Problem 60 - Elementary and Intermediate Algebra

Question

Divide, and then simplify, if possible. \(\frac{x^{2}-16}{x-4} \div(3 x+12)\)

Step-by-Step Solution

Answer

The simplified result is \( \frac{1}{3} \).

Step 1: Identify the expression to divide

The problem asks us to evaluate \( \frac{x^{2}-16}{x-4} \div (3x + 12) \). To divide by a fraction or term, multiply by its reciprocal.

Step 2: Factor where possible

First, look to factor the numerator \( x^2 - 16 \) as it is a difference of squares. This gives us \( (x-4)(x+4) \). Also, factor \( 3x + 12 \) as \( 3(x+4) \).

Step 3: Express division as multiplication

Convert the division statement into a multiplication by the reciprocal. The expression \( \frac{x^{2}-16}{x-4} \div (3x + 12) \) becomes \( \frac{(x-4)(x+4)}{x-4} \times \frac{1}{3(x+4)} \).

Step 4: Simplify by canceling common factors

Cancel common factors across fractions. \( x-4 \) in the numerator and denominator cancel out, and \( x+4 \) also cancels out with the numerator and denominator, leaving you with \( \frac{1}{3} \).

Step 5: Write final simplified solution

The division simplifies to \( \frac{1}{3} \) after canceling all common factors.

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making complex expressions easier to understand and work with. This involves reducing the expression to its simplest form by identifying and eliminating any unnecessary factors or terms.

• Start by looking for common factors or terms that you can divide out.
• This might involve factoring polynomials (we'll cover this in detail shortly) or canceling out similar terms found both in numerators and denominators.

In our example, the expression initially presented complexity due to the interaction between the two polynomial terms. By converting the division into multiplication, it became easier to cancel terms like \(x-4\) and \(x+4\), eventually leading to a simplified result.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This technique is crucial when simplifying expressions, particularly when different terms need to be canceled out to simplify an expression.
To factor the given polynomials, follow these steps:

• Recognize the structure of the polynomial: A common strategy is identifying the difference of squares, like in \(x^2 - 16\). This can be rewritten as \((x-4)(x+4)\).
• Factor out common terms: For another polynomial, \(3x + 12\), factor out the greatest common factor (GCF), resulting in \3(x+4)\.

These factored forms were crucial to proceed with simplifying the expression, allowing the division process to transform into multiplication, setting the stage for cancellation.

Difference of Squares

The difference of squares is a specific pattern you often encounter when working with polynomials. It follows the structure \(a^2 - b^2 = (a-b)(a+b)\), representing a subtraction of two square terms.
To utilize this:

• Identify terms that are perfect squares: For instance, \(x^2\) and \(16\) are perfect square terms in the expression \(x^2 - 16\).
• Apply the difference of squares formula to factor \(x^2 - 16\) into \((x-4)(x+4)\), which makes the expression easier to manipulate and simplify in the division process.

This key algebraic technique not only simplifies the expression but also allows more straightforward execution of division and multiplication tasks in algebra.