# What is synthetic division and how to find it?

# Synthetic Division

Synthetic division is a faster and more efficient alternative to the traditional long-division method. Finding the zeros of polynomials and simplifying polynomial equations are both possible with the help of synthetic division.

This technique is particularly useful when you need to find the zeros of a polynomial, or when you need to simplify the expression of a rational function. The synthetic division algorithm involves arranging the coefficients of the polynomial in a certain way and performing a series of simple arithmetic operations.

The process begins with the division of the coefficient of the highest-degree term of the polynomial by the divisor, followed by the multiplication of the divisor by the result, and then the subtraction of the resulting product from the next coefficient in the polynomial.

Until the coefficients have been used we can repeat them again and again. In this, we will discuss the definition of synthetic division, mathematical representation of synthetic division, needs of synthetic division, Advantages, finding step of synthetic division also with the help of example topic will be explained.

## Definition of synthetic division

To divide a polynomial by a linear factor more quickly, use synthetic division, this entails using the polynomial’s coefficients in a sequence of basic arithmetic operations.

The result of synthetic division is the quotient of the polynomial division, which is a polynomial of one degree lower than the original polynomial. This technique is particularly useful for finding the zeros of a polynomial and simplifying rational functions.

### Formula

Mathematically representation of the synthetic division

**P(x)/Q(x) = P(x)/(x-a) = Q + [R /(x-a)]**

A synthetic division calculator allows you to calculate the polynomial division with the help of synthetic division according to the above formula.

## Advantages of Synthetic division

There are several advantages of synthetic division, which is a shortcut method for dividing polynomials by a linear factor. Here are a few advantages:

**Faster and more efficient:**Synthetic division is much faster and more efficient than long division when dividing a polynomial by a linear factor. It involves fewer steps and eliminates the need for writing out the full division process, which can be time-consuming and prone to errors.**Easy to use:**Synthetic division is easy to learn and use, even for those who are not familiar with long division of polynomials. The process involves basic arithmetic operations and requires only a small amount of prior knowledge.**No need for complex calculations:**Synthetic division only requires basic arithmetic calculations, such as multiplication and addition, which can be easily performed mentally or using a calculator. This makes the process less prone to errors and easier to perform accurately.**Provides a clear result:**The result of synthetic division is immediately clear and easy to interpret, as it provides the quotient and remainder of the division in a simple format. This makes it easier to check the accuracy of the result and perform further calculations.

## Steps

Steps include finding the Polynomial division by Synthetic division

Step 1 | If any terms are missing, substitute a coefficient of 0 for them in the polynomial’s decreasing order of degree. |

Step 2 | Identify the linear factor of the form (x-a) that you want to divide the polynomial by, and write the value of outside the division symbol. |

Step 3 | Write the coefficients of the polynomial in a row, separated by spaces, under the division symbol. |

Step 4 | Bring down the first coefficient of the polynomial, and write it directly below the division symbol. |

Step 5 | Multiply the value outside the division symbol by the coefficient just brought down, and write the result below the next coefficient of the polynomial. |

Step 6 | Add the two numbers just written, and write the result below the next vertical line. |

Step 7 | Repeat steps 5 and 6 for each subsequent coefficient of the polynomial, until all coefficients have been processed. |

Step 8 | The number to the right of the final vertical line is the remainder, and the numbers to the left are the coefficients of the quotient. |

Step 9 | Write the quotient and remainder in the form of a polynomial expression, as follows: The quotient is the polynomial that results from the coefficients to the left of the final vertical line, written in descending order of degree.The remainder is a constant term that is written after the quotient, with a minus sign if the remainder is negative. |

## How to find the synthetic division?

**Example**

Divide 6x^{3}+4x^{2}+8x+15/x+2

Solve it by using synthetic division

**Solution:**

Given

6x^{3}+4x^{2}+8x+15/x+2

Step 1:

Write the polynomial in descending order of degree, with any missing terms represented by a coefficient of zero:

6x^{3}+4x^{2}+8x+15

Step 2:

Write the divisor, x + 2, to the left of the polynomial, and set up the division symbol:

-2 â”‚ 6 4 8 15

Step 3:

Bring down the first coefficient, 4, below the division symbol:

-2 6 4 8 15

6

Step 4:

Multiply the divisor, -2, by the coefficient just brought down, 4, and write the result below the next coefficient:

-2 6 4 8 15

-12

6 -8

Step 5:

Add the two numbers just written, 4 and -8, and write the result below the next vertical line:

-2 6 4 8 15

-12 16

6 -8 24

Step 6:

Multiply the divisor, -2, by the result of the previous step, -6, and write the result below the next coefficient:

-2 6 4 8 15

-12 16 -48

6 -8 24 -33 R

Step 7:

Therefore, the result obtained after simplification is 6x^{3}+4x^{2}+8x+15/x+2 is 6x^{2}-8x+24 and the remainder is **-33.**

## Frequently asked question

**Question 1:**

**What is synthetic division?**

**Answer:**

Synthetic division is a shorthand method for dividing a polynomial by a linear factor of the form (x – a), where “a” is a constant. It is a quicker and more efficient way of performing polynomial division compared to long division.

**Question 2:**

**What is the purpose of synthetic division?**

**Answer:**

The purpose of synthetic division is to simplify the process of dividing polynomials, particularly when dividing by linear factors. It can be used to find factors of polynomials, evaluate functions, and solve equations.

**Question 3:**

**When should I use synthetic division?**

**Answer:**

Synthetic division should be used when dividing a polynomial by a linear factor of the form (x – a). It is particularly useful when the polynomial has a high degree or many terms, as it simplifies the process and reduces the chance of errors.

## Conclusion

In this we have discussed the definition of synthetic division, mathematical representation of synthetic division, needs of synthetic division, Advantages, finding step of synthetic division also with the help of example topic will be explained. After studying this article any one can defend this topic.