Synthetic Division Calculator: Divide Polynomials Easily

Synthetic Division Calculator

Divide polynomials using the synthetic division method. This calculator shows step-by-step work and provides the quotient and remainder.

Enter in format: ax^n+bx^(n-1)+…+c (e.g., x^3-4x^2+5x-2)
Enter in format: x-a or x+a (e.g., x-2)

What is Synthetic Division?

Synthetic division is a simplified method for dividing polynomials by linear binomials of the form (x – c) or (x + c). It’s a shorthand technique that reduces the polynomial long division process to a compact algorithm involving only the coefficients of the polynomial. This method is particularly useful when evaluating polynomials, finding polynomial roots, and determining whether a binomial is a factor of a polynomial.

The synthetic division process works by arranging the coefficients of the dividend polynomial in descending order and performing a series of multiplications and additions. The result provides both the quotient polynomial and the remainder in a single, streamlined calculation.

While traditional polynomial long division requires writing out all terms and performing multiple steps, synthetic division simplifies this process significantly, making it faster and less prone to errors. However, it’s important to note that synthetic division only works for division by linear factors where the coefficient of x is 1.

How to Use the Synthetic Division Calculator

How to Perform Synthetic Division with Our Calculator

Step 1: Enter the Dividend Polynomial

Type your polynomial in the “Dividend” field using standard algebraic notation. For example, enter “x^3-4x^2+5x-2” for the polynomial x³-4x²+5x-2. Make sure to include all terms, even those with zero coefficients.

Step 2: Enter the Divisor

Input the linear divisor in the “Divisor” field in the format “x-a” or “x+a”. For example, enter “x-2” to divide by (x-2). Remember that synthetic division only works with linear factors where the coefficient of x is 1.

Step 3: Click Calculate

Press the Calculate button to perform the synthetic division. The calculator will process your input and display the results immediately.

Step 4: Review the Step-by-Step Solution

Examine the detailed synthetic division process shown in the results section. The calculator displays the synthetic division table with all intermediate steps, helping you understand how each value was calculated.

Step 5: Interpret the Results

The final result shows the quotient polynomial and remainder. If the remainder is zero, the divisor is a factor of the original polynomial. The calculator also provides an explanation of what the result means in the context of polynomial division.

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