Problem 83 - Algebra and Trigonometry

Question

77–84 ? Factor the expression completely. Begin by factoring out the lowest power of each common factor. $$ 2 x^{1 / 3}(x-2)^{2 / 3}-5 x^{4 / 3}(x-2)^{-1 / 3} $$

Step-by-Step Solution

Answer

The factored form is \(x^{1/3}(x-2)^{-1/3}(-3x - 4)\).

Step 1: Identify the Greatest Common Factor (GCF)

Identify the lowest power of each common factor in the given expression. Here, the two terms are \(2x^{1/3}(x-2)^{2/3}\) and \(-5x^{4/3}(x-2)^{-1/3}\). The lowest power of \(x\) is \(x^{1/3}\) and the lowest power of \((x-2)\) is \((x-2)^{-1/3}\).

Step 2: Factoring Out the GCF

Factor out the greatest common factor, which is \(x^{1/3}(x-2)^{-1/3}\), from each term in the expression. This gives: \[ x^{1/3}(x-2)^{-1/3} \left[ 2(x-2) - 5x^{3/3} \right] \].

Step 3: Simplify the Expression Inside the Brackets

Expand and simplify the expression within the brackets: \[2(x-2) - 5x = 2x - 4 - 5x\]. This simplifies further to \(-3x - 4\).

Step 4: Write the Factored Expression

Combine the results from the previous steps to give the completely factored expression: \[x^{1/3}(x-2)^{-1/3}(-3x - 4)\].

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a crucial concept when it comes to factoring algebraic expressions. It represents the largest factor shared by all terms in an expression, which can be factored out to simplify the expression.
For example, in the given expression:\[2x^{1/3}(x-2)^{2/3} - 5x^{4/3}(x-2)^{-1/3}\]we identify the common factors among the terms. This involves examining the powers of both \(x\) and \((x-2)\).

• The lowest power for \(x\) is \(x^{1/3}\).
• The lowest power for \((x-2)\) is \((x-2)^{-1/3}\).

By extracting these factors, we simplify our expression's structure, making further operations like distribution or simplification easier. Understanding GCF is not just about numbers but also includes algebraic expressions concerning variable bases and their powers.

Simplifying Expressions

Simplifying expressions is the process of transforming an algebraic expression into its simplest form. This often involves combining like terms, eliminating unnecessary factors, and conducting basic arithmetic operations.
Once the GCF is factored out, as shown previously, focus is shifted to simplifying what's left from the factorization.\[x^{1/3}(x-2)^{-1/3} \left[ 2(x-2) - 5x \right]\]Inside the brackets, we have:

• Distribute the 2 into \((x-2)\), leading to \(2x - 4\).
• Next, subtract \(5x\) from \(2x\), simplifying to \(-3x\).
• Combine terms: \(2x - 4 - 5x = -3x - 4\).

Simplifying enables us to understand the core structure and solution of the problem, thereby making algebra much more straightforward to manage.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. They can also include exponents and roots, as seen in complex expressions like:\[2x^{1/3}(x-2)^{2/3} - 5x^{4/3}(x-2)^{-1/3}\]Manipulating these expressions requires a solid understanding of algebraic rules and properties.
Algebraic expressions serve several functions:

• Model real-life situations such as calculating interest or projecting growth rates.
• Allow for expressions of complex relationships in a simplified format.
• Enable finding solutions to puzzles and equations by setting expressions equal to one another or certain values.

Understanding how to work with algebraic expressions, including factoring and simplifying, is fundamental in solving algebraic equations effectively.