Understanding Polynomial Division and Remainders

In this article, we will explore the process of finding the remainder of a polynomial when it is divided by a given polynomial. Specifically, we will solve the problem of finding the values p and q for the polynomial P(x) in the context of the division by x^2 - 5x 6.

Problem Statement

We are given that the remainder of the polynomial P(x) when divided by x - 2 is 3, and when divided by x - 3 is 2. We need to find the remainder when P(x) is divided by x^2 - 5x 6, which factors into (x - 2)(x - 3). The remainder is of the form P(x) px q.

Step-by-Step Solution

The problem is boiled down to setting up and solving a system of equations based on the given conditions. Here is a detailed breakdown of the solution process:

Step 1: Understand the Given Remainders

We know the following:

The remainder of P(x) when divided by x - 2 is 3: The remainder of P(x) when divided by x - 3 is 2:

This translates to:

P(2) 3 P(3) 2

Step 2: Formulate the Remainder

When P(x) is divided by x^2 - 5x 6, the remainder is a linear polynomial px q since the divisor is a quadratic polynomial.

Step 3: Set Up the System of Equations

Using the fact that the polynomial x^2 - 5x 6 factors into (x - 2)(x - 3), we can express the system of equations as:

P(2) 2p q 3 (from P(2) 3) P(3) 3p q 2 (from P(3) 2)

Step 4: Solve the System of Equations

Subtract the first equation from the second equation to eliminate q: [ (3p q) - (2p q) 2 - 3 ] This simplifies to: [ p -1 ] Now substitute p -1 back into the first equation to find q: [ 2(-1) q 3 ] [ -2 q 3 ] [ q 5 ] Thus, the values of p and q are:

boxed{-1} and boxed{5}

Conclusion

In this solution, we effectively used the properties of polynomial division and the remainder theorem to find the values of p and q. This method involves setting up and solving a system of linear equations based on the given remainders of the polynomial when divided by specific linear factors.

Related Keywords: polynomial division, remainder theorem, system of equations