Problem 12 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Factor \(b^{x}\) out of the following expressions. Check your answer by multiplying out. $$ 3 b^{2 x+1}-4 b^{2 x-1} $$

Step-by-Step Solution

Answer

The factorized form of the expression is \(b^{x}(3b^{x+1}-4b^{x-1})\).

Step 1: Identify Common Factor

The algebraic expression given here is \(3 b^{2x+1}-4 b^{2x-1}\). The common factor in both terms is \(b^{x}\). Note that \(b^{2x+1}\) is equivalent to \(b^{x} \times b^{2x}\), and \(b^{2x-1}\) is equivalent to \(b^{x} \times b^{x-1}\).

Step 2: Factor out the Common Term

Factor out \(b^{x}\). Replace each term of the original expression by what’s left when you take out the common factor. This gives the factored form as \(b^{x}(3b^{x+1}-4b^{x-1})\).

Step 3: Expand the factored expression

To verify the factoring, go through the process in reverse by multiplying \(b^{x}\) to each term within the parentheses. This gives \(3b^{2x+1}-4b^{2x-1}\). Hence, the correctness of the solution is validated.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases representing numbers and operations. They include variables, constants, and operators. In the context of the expression given, we noticed terms like \(3b^{2x+1}\) and \(-4b^{2x-1}\). Here, \(b\) is a variable raised to different powers, and coefficients such as 3 and -4 are multipliers of these terms. Understanding how to manipulate these expressions is crucial in algebra. It involves simplifying them or factoring them to reveal their underlying structure. Recognizing patterns and common terms will make it easier to work with these expressions.

Identifying the Common Factor

A common factor is a term that appears in multiple parts of an expression. It allows you to simplify by pulling out the shared term from each part. The goal is to rewrite the expression in a more manageable form. To find the common factor, look for terms that all the parts share. In this exercise, the expression \(3 b^{2x+1} - 4 b^{2x-1}\) both have \(b^{x}\) as a common factor. Once identified, the common factor can be factored out. This simplifies the expression and reveals a new perspective on its form.

Multiplying Out the Expression

Once an expression is factored, multiplying out is a reverse process to verify correctness. It involves distributing the extracted common factor back into the expression. For our example, to check if \(b^{x}(3b^{x+1} - 4b^{x-1})\) is correct, multiply \(b^{x}\) with each term inside the parentheses:

• \(b^{x} \times 3b^{x+1} = 3b^{2x+1}\)
• \(b^{x} \times -4b^{x-1} = -4b^{2x-1}\)

By performing these multiplications, we return to the original expression \(3 b^{2x+1} - 4 b^{2x-1}\), thus validating the factorization.

Verifying the Factoring Process

Verifying factoring involves ensuring that the factorization process has been done correctly. After factoring, the expression should be expandable back to its original form. This checks your work and ensures accuracy. In our case, once we factor out \(b^{x}\), multiplying it back into the resulting terms should reproduce the original \(3 b^{2x+1} - 4 b^{2x-1}\). Always take this step in factoring to confirm the results. It not only confirms that the solution is accurate but also reinforces understanding of the process's mechanics.