Problem 156 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The expression is factored as \((2x+3)^3(x+1)^3(x+2)\).

Step 1: Identify Common Factor

Observe the given expression: \((2x+3)^4(x+1)^4 - (2x+3)^3(x+1)^3\). The common factor in both terms is \((2x+3)^3(x+1)^3\). Factor it out from the expression.

Step 2: Factor Out the Common Term

Rewrite the expression by factoring out the common term \((2x+3)^3(x+1)^3\) from both parts. The expression becomes: \[(2x+3)^3(x+1)^3((2x+3) - (x+1))\].

Step 3: Simplify the Remaining Expression

Within the brackets, simplify \((2x+3) - (x+1)\). Distribute the negative sign and simplify: \((2x+3) - (x+1) = 2x + 3 - x - 1 = x + 2\).

Step 4: Combine All Parts

Substitute back the simplified expression into the overall factored form. The expression now is: \((2x+3)^3(x+1)^3(x+2)\).

Step 5: Verify Factorization

Check if the expression \((2x+3)^3(x+1)^3(x+2)\) can be factored further. Since each factor is in its simplest form, the factorization is complete.

Key Concepts

Common Factor

Finding the common factor in an algebraic expression is the first and often one of the most critical steps in polynomial factorization. A common factor is simply a term or combination of terms that is shared among different parts of an expression. In essence, it is like finding a number of goods in common among several groups of items. In our example, we have noticed that both terms of the expression

• \((2x+3)^4(x+1)^4\)
• \((2x+3)^3(x+1)^3\)

contain the smaller power of the repeated factors:

• \((2x+3)^3\)
• \((x+1)^3\)

Thus, we pulled out these common factors to assist us in simplifying the rest of the expression. Recognizing a common factor simplifies the factorization process because it reduces the complexity of the expression, making it easier to identify other factors or perform further calculations.

Simplification

Simplification in algebra refers to reducing an expression to its simplest form. This process usually involves combining like terms or reducing the terms within an expression through basic arithmetic operations. In the exercise, we observed simplification in action when addressing the term inside our parentheses after factoring out the common factors: \[(2x+3) - (x+1)\].Here, we took each like component (the constant terms and the variable terms) and combined them to create a cleaner expression. The process was as follows:

• First, distribute the negative sign: \((2x+3) - x - 1\).
• Next, combine like terms: \(2x - x\) results in \(x\) and \(3 - 1\) results in \(2\).

Ultimately, the simplified expression that fits within our factorized framework was \((x + 2)\). Simplifying is crucial for clarity, helping to ensure that the final expression, in this case, the factored polynomial, is as manageable as possible.

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and mathematical operations (such as addition and multiplication) that together form a mathematical phrase. It's important to understand that expressions can represent relationships and allow us to predict outcomes. In our example, the algebraic expression was \[(2x+3)^4(x+1)^4 - (2x+3)^3(x+1)^3\].Even though it looks complex, by breaking it down into its parts — common factors and remaining terms — we can handle it more easily. The final form after factoring was \[(2x+3)^3(x+1)^3(x+2)\].This showcases how expressions can be transformed through operations like factorization, revealing patterns and simplifying calculations. Recognizing an expression's components and operations at play aids in manipulating it effectively, offering insight into solving larger algebraic problems.

Factorization Verification

Factorization verification involves checking whether an expression is correctly factorized. This is an important step as it confirms your solution's accuracy. After completing the factorization of our initial algebraic expression into\((2x+3)^3(x+1)^3(x+2)\), one must ensure that no further factorization is possible. Here's how to verify factorization:

• Each factor, \((2x+3)^3\), \((x+1)^3\), and \((x+2)\), should be in its simplest form.
• No common factors should remain within or between the parts.Checking individual factors’ simplest forms ensures that the process has been applied correctly.

For additional validation, expanding the expression back to its original form would also confirm the accuracy. By doing so, you verify that each step reduced and transformed the expression correctly, guaranteeing the factorization is perfect — informative best practices for algebraic manipulation and solution confirmation.