# 7.13: Long Division and Synthetic Division

Difficulty Level: Advanced Created by: CK-12
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Practice Long Division of Polynomials

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Can you divide the following polynomials?

\begin{align*}\frac{x^2-5x+6}{x-2}\end{align*}

### Guidance

##### Polynomial Long Division

Whenever you want to divide a polynomial by a polynomial, you can use a process called polynomial long division. This process is similar to long division for regular numbers. Look at the example below:

\begin{align*}\frac{(x^2+3x+2)}{(x+1)}\end{align*}

This is the same as the division problem below:

Step 1: Divide the first term in the numerator (\begin{align*}x^2\end{align*}) by the first term in the denominator (\begin{align*}x\end{align*}). Put this result above the division bar in your answer. In this case, \begin{align*}\frac{x^2}{x} = x\end{align*}.

Step 2: Multiply the denominator (\begin{align*}x+1\end{align*}) by the result from Step 1 (\begin{align*}x\end{align*}), and put the new result below your numerator. Then, subtract to get your new polynomial. This is the same process as in regular number long division!

Step 3: Divide the first term in the new polynomial (\begin{align*}2x\end{align*}) by the first term in the denominator (\begin{align*}x\end{align*}). Put this result above the division bar in your answer. Multiply, subtract, and repeat this process until you cannot repeat it anymore.

Therefore: \begin{align*}\frac{(x^2+3x+2)}{(x+1)}=(x+2)\end{align*}

##### Synthetic Division

Synthetic division is another method of dividing polynomials. It is a shorthand of long division that only works when you are dividing by a polynomial of degree 1. Usually the divisor is in the form \begin{align*}(x\pm a)\end{align*}. In synthetic division, unlike long division, you are only concerned with the coefficients in the polynomials. Consider the same example as above:

Step 1: Write the coefficients in an upside down division sign.

Step 2: Put the opposite of the number from the divisor to the left of the division symbol. In this case, the divisor is \begin{align*}x+1\end{align*}, so you will use a \begin{align*}-1\end{align*}.

Step 3: Take your leading coefficient and bring it down below the division symbol.

Step 4: Multiply this number by the number to the left of the division symbol and place it in the next column. Add the two numbers together and place this new number below the division sign.

Step 5: Multiply this second number by the number to the left of the division symbol and place it into the third column. Add the two numbers together and place this new number below the division sign.

The numbers below the division sign represent your coefficients. Therefore, \begin{align*}\frac{(x^2+3x+2)}{(x+1)}=(x+2)\end{align*}.

#### Example A

Use long division to divide: \begin{align*}\frac{x^2+6x-7}{x-1}\end{align*}

Solution: Step 1: Divide the first term in the numerator by the first term in the denominator, put this in your answer. Therefore \begin{align*}\frac{x^2}{x} = x\end{align*}.

\begin{align*}(x-1) \overset{{\color{red}x}}{|\overline{x^2+6x-7}}\end{align*}

Step 2: Multiply the denominator by this number (variable) and put it below your numerator, subtract and get your new polynomial.

\begin{align*}&(x-1)\overset{{\color{red}x}}{|\overline{x^2+6x-7}}\\ &\qquad \quad \underline{\;\;\; x^2-x \;\;\;\;\;\;\;\;}\\ & \qquad \qquad \qquad \ 7x\end{align*}

Step 3: Repeat the process until you cannot repeat it anymore.

\begin{align*}&(x-1)\overset{{x\color{red}+7}}{|\overline{x^2+6x-7}}\\ &\qquad \quad \underline{\;\;\; x^2-x \;\;\;\; \downarrow \;}\\ & \qquad \qquad \qquad 7x-7\\ &\qquad \quad \underline{\;\;\;\;\;\;\;\;\;\; 7x-7 \;}\\ &\qquad \qquad \qquad \qquad \ 0\end{align*}

Therefore: \begin{align*}\frac{x^2+6x-7}{x-1} = (x + 7)\end{align*}

#### Example B

Use long division to divide: \begin{align*}\frac{2x^2+7x+5}{2x+5}\end{align*}

Solution: Step 1: Divide the first term in the numerator by the first term in the denominator; put this in your answer. Therefore \begin{align*}\frac{2x^2}{2x} = x\end{align*}.

\begin{align*}(2x+5) \overset{{\color{red}x}}{|\overline{2x^2+7x+5}}\end{align*}

Step 2: Multiply the denominator by this number (variable) and put it below your numerator, subtract and get your new polynomial.

\begin{align*}&(2x+5)\overset{{\color{red}x}}{|\overline{2x^2+7x+5}}\\ &\qquad \quad \ \underline{\;\;\; 2x^2+5x \;\;\;\;\;\;}\\ & \qquad \qquad \qquad \quad 2x\end{align*}

Step 3: Repeat the process until you cannot repeat it anymore.

\begin{align*}&(2x+5)\overset{{x\color{red}+1}}{|\overline{2x^2+7x+5}}\\ &\qquad \quad \ \underline{\;\;\; 2x^2+5x \;\;\; \downarrow \;}\\ & \qquad \qquad \qquad \quad 2x+5\\ &\qquad \quad \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\; 2x+5 \;\;}\\ &\qquad \qquad \qquad \qquad \quad \ 0\end{align*}

Therefore: \begin{align*}\frac{2x^2+7x+5}{2x+5}=(x+1)\end{align*}

#### Example C

Use synthetic division to divide: \begin{align*}(x-1) |\overline{3x^2+x-4}\end{align*}

Solution: Step 1: Write the coefficients in an upside down division sign.

