Remainder Theorem Calculator

Work out remainder theorem instantly with clear inputs, formula shown and shareable results.

About Remainder Theorem Calculator

The remainder theorem says dividing P(x) by (x − a) leaves remainder P(a). Synthetic division produces both the quotient and that remainder in a single pass.

Formula

Remainder theorem

P(x) = (x − a)·Q(x) + P(a)

Worked example

With the default values below, Remainder Theorem Calculator returns Remainder P(a) of 9.00000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Polynomial coefficients, highest power first: 2, -3, 0, 5
Divide by (x − a), with a =: 2

Results

Remainder P(a): 9.00000000
Quotient polynomial: 2x^2 + x + 2
Polynomial entered: 2x^3 - 3x^2 + 5
Degree: 3

Step by step

1Values used
2Remainder theorem
3Remainder P(a)= 9.00000000
4Quotient polynomial= 2x^2 + x + 2
5Polynomial entered= 2x^3 - 3x^2 + 5
6Degree= 3

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Divide 2x³ − 3x² + 5 by (x − 2).

P(2) = 16 − 12 + 5 = 9, so the remainder is 9 and the quotient is 2x² + x + 2.

What about dividing by (x + a)?

Write it as (x − (−a)) and evaluate at −a instead.