Remainder Theorem Calculator

Remainder Theorem Calculator

Calculate the remainder of a polynomial division instantly.

Enter values from highest power to constant (e.g., x² – 4x + 3 is 1, -4, 3)

If dividing by (x – 2), enter 2. If dividing by (x + 3), enter -3.

What is the Remainder Theorem?

The Remainder Theorem is a fundamental principle in algebra that links the division of polynomials to basic substitution. It states that when a polynomial \(P(x)\) is divided by a linear factor of the form \((x – a)\), the remainder of that division is exactly equal to \(P(a)\).

This theorem is incredibly useful because it allows you to find the remainder of long polynomial divisions without having to perform the tedious long division or synthetic division process.

The Remainder Theorem Formula

If you have a polynomial:

P(x) = a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀

And you divide it by (x – c), the remainder R is:

R = P(c)

Example Calculation

Let's say we want to divide \(P(x) = x^2 – 4x + 3\) by \((x – 2)\).

• Step 1: Identify the value of 'a'. In \((x – 2)\), \(a = 2\).
• Step 2: Substitute \(x = 2\) into the polynomial.
• Step 3: \(P(2) = (2)^2 – 4(2) + 3\)
• Step 4: \(P(2) = 4 – 8 + 3 = -1\)

Therefore, the remainder is -1.

Why is this useful?

Beyond simply finding remainders, this theorem is the basis for the Factor Theorem. If \(P(a) = 0\), then the remainder is zero, which means \((x – a)\) is a perfect factor of the polynomial. This is essential for solving higher-degree equations and graphing functions.

How to use this calculator

1. Enter Coefficients: List the numbers in front of the variables. For \(x^3 + 5\), you must enter `1, 0, 0, 5` (representing \(1x^3 + 0x^2 + 0x + 5\)).
2. Enter the Divisor: If your divisor is \((x – 5)\), enter `5`. If it is \((x + 5)\), enter `-5`.
3. Result: The calculator immediately evaluates the function at that point to give you the remainder.