Can you complete the following division problem with a polynomial and a monomial? How does this relate to factoring?

\begin{align*}4e^4+6e^3-10e^2 \div 2e\end{align*}

### Division of a Polynomial by a Monomial

Recall that a monomial is an algebraic expression that has only one term. So, for example, \begin{align*}x\end{align*}, 8, –2, or \begin{align*}3ac\end{align*} are all monomials because they have only one term. The term can be a number, a variable, or a combination of a number and a variable. A polynomial is an algebraic expression that has more than one term.

When dividing polynomials by monomials, it is often easiest to separately divide each term in the polynomial by the monomial. When simplifying each mini-division problem, don't forget to use exponent rules for the variables. For example,

\begin{align*}\frac{8x^5}{2x^3}=4x^2\end{align*}

Remember that a fraction is just a division problem!

#### Let's divide the following polynomials:

- \begin{align*}(14s^2-21s+42)\div(7)\end{align*}

This is the same as \begin{align*}\frac{14s^2-21s+42}{7}\end{align*}. Divide each term of the polynomial numerator by the monomial denominator and simplify.

\begin{align*}\frac{14s^4}{7}=2s^4\end{align*}

\begin{align*}\frac{-21s}{7}=-3s\end{align*}

\begin{align*}\frac{42}{7}=6\end{align*}

Therefore, \begin{align*}(14s^2-21s+42)\div(7)=2s^4-3s+6\end{align*}.

- \begin{align*}\frac{3w^3-18w^2-24w}{6w}\end{align*}

Divide each term of the polynomial numerator by the monomial denominator and simplify. Remember to use exponent rules when dividing the variables.

\begin{align*}\frac{3w^3}{6w}=\frac{w^2}{2}\end{align*}

\begin{align*}\frac{-18w^2}{6w}=-3w\end{align*}

\begin{align*}\frac{-24w}{6w}=-4\end{align*}

Therefore, \begin{align*}\frac{3w^3-18w^2-24w}{6w}=\frac{w^2}{2}-3w-4\end{align*}.

- \begin{align*}(-27a^4b^5+81a^3b^4-18a^2b^3)\div(-9a^2b)\end{align*}

This is the same as \begin{align*}\frac{-27a^4b^5+81a^3b^4-18a^2b^3}{-9a^2b}\end{align*}. Divide each term of the polynomial numerator by the monomial denominator and simplify. Remember to use exponent rules when dividing the variables.

\begin{align*}\frac{-27a^4b^5}{-9a^2b}=3a^2b^4\end{align*}

\begin{align*}\frac{81 a^3b^4}{-9a^2b}=-9ab^3\end{align*}

\begin{align*}\frac{-18a^2b^3}{-9a^2b}=2b^2\end{align*}

Therefore, \begin{align*}(-27a^4b^5+81a^3b^4-18a^2b^3) \div (-9a^2b)=3a^2b^4-9ab^3+2b^2\end{align*}.

### Examples

#### Example 1

Earlier, you were asked complete the following division problem:

\begin{align*}4e^4+6e^3-10e^2 \div 2e\end{align*}

This process is the same as factoring out a \begin{align*}2e\end{align*} from the expression \begin{align*}4e^4+6e^3-10e^2\end{align*}.

\begin{align*}\frac{4 e^4}{2e}=2e^3\end{align*}

\begin{align*}\frac{6e^3}{2e}=3e^2\end{align*}

\begin{align*}\frac{-10e^2}{2e}=-5e\end{align*}

Therefore, \begin{align*}4e^4+6e^3-10e^2 \div 2e=2e^3+3e^2-5e\end{align*}.

#### Example 2

Complete the following division problem.

\begin{align*}(3a^5-5a^4+17a^3-9a^2)\div(a)\end{align*}

\begin{align*}(3a^5-5a^4+17a^3-9a^2) \div (a)=3a^4-5a^3+17a^2-9a\end{align*}

#### Example 3

\begin{align*}(-40n^3-32n^7+88n^{11}+8n^2)\div(8n^2)\end{align*}

\begin{align*}(-40n^3-32n^7+88n^{11}+8n^2)\div(8n^2)=-5n-4n^5+11n^9+1\end{align*}

#### Example 4

\begin{align*}\frac{16m^6-12m^4+4m^2}{4m^2}\end{align*}

\begin{align*}\frac{(16m^6-12m^4+4m^2)}{(4m^2)}=4m^4-3m^2+1\end{align*}

### Review

Complete the following division problems.

- \begin{align*}(6a^3+30a^2+24a) \div 6\end{align*}
- \begin{align*}(15b^3+20b^2+5b) \div 5\end{align*}
- \begin{align*}(12c^4+18c^2+6c) \div 6c\end{align*}
- \begin{align*}(60d^{12}+90d^{11}+30d^8) \div 30d\end{align*}
- \begin{align*}(33e^7+99e^3+22e^2) \div 11e\end{align*}
- \begin{align*}(-8a^4+8a^2) \div (-4a)\end{align*}
- \begin{align*}(-3b^4+6b^3-30b^2+15b) \div (-3b)\end{align*}
- \begin{align*}(-40c^{12}-20c^{11}-25c^9-30c^3) \div 5c^2\end{align*}
- \begin{align*}(32d^{11}+16d^7+24d^4-64d^2) \div 8d^2\end{align*}
- \begin{align*}(14e^{12}-18e^{11}-12e^{10}-18e^7) \div -2e^5\end{align*}
- \begin{align*}(18a^{10}-9a^8+72a^7+9a^5+3a^2) \div 3a^2\end{align*}
- \begin{align*}(-24b^9+42b^7+42b^6) \div -6b^3\end{align*}
- \begin{align*}(24c^{12}-42c^7-18c^6) \div -2c^5\end{align*}
- \begin{align*}(14d^{12}+21d^9+42d^7) \div -7d^4\end{align*}
- \begin{align*}(-40e^{12}+30e^{10}-10e^4+30e^3+80e) \div -10e^2\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.12.