{\textcolor{blue}{y}} + \cfrac{6y}{\textcolor{blue}{y}} + \cfrac{-4y^2}{\textcolor{blue}{y}}\)

Finally, simplify the exponents on the \(y\):

\( = 2x + 6-4y\)

The final answe is:

\( = 2x - 4y + 6\)

Sometimes, the exponents on the denominator will be larger than the exponent on the numberator. For these scenarios, leave the term in
the denominator with a **positive** exponent:

\( \cfrac{12xy^3}{-2x^2y^2} \)

= \( \cfrac{\textcolor{red}{12}\textcolor{green}{x}\textcolor{brown}{y^3}}{\textcolor{red}{-2}\textcolor{green}{x^2}\textcolor{brown}{y^2}} \)

= \( \cfrac{\textcolor{red}{-6}\textcolor{brown}{y}}{\textcolor{green}{x}} \)