# Synthetic division

Synthetic division is a simplified method of polynomial division, especially useful for dividing by polynomials of the form x - c. This method is faster and less cumbersome than long division, making it ideal for students and mathematicians alike. To perform synthetic division, write down the coefficients of the polynomial, apply the root of the divisor, and systematically solve for the quotient and remainder.

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## What is Synthetic Division

Synthetic division is a simplified method of dividing polynomials. It is particularly useful when you need to divide a polynomial by a binomial of the form $$x - c$$. This technique is faster and more efficient compared to the traditional long division method.

### Definition of Synthetic Division

Synthetic division is a shorthand method of performing polynomial division, especially when dividing by a linear factor. The process uses only the coefficients of the polynomials, making it quicker and easier than long division.

Synthetic division is especially easy when the divisor is a binomial of the form $$x - c$$.

Example: Let's divide the polynomial $$2x^3 - 6x^2 + 2x - 1$$ by $$x - 3$$ using synthetic division.1. Write down the coefficients of the dividend: 2, -6, 2, -1.2. Place the zero of the divisor (3) on the left side.

 3 | 2 -6 2 -1 | 6 0 6 | 2 0 2 5
So, the quotient is $$2x^2 + 0x + 2$$ and the remainder is 5.

### How Synthetic Division Differs from Long Division

Synthetic division is distinguished from long division by its simplicity and efficiency. While synthetic division uses only the coefficients of the polynomials, long division involves writing out the full polynomial expressions.In long division,

• You divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by the result and subtract it from the dividend.
• Repeat the steps with the newly formed polynomial until you reach a remainder.
Synthetic division involves less writing and fewer steps:
• List only the coefficients of the dividend.
• Use the zero of the linear divisor.
• Perform simple arithmetic operations to get the quotient and remainder.
This makes synthetic division a more straightforward and faster method, making it ideal for specific types of polynomial division.

Despite its efficiency, synthetic division has some limitations. It's only applicable when dividing by a linear polynomial of the form $$x - c$$. When the divisor is a higher-degree polynomial or not in the correct form, you would need to resort to traditional long division. Additionally, synthetic division is greatly facilitated by its reliance on the “zero” of the divisor, streamlining the calculation process.

## Synthetic Division of Polynomials

Synthetic division is a simplified method of dividing polynomials. It streamlines the process by focusing only on the coefficients, making it a quicker alternative to traditional long division, especially for dividing polynomials by binomials of the form $$x - c$$.

### Step-by-Step Process of Synthetic Division of Polynomials

To understand synthetic division, follow these clear steps:

• Step 1: Write down the coefficients of the polynomial you wish to divide.
• Step 2: Identify the zero of the divisor binomial $$x - c$$. This will be $$c$$.
• Step 3: Set up the synthetic division by placing the zero on the left and drawing a horizontal line to separate the calculations.
• Step 4: Bring down the first coefficient to the bottom row.
• Step 5: Multiply the zero of the divisor by the number you just brought down and write the result in the next column of the top row.
• Step 6: Add this result to the coefficient directly above it and write the sum in the bottom row.
• Step 7: Repeat steps 5 and 6 until all coefficients have been processed.
• Step 8: The last number in the bottom row represents the remainder, while the other numbers give the coefficients of the quotient polynomial.

Example:Let's divide the polynomial $$2x^3 - 6x^2 + 2x - 1$$ by $$x - 3$$ using synthetic division.1. Write down the coefficients of the dividend: 2, -6, 2, -1.2. Place the zero of the divisor (3) on the left side.

 3 | 2 -6 2 -1 | 6 0 6 | 2 0 2 5
So, the quotient is $$2x^2 + 0x + 2$$ and the remainder is 5.

### Benefits of Using Synthetic Division for Polynomials

Synthetic division offers several key benefits:

• Simplicity: The process involves straightforward arithmetic operations, making it easier to execute.
• Efficiency: It requires fewer steps and less writing, significantly speeding up polynomial division.
• Reduced Complexity: By focusing only on the coefficients, you eliminate the need to manipulate polynomial terms, reducing the potential for errors.
• Ideal for Linear Divisors: It is particularly advantageous when dividing by binomials of the form $$x - c$$.

For more complex divisors, involving higher degree polynomials, long division is still more suitable.

## Synthetic Division Technique Explained

Synthetic division is a method used to divide polynomials, particularly useful for dividing by a binomial of the form $$x - c$$. This technique simplifies the process by focusing solely on the coefficients of the polynomials.

### Key Components of the Synthetic Division Technique

Synthetic Division: A shortcut method for dividing a polynomial by a binomial of the form $$x - c$$, concentrating on the coefficients rather than the entire polynomial expressions.

Understanding the key components of synthetic division helps in implementing the technique effectively. Here are the primary steps to follow:

• Coefficients: The numerical factors of the terms in the polynomial.
• Zero of the Divisor: The value of $$c$$ in the binomial $$x - c$$.
• Arithmetic Operations: Consists of multiplication and addition performed on the coefficients.

Example:Divide the polynomial $$2x^3 - 6x^2 + 2x - 1$$ by $$x - 3$$ using synthetic division.1. List the coefficients of the dividend: 2, -6, 2, -1.2. Place the zero of the divisor (3) on the left.

 3 | 2 -6 2 -1 | 6 0 6 | 2 0 2 5
The quotient is $$2x^2 + 0x + 2$$ and the remainder is 5.

