Remainder Theorem

The Remainder Theorem is a key concept in algebra that provides a simple and efficient way to find the remainder when a polynomial is divided by a linear divisor. It plays an important role in polynomial division, which is a fundamental operation in algebraic manipulations and problem-solving. The theorem is particularly useful in simplifying polynomial expressions and solving equations where division by a polynomial is involved.

What is the Remainder Theorem?

The Remainder Theorem states that when a polynomial 𝑓(𝑥) is divided by a linear divisor of the form (𝑥 − 𝑎), the remainder of the division is simply 𝑓(𝑎). In other words, the remainder when a polynomial is divided by (𝑥 − 𝑎) is the value of the polynomial evaluated at 𝑥 = 𝑎. 

Mathematically, the Remainder Theorem can be expressed as: 

If 𝑓(𝑥) is divided by (𝑥 − 𝑎), then the remainder is 𝑓(𝑎). 

This theorem is incredibly useful because it gives a quick way to find the remainder of polynomial division without needing to perform the entire long division process.

Statement of the Remainder Theorem

Given a polynomial 𝑓(𝑥) of degree 𝑛, when 𝑓(𝑥) is divided by (𝑥 − 𝑎), the remainder 𝑅 is given by: 

𝑓(𝑥) = (𝑥 − 𝑎) ⋅ 𝑞(𝑥) + 𝑅 where: 

• 𝑓(𝑥) is the polynomial being divided,

 • (𝑥 − 𝑎) is the divisor, • 𝑞(𝑥) is the quotient, 

• 𝑅 is the remainder. 

According to the Remainder Theorem, the remainder 𝑅 is simply 𝑓(𝑎), so we have: 

𝑓(𝑎) = 𝑅 

Thus, to find the remainder when dividing 𝑓(𝑥) by (𝑥 − 𝑎), you just need to evaluate 𝑓(𝑎).

Example of Applying the Remainder Theorem

Let's apply the Remainder Theorem to a specific example to see how it works in practice. 

Example: 

Divide 𝑓(𝑥) = 2𝑥³ − 3𝑥² + 4𝑥 − 5 by 𝑥 − 2. 

Solution: To find the remainder, we evaluate 𝑓(𝑥) at 𝑥 = 2: 

𝑓(2) = 2(2)³ − 3(2)² + 4(2) − 5 

First, calculate the powers of 2: 

𝑓(2) = 2(8) − 3(4) + 4(2) − 5 

𝑓(2) = 16 − 12 + 8 − 5 

𝑓(2) = 7 

Thus, the remainder when dividing 2𝑥³ − 3𝑥² + 4𝑥 − 5 by 𝑥 − 2 is 7.

Proof of the Remainder Theorem:

Let's briefly prove the Remainder Theorem using polynomial division. 

Given a polynomial 𝑓(𝑥), we divide it by (𝑥 − 𝑎). According to the division algorithm for polynomials, we can write: 

𝑓(𝑥) = (𝑥 − 𝑎) ⋅ 𝑞(𝑥) + 𝑅 

where 𝑞(𝑥) is the quotient and 𝑅 is the remainder. Since we are dividing by a linear polynomial (𝑥 − 𝑎), the degree of the remainder 𝑅 must be less than the degree of (𝑥 − 𝑎), which is 1. Therefore, 𝑅 must be a constant. 

Now, substitute 𝑥 = 𝑎 into the equation: 

𝑓(𝑎) = (𝑎 − 𝑎) ⋅ 𝑞(𝑎) + 𝑅 𝑓(𝑎) = 0 

⋅ 𝑞(𝑎) + 𝑅 𝑓(𝑎) = 𝑅 

Thus, the remainder when dividing 𝑓(𝑥) by (𝑥 − 𝑎) is 𝑓(𝑎), proving the Remainder Theorem.

Applications of the Remainder Theorem:

1. Evaluating Polynomial Remainders Quickly: As seen in the example above, the Remainder Theorem allows us to quickly find the remainder of a polynomial division by simply evaluating the polynomial at a specific value of 𝑥, rather than performing full polynomial long division.

2. Solving Polynomial Equations: The Remainder Theorem can help solve polynomial equations. For instance, if a polynomial 𝑓(𝑥) is divided by (𝑥 − 𝑎) and the remainder is zero, then 𝑥 = 𝑎 is a root of the polynomial.

3. Finding Factors of Polynomials: If 𝑓(𝑎) = 0, it implies that 𝑥 − 𝑎 is a factor of the
polynomial 𝑓(𝑥). This can be useful in factoring polynomials or finding roots of
equations.

4. Testing for Divisibility: The Remainder Theorem is also helpful in testing
whether one polynomial is divisible by another. If 𝑓(𝑎) = 0, then 𝑓(𝑥) is divisible
by (𝑥 − 𝑎), and 𝑥 = 𝑎 is a root of the polynomial.

The Factor Theorem: 

The Factor Theorem is a direct consequence of the Remainder Theorem. It states that if 𝑓(𝑎) = 0, then 𝑥 − 𝑎 is a factor of the polynomial 𝑓(𝑥). In other words, if the remainder is zero, the divisor is a factor of the polynomial. If 𝑓(𝑎) = 0, then (𝑥 − 𝑎) is a factor of 𝑓(𝑥).

Example of the Factor Theorem Suppose we have the polynomial 𝑓(𝑥) = 𝑥² − 5𝑥 + 6 and want to check if 𝑥 − 2 is a factor of 𝑓(𝑥). We apply the Remainder Theorem by evaluating 𝑓(𝑥) at 𝑥 = 2: 

𝑓(2) = (2)² − 5(2) + 6 

𝑓(2) = 4 − 10 + 6 = 0 

Since 𝑓(2) = 0, according to the Factor Theorem, 𝑥 − 2 is a factor of 𝑓(𝑥).