Problem 62 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{d^{2}-16 c^{2}}{4 c-d} $$

Step-by-Step Solution

Answer

The simplified expression is \(-d - 4c\).

Step 1: Recognize the Expression as a Difference of Squares

The numerator of the rational expression \(d^{2} - 16c^{2}\) is a difference of squares. Recall that a difference of squares takes the form \(a^{2} - b^{2}\), which can be factored into \((a - b)(a + b)\). In this expression, \(d^{2}\) is \(a^{2}\) and \(16c^{2}\) is \(b^{2}\) where \(b = 4c\).

Step 2: Factor the Difference of Squares

Using the difference of squares formula, factor \(d^{2} - 16c^{2}\) to become \((d - 4c)(d + 4c)\). Rewriting the expression provides: \[\frac{(d - 4c)(d + 4c)}{4c - d}\]

Step 3: Recognize Equivalent Expressions in the Denominator

Note that the denominator \(4c - d\) can be rewritten as \(-1 \cdot (d - 4c)\) or \(-(d - 4c)\). By recognizing and factoring \(-1\) from \(4c - d\), the expression becomes: \[\frac{(d - 4c)(d + 4c)}{-(d - 4c)}\]

Step 4: Simplify by Canceling Common Factors

Notice the common factor of \((d - 4c)\) in both the numerator and denominator. Cancel this common factor, paying attention to the negative sign introduced in the previous step: \[-(d + 4c)\] After simplification, we find that the expression is simplified to \(-d - 4c\).

Key Concepts

Difference of Squares

When we talk about the difference of squares, we're referring to a specific algebraic expression. The concept is best understood with the formula: \(a^2 - b^2\). This pattern is called a 'difference' because it involves subtraction, and 'squares' since it deals with squared terms. For example, if you have \(d^2 - 16c^2\), that can be set in the form \(a^2 - b^2\) where \(a = d\) and \(b = 4c\). Recognizing a difference of squares allows you to factor the expression simply into \((a-b)(a+b)\).

• Key Pattern: \(a^2 - b^2 = (a-b)(a+b)\).
• In our example: \(d^2 - 16c^2 = (d-4c)(d+4c)\).

Identifying this structure in expressions prompts easier simplifications, which is crucial for handling rational expressions.

Factoring

Factoring is like reverse multiplication. It's the process of breaking down an expression into simpler "building blocks" or factors that, when multiplied together, give you the original expression. In the context of this exercise, once we identify the difference of squares structure in \(d^2 - 16c^2\), we apply the formula to factor it into \((d-4c)(d+4c)\).

• This makes the expression more manageable and exposes common terms that might cancel out later.
• Factoring can simplify complex expressions greatly, especially when simplifying rational expressions.
• It's crucial for uncovering potential cancellations with terms in the denominator.

By transforming the original problem into a product of its factors, we make subsequent steps towards simplifying much clearer and approachable.

Simplification of Algebraic Expressions

The simplification of algebraic expressions involves reducing expressions to their simplest form. After factoring the difference of squares in the numerator, we end up with the expression: \(\frac{(d-4c)(d+4c)}{4c-d}\). Here, simplification isn't just about factors but also recognizing equivalent expressions. The denominator \(4c - d\) can be seen as \(-(d - 4c)\), allowing you to rewrite the fraction as \(\frac{(d-4c)(d+4c)}{-(d-4c)}\). Identify common factors between the numerator and denominator to simplify.

• Cancel out \(d-4c\) from both sides, taking care of the negative involved in the denominator.
• The result \(-(d+4c)\) is the simplest form of the expression.

This step highlights the importance of the negative sign in reversal operations, ensuring accuracy in the final simplified expression.