Synthetic Division Calculator
Enter the coefficients of the polynomial and the root (divisor) value.
Example: For x³ - 6x² + 11x - 6, enter 1, -6, 11, -6 as coefficients.
Result
Remainder: 0
Equation Preview:
What is Synthetic Division Calculator?
A Synthetic Division Calculator is a fast, structured tool for dividing a polynomial \(P(x)\) by a linear divisor \(x-r\). Instead of long polynomial division, it uses a compact table of coefficients to compute the quotient and remainder with fewer operations. The method relies on the Remainder and Factor Theorems and is especially useful for evaluating polynomials, finding zeros, and simplifying rational expressions.
The synthetic scheme transforms coefficients by the recurrence
yielding the quotient coefficients \(b_0,b_1,\dots,b_{n-1}\) and remainder \(b_n\).
About the Synthetic Division Calculator
This calculator automates the entire process: it aligns the polynomial degrees, fills missing powers with zero coefficients, applies the synthetic recurrence, and clearly reports the quotient and remainder. It also verifies the Remainder Theorem (that \(P(r)=\) remainder) and flags when the remainder is zero so you know the divisor is a factor by the Factor Theorem. You can use it to test candidate roots from the Rational Root Theorem, factor polynomials step-by-step, or simplify rational functions.
How to Use this Synthetic Division Calculator
- Enter the polynomial \(P(x)\) in descending powers. Include zeros for missing powers if needed.
- Enter the linear divisor \(x-r\) by providing the value \(r\) (note: \(x+3\Rightarrow r=-3\)).
- Click Calculate to run the synthetic recurrence and display quotient and remainder.
- Review the step-by-step table and the verification \(P(r)=\) remainder.
- Optionally repeat with the quotient to factor further or test new candidate roots.
Worked Example
Divide \(P(x)=2x^{3}-6x^{2}+2x-1\) by \(x-3\) \((r=3)\).
The calculator’s concise steps reduce algebraic errors, making polynomial division, factor testing, and root finding faster and clearer for students and professionals.