Can you divide the polynomial by the monomial? How does this relate to factoring?

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

### Watch This

James Sousa: Dividing Polynomials by Monomials

### Guidance

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!

#### Example A

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

**Solution:** 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*}.

#### Example B

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

**Solution:** 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*}.

#### Example C

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

**Solution:** 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*}.

#### Concept Problem Revisited

Can you divide the polynomial by the monomial? How does this relate to factoring?

\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*}.

### Vocabulary

- Divisor
- A
is the expression in the denominator of a fraction.*divisor*

- Monomial
- A
is an algebraic expression that has only one term. \begin{align*}x\end{align*}, 8, –2, or \begin{align*}3ac\end{align*} are all monomials because they have only one term.*monomial*

- Polynomial
- A polynomial is an algebraic expression that has more than one term.

### Guided Practice

Complete the following division problems.

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

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

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

**Answers:**

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

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

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

### Practice

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*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 7.12.