Polynomial Long Division Calculator
The polynomial long division calculator accepts descending-order coefficient lists, reduces exact fractions, handles missing powers with zero coefficients and reports every leading-term elimination.
Polynomial Long Division Calculator inputs
The polynomial long division calculator accepts descending-order coefficient lists, reduces exact fractions, handles missing powers with zero coefficients and reports every leading-term elimination.
Enter coefficients from highest power to constant. Use 0 for missing powers and forms such as 3/4 for fractions.
Polynomial Long Division Calculator results
How to use the polynomial long division calculator
The polynomial long division calculator accepts descending-order coefficient lists, reduces exact fractions, handles missing powers with zero coefficients and reports every leading-term elimination. The sections below explain the data, assumptions and limitations so the answer can be checked rather than accepted as a generic black-box output.
Coefficient-list input
The polynomial long division calculator uses coefficient lists in descending power order. For x cubed minus 4x plus 1, enter 1 0 -4 1; the zero preserves the missing x-squared position. This format is unambiguous and avoids evaluating arbitrary code. Integers and reduced or unreduced fractions such as 3/4 are accepted.
Exact rational arithmetic
Every addition, subtraction, multiplication and division uses numerator-denominator pairs reduced by the greatest common divisor. The calculator does not convert one third to a rounded decimal and then carry that error through later steps. Quotient and remainder therefore remain exact even when the leading divisor coefficient does not divide evenly.
How each long-division step works
The leading remaining term is divided by the leading divisor term to produce the next quotient coefficient. A shifted copy of the divisor is multiplied by that coefficient and subtracted from the running remainder. The polynomial long division calculator records the factor and shift for each elimination, continuing until the remainder degree is lower than the divisor degree.
Missing terms and zero leading entries
Leading zero coefficients are removed, while zeros inside the list remain meaningful placeholders. A zero polynomial divisor is rejected. When the dividend degree is already lower than the divisor degree, the quotient is zero and the full dividend is returned as the remainder. These cases are handled directly rather than forcing an unnecessary step.
Verification identity
A correct result satisfies dividend equals divisor times quotient plus remainder. The page presents those four polynomials together, making substitution or independent multiplication straightforward. The remainder must be zero or have lower degree than the divisor. If a hand calculation differs, check whether a zero placeholder for a missing power was omitted.
Scope and notation
This polynomial long division calculator handles one-variable coefficient arithmetic. It does not factor the polynomial, solve for roots or interpret a multivariable expression. Coefficient input is intentionally separate from symbolic parsing so signs, powers and fractions remain predictable. Use the output as a worked algebra check and retain exact fractions until a decimal is specifically required.
Polynomial Long Division Calculator questions
How do I enter a missing x term?
Enter zero in that coefficient position.
Can coefficients be fractions?
Yes. Use forms such as -3/5.
What if the remainder is not zero?
That is valid when its degree is lower than the divisor degree.