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# Division of Polynomials

## Using long division to divide polynomials

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Practice Division of Polynomials
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Division of a Polynomial by a Monomial

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

4e4+6e310e2÷2e\begin{align*}4e^4+6e^3-10e^2 \div 2e\end{align*}

### Guidance

Recall that a monomial is an algebraic expression that has only one term. So, for example, x\begin{align*}x\end{align*}, 8, –2, or 3ac\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,

8x52x3=4x2
.

Remember that a fraction is just a division problem!

#### Example A

What is (14s221s+42)÷(7)\begin{align*}(14s^2-21s+42)\div(7)\end{align*}?

Solution: This is the same as 14s221s+427\begin{align*}\frac{14s^2-21s+42}{7}\end{align*}. Divide each term of the polynomial numerator by the monomial denominator and simplify.

• 14s47=2s4\begin{align*}\frac{14s^4}{7}=2s^4\end{align*}
• 21s7=3s\begin{align*}\frac{-21s}{7}=-3s\end{align*}
• 427=6\begin{align*}\frac{42}{7}=6\end{align*}

Therefore, (14s221s+42)÷(7)=2s43s+6\begin{align*}(14s^2-21s+42)\div(7)=2s^4-3s+6\end{align*}.

#### Example B

What is 3w318w224w6w\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.

• 3w36w=w22\begin{align*}\frac{3w^3}{6w}=\frac{w^2}{2}\end{align*}
• 18w26w=3w\begin{align*}\frac{-18w^2}{6w}=-3w\end{align*}
• 24w6w=4\begin{align*}\frac{-24w}{6w}=-4\end{align*}

Therefore, 3w318w224w6w=w223w4\begin{align*}\frac{3w^3-18w^2-24w}{6w}=\frac{w^2}{2}-3w-4\end{align*}.

#### Example C

What is (27a4b5+81a3b418a2b3)÷(9a2b)\begin{align*}(-27a^4b^5+81a^3b^4-18a^2b^3)\div(-9a^2b)\end{align*}?

Solution: This is the same as 27a4b5+81a3b418a2b39a2b\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.

• 27a4b59a2b=3a2b4\begin{align*}\frac{-27a^4b^5}{-9a^2b}=3a^2b^4\end{align*}
• 81a3b49a2b=9ab3\begin{align*}\frac{81 a^3b^4}{-9a^2b}=-9ab^3\end{align*}
• 18a2b39a2b=2b2\begin{align*}\frac{-18a^2b^3}{-9a^2b}=2b^2\end{align*}

Therefore, (27a4b5+81a3b418a2b3)÷(9a2b)=3a2b49ab3+2b2\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*}.

### 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*}

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*}

### Explore More

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*}

### Vocabulary Language: English

Denominator

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

Dividend

In a division problem, the dividend is the number or expression that is being divided.
divisor

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

Polynomial long division is the standard method of long division, applied to the division of polynomials.
Rational Expression

Rational Expression

A rational expression is a fraction with polynomials in the numerator and the denominator.
Rational Root Theorem

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

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

Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used.