Practice Problem 1a: Divide using synthetic division.

Practice Problem 2a: Given the function f ( x ), use the Remainder Theorem to find f (-1).

Practice Problem 3a: Solve the given equation given that 1/2 is a zero (or root) of .

Last revised on March 15, 2012 by Kim Seward. All contents copyright (C) 2002 - 2012, WTAMU and Kim Seward. All rights reserved.

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## Synthetic Division

In these lessons, we will look at Synthetic Division, which is simplified form of long division.

Related Pages Long Division Of Polynomials More Lessons for Algebra Math Worksheets

## What is Synthetic Division?

Example: Evaluate ( x 3 – 8 x + 3) ÷ ( x + 3) using synthetic division

Solution: ( x 3 – 8 x + 3) is called the dividend and ( x + 3) is called the divisor.

Step 1: Write down the constant of the divisor with the sign changed –3

Step 3: Bring down the first coefficient.

Step 4: Multiply (1)( –3) = –3 and add to the next coefficient.

Repeat Step 4 for all the coefficients

We find that ( x 3 – 8 x + 3) ÷ ( x + 3) = x 2 – 3 x + 1

Example: Divide using synthetic division

(2x 3 + 6x 2 - 17x + 15) ÷ (x + 5)

(16x 3 - 2 + 14x - 12x 2 ) ÷ (2x + 1)

Divide a Trinomial by a Binomial Using Synthetic Division

Example: (x 3 - 2x 2 + 3x - 4) ÷ (x - 2)

Example: (x 4 - x 2 + 5) ÷ (x + 3)

## Module 9: Power and Polynomial Functions

Synthetic division, learning outcomes.

To illustrate the process, recall the example at the beginning of the section.

The final form of the process looked like this:

## A General Note: Synthetic Division

## How To: Given two polynomials, use synthetic division to divide

- Write k for the divisor.
- Write the coefficients of the dividend.
- Bring the leading coefficient down.
- Multiply the leading coefficient by k . Write the product in the next column.
- Add the terms of the second column.
- Multiply the result by k . Write the product in the next column.
- Repeat steps 5 and 6 for the remaining columns.
- Use the bottom numbers to write the quotient. The number in the last column is the remainder and has degree 0, the next number from the right has degree 1, the next number from the right has degree 2, and so on.

## Example: Using Synthetic Division to Divide a Second-Degree Polynomial

Use synthetic division to divide [latex]5{x}^{2}-3x - 36[/latex] by [latex]x - 3[/latex].

Begin by setting up the synthetic division. Write k and the coefficients.

Bring down the leading coefficient. Multiply the leading coefficient by k .

## Analysis of the Solution

[latex]\left(x - 3\right)\left(5x+12\right)+0=5{x}^{2}-3x - 36[/latex]

## Example: Using Synthetic Division to Divide a Third-Degree Polynomial

Use synthetic division to divide [latex]4{x}^{3}+10{x}^{2}-6x - 20[/latex] by [latex]x+2[/latex].

## Example: Using Synthetic Division to Divide a Fourth-Degree Polynomial

The result is [latex]-9{x}^{3}+{x}^{2}+8x+8+\frac{2}{x - 1}[/latex].

Use synthetic division to divide [latex]3{x}^{4}+18{x}^{3}-3x+40[/latex] by [latex]x+7[/latex].

[latex]3{x}^{3}-3{x}^{2}+21x - 150+\frac{1,090}{x+7}[/latex]

https://www.myopenmath.com/multiembedq.php?id=29483&theme=oea&iframe_resize_id=mom1

## Example: Using Polynomial Division in an Application Problem

To solve for h , first divide both sides by 3 x .

Now solve for h using synthetic division.

[latex]h=\frac{{x}^{3}-{x}^{2}-11x+18}{x - 2}[/latex]

## Contribute!

- Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
- Question ID 29483. Authored by : McClure,Caren. License : Other . License Terms : IMathAS Community License CC-BY + GPL
- College Algebra. Authored by : Abramson, Jay et al.. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

## Synthetic Division Method

## Things to Remember:

- Make sure the dividend is in standard form. That means the powers are in decreasing order.
- The divisor must be in the form x - \left( c \right) .

## Examples of How to Divide Polynomials Using the Synthetic Division

Example 1 : Divide the polynomial below.

Let us re-examine the given problem and make the necessary adjustments, if necessary.

The divisor needs to be rewritten as

Directly to the left side, place the value of c = - 2 inside the “box”.

Finally, construct a horizontal line just below the coefficients of the dividend.

1. Drop the first coefficient below the horizontal line.

3. Add the column of numbers, then put the sum directly below the horizontal line.

4. Repeat the process until you run out of columns to add.

See the animated solution below:

So how do we present our final answer?

More so, the exponents of the variables of the quotient are all reduced by 1 .

Example 2 : Divide the polynomial.

From this point, I can now set up the numbers to continue with the process.

So putting the final answer in the form

Example 3 : Divide the polynomial below.

\left( { - 2{x^4} + x} \right) \div \left( {x - 3} \right)

The “new and improved” problem should look like this:

From here, proceed with the steps as usual.

Okay then, the final answer for this is

Example 4 : Divide the polynomial below.

\left( { - {x^5} + 1} \right) \div \left( {x + 1} \right)

Observe the dividend and you should agree that the missing parts are {x^4} , {x^3} , {x^2} , and x .

Rewriting the original problem that is synthetic-division ready, we get…

We populated the missing x ‘s with zeros and explicitly solve for c = -1 .

