How this calculator works
Synthetic division is a compact form of division by a monic linear polynomial. Bring down the first coefficient, multiply it by r, add to the next coefficient, and repeat. All but the final entry are quotient coefficients. The final entry equals P(r) by the remainder theorem. Include zero coefficients for every missing power.
Formula
b_0 = a_0; b_j = a_j + r*b_(j-1). The final b is the remainder; preceding b values are the quotient coefficients.
Worked example
Divide x^3 - 6x^2 + 11x - 6 by x - 1. The bottom row is 1, -5, 6, 0, so the quotient is x^2 - 5x + 6 and the remainder is zero.
Advertisement
Frequently asked questions
Can the divisor be 2x - 3?
This tool uses x - r only. Divide by x - 1.5, then divide the quotient by 2 to account for 2x - 3; the remainder is unchanged.
How does this test a root?
A zero remainder means x - r divides the polynomial, so r is a root. Near-zero decimal residuals may need closer numerical analysis.
Why are internal zeros required?
Each synthetic step corresponds to one descending power. Omitting a zero would shift the remaining coefficients to the wrong powers.