For larger polynomials or more complex divisors, traditional long division can be used.

### Common Missteps in Synthetic Division Technique

While synthetic division simplifies polynomial division, there are some common errors to watch out for:

• Incorrect Zero of the Divisor: Ensure that you use the correct zero $$c$$ when working with the divisor $$x - c$$.
• Missing Coefficients: Don't forget to include a zero for any missing terms in the polynomial, such as a missing $$x^2$$ term in the coefficients list.
• Arithmetic Errors: Be vigilant with addition and multiplication steps to avoid calculation errors.

In complex scenarios, such as dividing by higher-degree polynomials, synthetic division falls short and traditional polynomial division methods must be relied upon. Despite these limitations, understanding and applying synthetic division can greatly help simplify many algebraic tasks.

## Synthetic Division Examples

In this section, you will explore various examples of synthetic division. This will help you understand how to apply the method effectively to both simple and complex polynomials.

### Simple Synthetic Division Example

Let's start with a basic example of synthetic division. We'll divide the polynomial $$2x^3 - 6x^2 + 2x - 1$$ by $$x - 3$$. The steps are as follows:

Example:Divide the polynomial $$2x^3 - 6x^2 + 2x - 1$$ by $$x - 3$$ using synthetic division.1. Write down the coefficients of the dividend: 2, -6, 2, -1.2. Place the zero of the divisor (3) on the left side.

 3 | 2 -6 2 -1 | 6 0 6 | 2 0 2 5
The quotient is $$2x^2 + 0x + 2$$ and the remainder is 5.

In this example, you can see that synthetic division reduces the polynomial division to simple arithmetic operations. This efficiency is valuable when dealing with complex polynomials.

### Complex Synthetic Division Example

Now, let's examine a more complex case involving a higher-degree polynomial. We'll divide $$4x^4 - 3x^3 + 2x^2 - x + 5$$ by $$x + 2$$. Follow these steps:

Example:Divide the polynomial $$4x^4 - 3x^3 + 2x^2 - x + 5$$ by $$x + 2$$ using synthetic division.1. Write down the coefficients of the dividend: 4, -3, 2, -1, 5.2. Place the zero of the divisor (-2) on the left.

 -2 | 4 -3 2 -1 5 | -8 22 -42 86 | 4 -11 24 -43 91
The quotient is $$4x^3 - 11x^2 + 24x - 43$$ and the remainder is 91.

When dealing with a negative divisor, ensure you place the correct zero in the synthetic division setup.

### How to Do Synthetic Division with Multiple Roots

Synthetic division can also be used when dealing with polynomials that need to be divided by binomials with multiple roots. Let’s explore how to handle such scenarios:

Example:Divide the polynomial $$x^3 - 6x^2 + 11x - 6$$ by $$(x - 1)(x - 2)$$ using synthetic division.1. First, perform synthetic division with the root 1:

 1 | 1 -6 11 -6 | 1 -5 6 | 1 -5 6 0
2. Now, take the coefficients 1, -5, and 6 (ignoring the remainder) and perform synthetic division again with the next root 2:
 2 | 1 -5 6 | 2 -6 | 1 -3 0
The quotient is $$x - 3$$ and the final remainder is 0.

When dividing by multiple roots, performing multiple rounds of synthetic division makes the process efficient. This approach can be extended to resolve complex polynomial equations with ease.

## Synthetic division - Key takeaways

• Synthetic division: A shorthand method for dividing polynomials, particularly useful for dividing by a binomial of the form x - c.
• Steps to perform synthetic division: Begin by writing the coefficients of the polynomial, identify the zero of the divisor, and then perform systematic arithmetic operations to derive the quotient and remainder.
• Example of synthetic division: To divide 2x3 - 6x2 + 2x - 1 by x - 3, write the coefficients (2, -6, 2, -1), place the zero (3), and use synthetic division to end up with a quotient of 2x2 + 0x + 2 and a remainder of 5.
• Benefits: Synthetic division is simpler and faster than traditional long division, involving fewer steps and writing, and focuses mainly on the coefficients of the polynomial.
• Limitations: This technique is only effective when the divisor is a linear polynomial of the form x - c and doesn't work for higher-degree polynomials or more complex divisors.

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Can synthetic division be used with complex numbers?
Yes, synthetic division can be used with complex numbers. The process is similar to that with real numbers, but complex arithmetic must be applied.
What is the purpose of synthetic division?
The purpose of synthetic division is to simplify the process of dividing a polynomial by a linear binomial, specifically of the form \$$x - c \$$, making it quicker and more efficient than long division.
How does synthetic division differ from long division?
Synthetic division is a simplified method of dividing polynomials, specifically useful when dividing by a linear factor, and is more efficient than long division. It reduces the process to basic arithmetic operations, avoiding variable notation, whereas long division involves carrying down terms and dealing with polynomial expressions throughout.
Can synthetic division be used for polynomials with coefficients that are not integers?
Yes, synthetic division can be used for polynomials with non-integer coefficients, including fractions and irrational numbers. The method remains the same regardless of the type of coefficients.
What are the steps involved in synthetic division?
1. Write down the coefficients of the dividend.2. Write the zero of the divisor using x - k form.3. Perform division by bringing down the first coefficient, multiplying and adding sequentially.4. The final row of numbers represents the coefficients of the quotient and the remainder.

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