Problem 68 - A Graphical Approach to College Algebra

Question

Factor, using the given common factor. Assume that all variables represent positive real numbers. $$(3 r+1)^{-2 / 3}+(3 r+1)^{1 / 3}+(3 r+1)^{4 / 3} ; \quad(3 r+1)^{-2 / 3}$$

Step-by-Step Solution

Answer

The expression factors to \((3r+1)^{-2/3} (9r^2 + 9r + 3)\).

Step 1: Identify the Common Factor

The common factor in the given expression is \((3r+1)^{-2/3}\). We need to factor this out from each term in the expression.

Step 2: Factor Out the Common Factor

Factor \((3r+1)^{-2/3}\) out of each term in the expression. Each term will be expressed in terms of this common factor. - For the first term, \((3r+1)^{-2/3}\) is factored out, leaving 1. - For the second term, \((3r+1)^{1/3}\) becomes \((3r+1)^{1/3 - (-2/3)} = (3r+1)^{3/3} = 3r+1\). - For the third term, \((3r+1)^{4/3}\) becomes \((3r+1)^{4/3 - (-2/3)} = (3r+1)^{6/3} = (3r+1)^2\).

Step 3: Write the Factored Expression

Combine the results from Step 2 into a single expression: \[(3r+1)^{-2/3} (1 + (3r+1) + (3r+1)^2)\].

Step 4: Simplify the Factored Terms

Simplify the terms inside the parentheses:- The terms inside are \(1 + (3r+1) + (3r+1)^2\).- Distribute and simplify: \(1 + 3r + 1 + (9r^2 + 6r + 1)\), which simplifies to \(9r^2 + 9r + 3\).The factored expression becomes:\[(3r+1)^{-2/3} (9r^2 + 9r + 3)\].

Key Concepts

Factoring Expressions

Factoring expressions is like finding the hidden pieces that come together to form the entire picture. When you look at an algebraic expression, the goal is to identify parts that are repeated or that appear in all the terms. Once those pieces are identified, they can be 'factored out'. This means simplifying the expression by writing it in terms of these common elements. In our example, we consider the expression:

• \((3 r+1)^{-2 / 3} + (3 r+1)^{1 / 3} + (3 r+1)^{4 /3}\)

Here, we notice that each term involves the expression \((3r+1)\) in different powers. A skilled eye will recognize that these powers can be aligned by factoring out \((3r+1)^{-2/3}\) from each term. Essentially, factoring boils down to pulling out the biggest common part of an expression to make it easier to work with, much like rewriting a complex sentence in simpler terms.

Common Factor

The concept of a common factor is central to simplifying expressions. In the realm of algebra, a common factor is a term that is present in each part of an expression. Think of it as a universal component or thread that runs through each term. In our original exercise, the common factor among all the terms \((3r+1)^{-2/3} + (3r+1)^{1/3} + (3r+1)^{4/3}\) is \((3r+1)^{-2/3}\).
This insight allows us to rewrite each part of the expression in terms of this factor. By doing this, each term becomes more manageable. It's like peeling away layers to reveal the core, making the expression simpler. The act of recognizing and using the common factor not only simplifies the work but also maintains the integrity of the original expression.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the process of reducing expressions to their simplest form, making them easier to work with and understand. This is particularly useful in solving equations and understanding the relationships within an expression.
Let's take the expression we factored: \[(3r+1)^{-2/3} (1 + (3r+1) + (3r+1)^2)\]. Now, simplify the terms inside the parentheses by performing operations such as distribution and combining like terms.

• The terms become: \[1 + 3r + 1 + (9r^2 + 6r + 1)\].
• By combining like terms, we arrive at: \[9r^2 + 9r + 3\].

This whole process shows how simplification not only reduces clutter but makes the mathematical relationships clearer. In practical scenarios, a more straightforward form can provide insight into trends and solutions that might not be evident in the original, more complex expression.