To calculate result you have to disable your ad blocker first.
Remainder Theorem Calculator
To use the remainder theorem calculator, Enter the polynomial equation, enter the value of the divisor, and Click Calculate.
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.
Perform the necessary mathematical operations to simplify the expression. The result will be the remainder when f(x) is divided by (x - k).
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
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.