ThePolynomial Remainder Theorem states that if a polynomial $f(x)$ is divided by a linear divisor of the form $x - c$, then the remainder of this division is simply $f(c)$.

$$f(x) = (x - c) \cdot Q(x) + R$$

where $Q(x)$ is the quotient polynomial and $R$ is the remainder, the theorem tells us that:

$$R = f(c)$$

The Polynomial Remainder Theorem states that if a polynomial $f(x)$ is divided by a linear divisor $(x - c)$, then the remainder of this division is simply $f(c)$. In other words:

$$f(x) = (x - c) Q(x) + R$$

where:

• $Q(x)$ is the quotient,
• $R$ is the remainder.

Since the divisor $(x - c)$ is a linear polynomial, the remainder must be a constant. The theorem asserts that this constant remainder is just $f(c)$.

Additionally, if the remainder $R$ is zero, then $x - c$ is a factor of the polynomial $f(x)$, which is related to the Factor Theorem.

Example:

Let’s say we have the polynomial:

$$f(x) = x^3 - 4x + 6$$

If we divide $f(x)$ by $(x - 2)$, then the remainder is found by evaluating $f(2)$:

$$f(2) = (2)^3 - 4(2) + 6 = 8 - 8 + 6 = 6$$

So, the remainder when dividing $f(x)$ by $(x - 2)$ is 6.

In this example, $Q=x^2+2x$. Thus, we can express the division as:

$$f(x) = (x - 2)(x^2 + 2x) + 6$$

This theorem is particularly useful in polynomial division and checking factors using the Factor Theorem, which is a direct consequence of the Remainder Theorem.