Synthetic Division Calculator

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 \(\ {-5}\)

Therefore, the final answer is:

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

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.