How do you find the divisor of a remainder theorem?

To find the divisor in the context of the Remainder Theorem, you're essentially trying to determine what expression, when dividing a polynomial P(x), leaves a specific remainder. The key aspect of the Remainder Theorem, as illustrated in the provided reference, is the relationship between the divisor and the value of the polynomial evaluated at a specific point.

Here's a breakdown:

The Remainder Theorem states: When a polynomial P(x) is divided by (x - a), the remainder is P(a).

Conversely, x - a is the divisor of P(x) if and only if P(a) = 0.

Here's how you can use this:

Finding the Divisor

1. If you know the root: If you know a value a such that P(a) = 0, then (x - a) is a divisor of P(x). For example, if P(2) = 0, then (x - 2) is a divisor.

2. If you know the remainder when dividing by (x-a): If you divide P(x) by (x-a) and obtain a remainder R, then P(a) = R. If the remainder R is equal to 0, then x-a is a factor.

Example

As given in the reference, if f(a) = a³ - 12a² - 42 is divided by (a - 3), the remainder is -123. In this case:

• P(a) = a³ - 12a² - 42
• Divisor is (a - 3), meaning a = 3
• P(3) = (3)³ - 12(3)² - 42 = 27 - 108 - 42 = -123

Since the remainder P(3) is -123 (and not 0), (a - 3) is not a factor of a³ - 12a² - 42.

In Summary

• The Remainder Theorem allows you to determine the remainder of a polynomial division without actually performing the long division.
• If P(a) = 0, then (x - a) divides P(x) evenly, meaning it's a divisor and there's no remainder.