Problem 155 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \((6 x+1)(8 x-3)^{4}-(6 x+1)^{2}(8 x-3)^{3}\)

Step-by-Step Solution

Answer

The completely factored expression is \(2(8x-3)^3(6x+1)(x-2)\).

Step 1: Identify Common Factor

Observe the given expression: \[(6x + 1)(8x - 3)^4 - (6x + 1)^2(8x - 3)^3\]Identify the common factor in both terms, which is \((8x - 3)^3(6x + 1)\).

Step 2: Factor Out Common Terms

Factor out the common terms from the expression. This results in:\[(8x - 3)^3(6x + 1) \left((8x - 3) - (6x + 1)\right)\]

Step 3: Simplify the Remaining Terms

Simplify the expression inside the parenthesis:\[((8x - 3) - (6x + 1)) = 8x - 3 - 6x - 1 = 2x - 4\].

Step 4: Fully Factor the Expression

Substitute the simplified term back into the expression:\[(8x - 3)^3(6x + 1)(2x - 4)\].Notice that \(2x - 4\) can be further factored as \(2(x - 2)\).

Step 5: Present the Completely Factored Expression

The final factorization of the given expression becomes:\[2(8x - 3)^3(6x + 1)(x - 2)\].

Key Concepts

Factoring Expressions

Factoring expressions is a fundamental skill in algebra that involves breaking down an expression into a product of simpler components. Think of it as the reverse of expanding expressions. Instead of multiplying out terms, you're piecing them together to see where they originally came from. Factoring makes solving equations easier and simplifies expressions for future use. In the given exercise, the expression \(6x + 1\)(8x - 3)^4 - (6x + 1)^2(8x - 3)^3\, a more manageable form was achieved through factoring. By identifying and extracting common terms using algebraic manipulations, we refine expressions to reveal hidden relationships between their components.

Common Factors

Finding common factors in an algebraic expression is akin to discovering a universal thread that runs through different pieces of an expression. A common factor is a term or expression shared by all terms in an equation. It can be factored out to simplify the expression or equation.

In the original exercise, the common factor detected was \((8x - 3)^3(6x + 1)\). By recognizing this factor, we create an opportunity to simplify the expression, reduce complexity, and solve problems more easily. Recognizing common factors is crucial; it acts like finding the key piece of a puzzle.

Polynomial Simplification

Polynomial simplification is the process of making polynomials easier to work with by reducing them to their simplest form. This involves using algebraic techniques to comb through expressions, clarifying conflicting terms, combining like terms, and eliminating redundancies.

In our exercise, simplification was necessary after factoring out the common terms, resulting in a smaller, more digestible polynomial. By handling the terms inside the parentheses \((8x - 3) - (6x + 1)\), we simplified this to \2x - 4\. Such steps make polynomial equations not only easier to solve but also help in understanding the inherent structure of the polynomial.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that form the foundation of algebra. They are the building blocks of more complex equations and functions. An algebraic expression can represent a quantity's relationship to other quantities and is used in a variety of mathematical applications.

The given problem required understanding the composition of the algebraic expression and using that understanding to simplify and factor it. By breaking down \(6x + 1\)(8x - 3)^4 - (6x + 1)^2(8x - 3)^3\ into its constituent factors and terms, algebraic rules were effectively applied to achieve a completely factored expression. Mastery of algebraic expressions is essential, as they form a crucial part of solving algebraic equations effectively.