# Synthetic Division Calculator

## Perform synthetic division step by step

The calculator will divide the polynomial by the binomial using synthetic division, with steps shown.

Related calculator: Polynomial Long Division Calculator

The Synthetic Division Calculator is an intuitive and efficient online tool designed for dividing a polynomial by a binomial. Whether you're a student trying to grasp the synthetic division method or a professional seeking quick solutions, our calculator is exactly what you need.

## How to Use the Synthetic Division Calculator?

### Input

In the specified input field, enter the polynomial you wish to divide. Ensure you've correctly typed the coefficients and degrees. Enter the divisor into the other input field.

### Calculation

Once you've entered all the data, click the "Calculate" button. The tool will quickly process the input and display the result.

### Result

The output will present the quotient and the remainder.

## What Is Synthetic Division of Polynomials?

Synthetic division is a shorthand method for dividing polynomials that simplifies the process significantly but can be used only when dividing by a linear factor. It is an alternative to the long division method, allowing for faster calculations while giving the same result.

## How to Do Synthetic Division?

This method requires fewer steps compared to the traditional long division approach. The only drawback is that the divisor should be a linear binomial.

The steps of the method are the following:

**Set Up the Coefficients:**- List the coefficients of the polynomial you wish to divide in descending order of their respective powers.
- If any terms are missing, represent them using a zero coefficient.

**Write Down the Divisor:**- Remember that the divisor should be in the form of $$$x-c$$$, where $$$c$$$ is a constant.

**Begin the Division:**- Drop down the leading coefficient of the polynomial; this starts your division.
- Multiply this coefficient by the constant term of the divisor with the opposite sign.
- Write this product under the next coefficient and add them.
- Continue multiplying the constant term of the divisor with the opposite sign by the obtained sum and add the result to the next coefficient in the line.

**Continue Till the End:**- Keep doing this process, moving left to right, until you've accounted for all the coefficients.

**Determine the Quotient and Remainder:**- The numbers you generate in the final row (except the last one) represent the coefficients of the quotient.
- The very last number in this sequence is the remainder. If the remainder is zero, the divisor is a factor of the polynomial.

For example, suppose you have the polynomial $$$p(x)=x^3-4x^2+5x-2$$$ and want to divide it by $$$x-2$$$.

Using synthetic division, you'll eventually determine that the quotient is $$$x^2-2x+1$$$ and the remainder is $$$0$$$, indicating $$$x-2$$$ is a factor of $$$x^3-4x^2+5x-2$$$.

Practicing synthetic division on various examples will enhance your understanding and speed. Over time, this method becomes intuitive, allowing you to perform division quickly.

## Why Choose Our Synthetic Division Calculator?

### Accuracy & Precision

Our calculator has been designed to provide accurate results every time, eliminating potential human errors that can occur with manual calculations.

### User-Friendly Interface

With its intuitive design, users of all skill levels can easily navigate and use our calculator.

### Speed

Our calculator delivers immediate results, saving you valuable time and effort.

### Step-by-Step Solution

Our tool returns the quotient and remainder and provides a detailed breakdown of each step, enhancing understanding and making it a valuable learning tool.

### FAQ

#### What is the use of synthetic division?

The synthetic division method is used for dividing polynomials by linear divisors. Additionally, it's a helpful tool for evaluating polynomials at a specific point, determining factors, and identifying their zeros. Its efficiency and simplicity make it a perfect choice for dividing a polynomial by a binomial.

#### Why is synthetic division important?

Synthetic division offers a quicker and simpler approach to dividing a polynomial by a first-degree binomial. It's also invaluable for evaluating polynomials and determining factors or zeros. Its efficiency can simplify the process of polynomial factoring and root-finding.

#### Can you always use the synthetic division method?

While the synthetic division method is highly efficient, it's used only for dividing a polynomial by a first-degree binomial, like $$$x-c$$$. For higher-degree divisors, only traditional polynomial long division can be used.

#### What are the types of polynomial division?

Polynomial division can be categorized mainly into two types:

- Long Division: This method is similar to arithmetic long division and also works with polynomials. It's versatile and can handle any polynomial division.
- Synthetic Division: This is a shorthand method suitable for dividing by a first-degree binomial.