91Ó°ÊÓ

Polynomials are expressions that consist of variables (also known as indeterminates), coefficients, and operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, a simple polynomial could be expressed as \( ax^n + bx^{n-1} + \, \ldots \, + c \), where each term consists of a coefficient and a variable raised to a power. Such expressions can vary from very simple to highly complex, depending on the number of terms and the powers involved.
Understanding polynomials is crucial in mathematics because they can represent a wide range of relationships and functions. They are often used to model problems in physics, economics, and engineering. In this exercise, the polynomial provided is rather straightforward due to its arithmetic form \( P_k = \frac{3}{(k+2)(k+3)} \).
When working with polynomials, especially in sequences, it's essential to be familiar with identifying the variable or index (in this case, \( k \)) and understanding how changing this variable affects the entire expression.

Substitution is a powerful technique in mathematics, used to transform a given expression by replacing variables with specific values or other expressions. This process is particularly helpful when dealing with sequences to find successive elements based on a pattern.
In the context of this exercise, we need to find \( P_{k+1} \) given \( P_k \). This requires substituting \( (k+1) \) into the original polynomial expression. Specifically, every instance of \( k \) in the expression \( P_k = \frac{3}{(k+2)(k+3)} \) is replaced with \( (k+1) \), resulting in:

• Change \( k \) to \( (k+1) \) in every factor of the polynomial denominator.
• This substitution leads to the new expression for \( P_{k+1} \): \( \frac{3}{((k+1)+2)((k+1)+3)} \).

By substituting correctly, we pave the way for further simplification, which is the final step to derive the required form of the polynomial sequence.

Simplification is the process of reducing a mathematical expression or equation to its simplest form. This means eliminating unnecessary components and combining like terms to present the expression as neatly and efficiently as possible.
After substitution, the next step in this exercise is simplifying the expression \( \frac{3}{((k+1)+2)((k+1)+3)} \). The key here is expanding the terms and performing any possible arithmetic or algebraic reduction.
Let's break down the simplification:

• First, calculate \((k+1) + 2\) to get \(k + 3\), and \((k+1) + 3\) to obtain \(k + 4\).
• The resulting polynomial expression is thus \( \frac{3}{(k+3)(k+4)} \).
• Always check if any further factorization or cancellation can simplify the expression more.

Successfully simplifying these steps makes the given sequence's expression cleaner, thereby making further analysis or calculation easier.