Polynomial Remainder Theorem

In this lesson, we will explore an important theorem in algebra that allows us to calculate the remainder of a division between polynomials without actually performing the division itself. This result is known as the Remainder Theorem and can only be applied in cases where the divisor polynomial takes the form .

This theorem is also referred to as the Little Bézout's Theorem, named after the French mathematician Étienne Bézout who discovered it.

Furthermore, we will see the proof of the remainder theorem and how to use it to quickly calculate the remainder in polynomial divisions.

A Note on Ruffini's Rule

In the previous lesson, we explored how to rapidly divide two polynomials when the divisor polynomial takes the form , employing the Ruffini's Rule.

Suppose we aim to divide the polynomial by the polynomial . We apply the rule in this manner:

The result is then:

while the remainder is:

Now, let's substitute , which is the opposite of the constant term of the divisor polynomial, into the dividend polynomial :

The result matches the remainder computed using Ruffini's rule. This isn't a coincidence but a general rule allowing us to calculate the remainder of a division between polynomials without having to perform the division itself. This principle is known as the Polynomial Remainder Theorem.

Polynomial Remainder Theorem

Definition

Polynomial Remainder Theorem (Little Bézout's Theorem)

Let be a polynomial of degree , with , and let be a polynomial of the form . Then, the remainder of the division between and is given by:

In other words, to calculate the remainder of the division between and , simply substitute into .

Proof

Proof of the Remainder Theorem

Given the polynomial and the polynomial , the division between and is given by:

where is the quotient and is the remainder. The expression above can be written as:

Substituting into the previous expression, we get:

But and . Therefore:

and thus:

Examples

Let's work through some examples to verify the remainder theorem.

Example

Calculate the remainder of the division between the polynomial and the polynomial .

Without carrying out the division, substitute into :

Therefore, the remainder is .

Example

Calculate the remainder of the division between the polynomial and the polynomial .

Without carrying out the division, substitute into :

Therefore, the remainder is .

In Summary

In this lesson, we explored how to calculate the remainder of a division between polynomials without having to perform the actual division, but only when the divisor polynomial is of the form . This result is known as the Polynomial Remainder Theorem.

In essence, the theorem tells us that to compute the remainder, it's enough to substitute , which is the opposite of the constant term of the divisor polynomial, into the dividend polynomial:

In the next lesson, we'll look into an important corollary of the remainder theorem called the Factor Theorem also known as the Ruffini's Theorem.

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