Example 5 : Divide the polynomial by a binomial.

Because the remainder equals zero, this means the divisor x - 5 is a factor of the dividend

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## How to Divide Polynomials Using Synthetic Division

Last Updated: May 23, 2021 References

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## You Might Also Like

- ↑ https://www.purplemath.com/modules/synthdiv.htm
- ↑ http://www.mesacc.edu/~scotz47781/mat120/notes/divide_poly/synthetic/synthetic_division.html
- ↑ http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U11_L2_T6_text_final.html
- PurpleMath.com - Great help with Algebra I and II
- Ruffini's Rule (Synthetic Division) on Wikipedia

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## Synthetic Division: The Process

The Process Worked Examples Finding Zeroes Factoring Polynomials

## What is synthetic division?

## MathHelp.com

## How are polynomial zeroes and factors related?

## How do you do synthetic division?

Put the test zero, in our case x = 1 , at the left, next to the (top) row of numbers:

This last carry-down value is the remainder.

URL: https://www.purplemath.com/modules/synthdiv.htm

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## Synthetic Division

## What is Synthetic Division?

We can represent the division of two polynomials in the form: p(x)/q(x) = Q(x) + R/(q(x))

## Synthetic Division of Polynomials Definition

p(x)/q(x) = p(x)/(x- a) = Quotient + (Remainder/(x - a))

p(x)/(x - a) = Q(x) + (R/(x - a))

The coefficients of p(x) are taken and divided by the zero of the linear factor.

## Synthetic Division Example

Step 1: Write the coefficients of the dividend inside the box and zero of x + 2 as the divisor.

Step 2: Bring down the leading coefficient 1 to the bottom row.

Step 3: Multiply -2 by 1 and write the product -2 in the middle row.

Step 4: Add 1 and -2 in the second column and write the sum -1 in the bottom row.

Step 5: Now, multiply -2 by -1 (obtained in step 4) and write product 2 below -2.

Step 6: Add -2 and 2 in the third column and write the sum 0 in the bottom row.

Please note that the last box in the bottom row gives the remainder.

The profit per apple is given by (x - 1).

## Synthetic Division vs Long Division

Using synthetic division, we can perform complex division and obtain the solutions easily.

## Synthetic Division Method

- Check whether the polynomial is in the standard form .
- Write the coefficients in the dividend's place and write the zero of the linear factor in the divisor's place.
- Bring the first coefficient down.

- Multiply it with the divisor and write it below the next coefficient.
- Add them and write the value below.

- Separate the last term thus obtained which is the remainder.
- Now group the coefficients with the variables to get the quotient.

## How to do Synthetic Division?

- Write the coefficients of the dividend and use the zero of the linear factor in the divisor's place.
- Bring the first coefficient down and multiply it with the divisor.
- Write the product below the 2nd coefficient and add the column.
- Repeat until the last coefficient. The last number is taken as the remainder.
- Take the coefficients and write the quotient.
- Note that the resultant polynomial is of one order less than the dividend polynomial.

Tips and Tricks on Synthetic Division:

- Write down the coefficients and divide them using the zero of the linear factor to obtain the quotient and the remainder. (P(x)/(x - a) = Q(x) + (R/(x - a))
- When we do synthetic division by (bx + a), we should get (Q(x)/b) as the quotient.
- Perform synthetic division only when the divisor is a linear factor.
- Perform multiplication and addition in the place of division and subtraction that is used in the long division method.

- Long division of polynomials
- Division Algorithm for Polynomials
- Dividing Two Polynomials
- Division of Polynomial by Linear Factor

## Synthetic Division Examples

Speed is given as the ratio of the distance to the time.

Speed = (9a 2 - 39a - 30)/(a - 5)

Answer: Speed is given by the expression 9a + 6.

Area (A) = length(l) × breadth(b)

Given A = 4x 2 - 1. This is of the form a 2 - b 2 = (a + b)(a - b)

This can be expressed as, A = (2x + 1)(2x - 1)

h = (V/A) = (8x 3 + 12x 2 - 2x - 3)/[(2x + 1)(2x - 1)]

Let's solve this by the synthetic division twice.

Answer: Height of the box = 2x + 3.

Example 3: Perform synthetic division to solve the following expression: (6x 2 + 7x - 20)/(2x + 5).

Let us have a look at the steps shown below,

Answer: Quotient for the given division of polynomials = 3x - 4.

go to slide go to slide go to slide

## IMAGES

## VIDEO

## COMMENTS

Synthetic Division of Polynomials. 1.9M views 5 years ago New ... This video contains plenty of examples and practice problems. … Show more.

Step 1: Set up the synthetic division. · Step 2: Bring down the leading coefficient to the bottom row. · Step 3: Multiply c by the value just

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How To: Given two polynomials, use synthetic division to divide · Write k for the divisor. · Write the coefficients of the dividend. · Bring the leading

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Write down the problem. For this example, you will be dividing x3 + 2x2 - 4x + 8 by x + 2. Write the first polynomial equation, the dividend, in the numerator

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Step 1: Divide p(x) with (x - 1): (4x^3 - 8x^2 - 20x + 24) / (x - 1) = 4x^2 - 4x - 24. There's no remainder, so x = 1 is indeed a root of p(x). Step 2. Factor

Synthetic Division Examples ... Example 1: The distance covered by Steve in his car is given by the expression 9a2 - 39a - 30. The time taken by him to cover this