# Division of Polynomials

## Using long division to divide polynomials

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Division of a Polynomial by a Monomial

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:

1. \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*}.

1. \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*}.

1. \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.

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

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

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### Vocabulary Language: English

TermDefinition
Denominator The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. $\frac{5}{8}$ has denominator $8$.
Dividend In a division problem, the dividend is the number or expression that is being divided.
divisor In a division problem, the divisor is the number or expression that is being divided into the dividend. For example: In the expression $152 \div 6$, 6 is the divisor and 152 is the dividend.
Polynomial long division Polynomial long division is the standard method of long division, applied to the division of polynomials.
Rational Expression A rational expression is a fraction with polynomials in the numerator and the denominator.
Rational Root Theorem The rational root theorem states that for a polynomial, $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, where $a_n, a_{n-1}, \cdots a_0$ are integers, the rational roots can be determined from the factors of $a_n$ and $a_0$. More specifically, if $p$ is a factor of $a_0$ and $q$ is a factor of $a_n$, then all the rational factors will have the form $\pm \frac{p}{q}$.
Remainder Theorem The remainder theorem states that if $f(k) = r$, then $r$ is the remainder when dividing $f(x)$ by $(x - k)$.
Synthetic Division Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used.