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# Synthetic Division Calculator

Enter the dividend and divisor to perform synthetic division on polynomial expression.

## Synthetic Division Calculator:

Use this synthetic division calculator to divide the polynomials by binomials. It processes step-by-step synthetic division to find the actual quotient and remainders.

## What Is Synthetic Division?

Synthetic division is a shortcut method for dividing a polynomial by linear factors where the leading coefficient is 1. If it's not equal to 1, then it requires modifying the dividend to make the leading coefficient 1 before using the method. This method provides you with the quotient (the result of the division) and the remainder. It uses the divisors of the form "(x + a)" and "(x - a)".

• Polynomials are considered dividends
• Linear factors of the form (ax + b) are divisors

### Synthetic Division Formula:

$$\frac{P(x)}{(x-a)} =\ Q(x) + \frac{R}{(x-a)}$$

Where:

• P(x): Dividend Polynomial of any order
• (x-a): Linear Factor of degree “1”
• Q(x): Quotient
• R: Remainder

## How To Do Synthetic Division?

• Write the coefficients of the dividend in descending order
• Write zeros of linear factors as the divisor
• Convert values in the synthetic division format
• Carry down the leading coefficient in the next column
• Multiply the leading coefficient by the divisor and put in next column
• Repeat steps 4 and 5 until the last coefficient
• The value in the last column is a remainder, the number from the right is the quotient

## Example:

Solve the given below values by using the synthetic division method.

• Dividend: $$\ 2x^3 - 5x^2 + 3x - 7$$
• Divisor: $$\ x - 2$$

### Solution (Step by Step):

Navigate with the steps to evalaute polynomials using synthetic division method:

$$\dfrac{2 x^{3} - 5 x^{2} + 3 x - 7}{x - 2}$$

Step #1: Write The Coefficients Of The Dividend

2, -5, 3, -7

Step #2: Write Zeros Of Linear Factors As The Divisor

x - 2 = 0

x = 2

Step #3: Write values in synthetic division format:

$$\begin{array}{c|rrrrr}& x^{3}&x^{2}&x^{1}&x^{0} \\2.0& 2&-5&3&-7 \\&&\\\hline&\end{array}$$

Step #4: Carry Down The Leading Coefficient In The Next Column

$$\begin{array}{c|rrrrr}2.0& 2&-5&3&-7 \\&&\\\hline&2\end{array}$$

Step #5: Multiply The Leading Coefficient By The Divisor

Using the synthetic division to find zeros, simply multiply the obtained value by the denominator and write the result into the next column.

$$\ 2 \times (2.0) = 4$$

Write The Outcome In The Next Column

$$\ \begin{array}{c|rrrrr}2.0&2&-5&3&-7\\&&4&\\\hline&2&\end{array}$$

Step #6: Repeat Steps 4 & 5 Until The Last Coefficient

$$\ \begin{array}{c|rrrrr}2.0&2&-5&3&-7\\&&4&-2&2&\\\hline&2&-1&1&-5&\end{array}$$

Step #7: Value in Last Column is Remainder, The Number From Right Is Quotient

The quotient is $$\ 2 x^{2} - x + 1$$, and the remainder is $$\ {-5}$$

$$\dfrac{2 x^{3} - 5 x^{2} + 3 x - 7}{x - 2} = {2 x^{2} - x + 1 - \dfrac{5}{x - 2} }$$

## Why Choose Our Synthetic Division Calculator?

• Fast & Easy Calculations: Our calculator performs quick calculations, saving valuable time
• Step-by-Step Solution: The calculator provides you with a detailed step-by-step division of polynomials
• Clear UI: It's simple interface allows to make straight-forward calculations, no technical expertise is required

## FAQ's:

### When Is The Synthetic Division Applicable?

This division method is applicable when the divisor of a polynomial is a linear factor in the form ax + b, where the highest power of x is 1. For this division if you need complete calculations, take help from this synthetic division solver that simplifies the division with given steps.

### Does Synthetic Division Work For All Polynomials?

No, the synthetic division method only works to divide polynomials by linear expressions (a binomial of the form x-c, in which c is represented as a constant).

### What Are Some Limitations of This Synthetic Division Calculator?

The calculator only works to divide polynomials using synthetic division as we already stated above.

References:
Form the source of courses.lumenlearning.com: Synthetic Division.