# Monomial Division

## 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:

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

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:

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

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

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