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.
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 \( =3x^3 + -2x^2 + x\).
\(\cfrac{4x^2y + 12xy -8xy^2}{2xy}\)
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:
\( \cfrac{12xy^3}{-2x^2y^2} \)
= \( \cfrac{\textcolor{red}{12}\textcolor{green}{x}\textcolor{brown}{y^3}}{\textcolor{red}{-2}\textcolor{green}{x^2}\textcolor{brown}{y^2}} \)
= \( \cfrac{\textcolor{red}{-6}\textcolor{brown}{y}}{\textcolor{green}{x}} \)