Monomial Division

By now we have seen how to add and subtract polynomials by collecting like terms and multiplying polynomials using 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.


Example

Simplify the expression \( \cfrac{15x^5 - 10x^4 + 5x^3}{5x^2}\).

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

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

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


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

Next, we can simplify each term. First, we can simplify the coefficients:

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

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

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


Then, we can simplify the exponents! Remember the exponent rules for dividing exponents with the same base:

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

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

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

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


Therefore, we can determine that the expression simplified is \( =3x^3 + -2x^2 + x\).


Simplify the following expressions:

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




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:

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

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

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

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