Step 2: Put the opposite of the number from the divisor to the left of the division symbol.

Step 3: Take your leading coefficient and bring it down below the division symbol.

Step 4: Multiply this number by the number to the left of the division symbol and place it in the next column. Add the two numbers together and place this new number below the division sign.

Step 5: Multiply this second number by the number to the left of the division symbol and place it into the third column. Add the two numbers together and place this new number below the division sign.

Therefore: \begin{align*}\frac{3x^2+x-4}{x-1} = (3x + 4)\end{align*}

#### Concept Problem Revisited

You can divide using long division or synthetic division.

\begin{align*}\frac{x^2-5x+6}{x-2}\end{align*}

Long Division:

Step 1: Divide the first term in the numerator by the first term in the denominator, put this in your answer. Therefore \begin{align*}\frac{x^2}{x} = x\end{align*}.

\begin{align*}(x-2) \overset{{\color{red}x}}{|\overline{x^2-5x+6}}\end{align*}

Step 2: Multiply the denominator by this number (variable) and put it below your numerator, subtract and get your new polynomial.

\begin{align*}&(x-2)\overset{{\color{red}x}}{|\overline{x^2-5x+6}}\\ &\qquad \quad \underline{\;\;\; x^2-2x \;\;\;\;\;\;}\\ & \qquad \qquad \qquad \ 3x\end{align*}

Step 3: Repeat the process until you cannot repeat it anymore.

\begin{align*}&(x-2)\overset{{x\color{red}-3}}{|\overline{x^2-5x+6}}\\ &\qquad \quad \underline{\;\;\;x^2-2x \;\;\; \downarrow \;}\\ & \qquad \qquad \qquad -3x+6\\ &\qquad \quad \underline{\;\;\;\;\;\;\;\;\;\; -3x+6 \;\;}\\ &\qquad \qquad \qquad \qquad \ 0\end{align*}

Therefore: \begin{align*}\frac{x^2-5x+6}{x-2} = (x - 3)\end{align*}

### Vocabulary

Dividend
The dividend is the number, variable, or expression you are dividing in a mathematical expression. In the expression \begin{align*}\frac{15}{4x}\end{align*}, 15 is the dividend.
Divisor
The divisor is the number, variable, or expression you are dividing by in a mathematical expression. In the expression \begin{align*}\frac{15}{4x}\end{align*}, \begin{align*}4x\end{align*} is the divisor.
Polynomial Long Division
Polynomial long division is a mathematical process similar to the long division process for numbers that allows you to divide polynomials.
Synthetic Division
Synthetic division is a mathematical process of dividing polynomials that works when dividing by a polynomial of degree 1.

### Guided Practice

1. Use long division to divide \begin{align*}5x^2+6x+1\end{align*} by \begin{align*}x+1\end{align*}.

2. Use synthetic division to divide \begin{align*}3x^2-2x-1\end{align*} by \begin{align*}x-1\end{align*}.

3. Use synthetic division to divide \begin{align*}3x^3+11x^2+4x-4\end{align*} by \begin{align*}x+1\end{align*}.

1. \begin{align*}\frac{5x^2+6x+1}{x+1}=(5x + 1)\end{align*}

2. \begin{align*}\frac{3x^2-2x-1}{x-1} = (3x + 1)\end{align*}

3. \begin{align*}\frac{3x^3+11x^2+4x-4}{x+1} = (3x^2 + 8x - 4)\end{align*}

### Practice

Use long division to divide each of the following:

1. \begin{align*}(x^2+7x+12)\div(x+3)\end{align*}
2. \begin{align*}(x^2+4x+3)\div(x+3)\end{align*}
3. \begin{align*}(a^2-4a-45)\div(a-9)\end{align*}
4. \begin{align*}(3x^2+5x-2)\div(3x-1)\end{align*}
5. \begin{align*}(2x^2-5x+2)\div(2x-1)\end{align*}

Use synthetic division to divide each of the following:

1. \begin{align*}(b^2-5b+6)\div(b-3)\end{align*}
2. \begin{align*}(x^2-6x+8)\div(x-4)\end{align*}
3. \begin{align*}(a^2-1)\div(a+1)\end{align*}
4. \begin{align*}(c^2-9)\div(c-3)\end{align*}
5. \begin{align*}(5r^2+2r-3)\div(r+1)\end{align*}

Divide each of the following:

1. \begin{align*}\frac{2x^3-7x^2-14x-5}{x-5}\end{align*}
2. \begin{align*}\frac{9x^4-15x^3+12x^2-11x-15}{3x^3+4x+3}\end{align*}
3. \begin{align*}\frac{6x^4+4x^3+9x^2+2x+3}{2x^2+1}\end{align*}
4. \begin{align*}\frac{x^4+4x^3+3x^2+x+1}{x+1}\end{align*}
5. \begin{align*}\frac{2x^3+7x^2-27x+18}{x+6}\end{align*}
6. \begin{align*}\frac{8x^3-2x^2+7x+5}{2x+1}\end{align*}
7. \begin{align*}\frac{3x^3-15x^2+4x-20}{x-5}\end{align*}
8. \begin{align*}\frac{9x^3+26x^2-48x+5}{x^2+3x-5}\end{align*}
9. \begin{align*}\frac{-x^3+13x+12}{x+3}\end{align*}
10. \begin{align*}\frac{x^3-2x^2-5x+10}{x-2}\end{align*}

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### Vocabulary Language: English

TermDefinition
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.

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