Remainder Theorem Calculator

Remainder Theorem Calculator 

Find what is remaining after a polynomial division with the remainder theorem calculator. See the complete calculation process below the result. 

The Remainder Theorem provides an easy way to determine the remainder of a polynomial division without actually performing the division operation. 

What is the remainder theorem?

The Remainder Theorem states: 

              "If a polynomial f(x) is divided by (x - k), then the remainder is f(k)." 

In simpler terms, if you want to know the remainder of a polynomial division by (x - k), all you have to do is substitute the value of k into the polynomial.

A fact about the remainder polynomial is that it is always one degree less than the polynomial which was the divisor. So, for a linear polynomial, the remainder is constant.

Limitations of the remainder theorem:

The remainder theorem is applicable in cases where the divisor polynomial has 1st degree only because that would make the value of the variable a constant value that can be put in the main polynomial to find the remainder.

For example: Consider a polynomial 8x^3 - 9. You want to find its remainders for two different divisors i.e. x - 4 and x^2 + 3x.

For x - 4, the value of x is 4 which can be put in the polynomial as 8(4)^3 - 9. The remainder is 503.

But for x^2 + 3x, the value of x is not going to extract a constant remainder from the polynomial. It will only make the calculation complex. 

How to find the remainder theorem?

Let's break down the process of finding the remainder of a polynomial division using the Remainder Theorem.

1: Identify the Polynomial and Divisor.

2: Identify 'k' from the Divisor.

Extract the value of 'k' from the divisor. The divisor takes the form (x - k), so 'k' is the number being subtracted from x. If your divisor is (x - 3), 'k' is 3. For the divisor (x + 2), remember that this is the same as (x - (-2)), so 'k' is -2.

3: Substitute 'k' into the Polynomial.

4: Simplify

Perform the necessary mathematical operations to simplify the expression. The result will be the remainder when f(x) is divided by (x - k).

Example:

Let's illustrate these steps with an example. Suppose a polynomial f(x) = 4x^2 - 5x + 2. Divide it by (x - 3).

Step 1: Identify the Polynomial and Divisor.

The polynomial, f(x), is 4x^2 - 5x + 2, and our divisor is (x - 3).

Step 2: Identify 'k' from the Divisor.

From the divisor (x - 3), identify that 'k' is 3.

Step 3: Substitute 'k' into the Polynomial.

f(3) = 4*(3)^2 - 5*3 + 2.

Step 4: Simplify

f(3) = 4*9 - 15 + 2 

      = 36 - 15 + 2 

      = 23.

Hence, the remainder when the polynomial 4x^2 - 5x + 2 is divided by (x - 3) is 23.

Proof of the remainder theorem:

Suppose you're dividing a polynomial f(x) by (x - k), then by the definition of polynomial division, f(x) can be expressed as follows:

Dividend = divisor * quotient + remainder

f(x) = (x - k) * q(x) + r

where q(x) is the quotient and r is the constant remainder, which does not depend on x.

According to the definition of polynomial division, r must be less than the degree of (x - k). Because (x - k) is a first-degree polynomial, the remainder r must be a zero-degree polynomial, or in other words, a constant.

We're interested in finding the value of r, the remainder when f(x) is divided by (x - k). To do this, we substitute k into the equation.

f(k) = (k - k) * q(k) + r

f(k) = 0 * q(k) + r

f(k) = r

So, the remainder r when f(x) is divided by (x - k) is indeed f(k), which is what the Remainder Theorem states.

This proof provides a clear mathematical justification for the Remainder Theorem, showing why it's true for all polynomials f(x) and all real numbers k.