Problem 5 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{x^{2}-16}{x-4} $$

Step-by-Step Solution

Answer

The expression simplifies to \( x + 4 \).

Step 1: Identify the Expression

The given expression is \( \frac{x^2 - 16}{x - 4} \). We need to simplify this by checking if the numerator can be factored.

Step 2: Factor the Numerator

The numerator \( x^2 - 16 \) is a difference of squares, which can be factored as \( (x - 4)(x + 4) \).

Step 3: Rewrite the Fraction

Replace the numerator with its factored form: \( \frac{(x - 4)(x + 4)}{x - 4} \).

Step 4: Cancel the Common Factor

Since \( x - 4 \) is a common factor in both the numerator and the denominator, we can cancel it out. This simplifies the fraction to \( x + 4 \).

Step 5: State the Simplified Expression

The simplified expression after canceling the common factor is \( x + 4 \).

Key Concepts

Difference of Squares

The difference of squares is a specific type of algebraic expression represented as \( a^2 - b^2 \), where \( a \) and \( b \) are any algebraic terms.
It is called a 'difference' because it involves subtraction, and 'of squares' because both terms are perfect squares. This expression factorizes neatly into \((a - b)(a + b)\).

• Identify perfect squares: Recognize the expressions like \( x^2 \) or numbers like 16, which can be rewritten as \( 4^2 \).
• Apply the formula: Transform \( x^2 - 16 \) into \((x - 4)(x + 4)\).

Understanding this helps you simplify expressions quicker, as it reduces them into products of factors. For instance, knowing that \( x^2 - 16 \) is a difference of squares is the key step in the given problem. It allows you to break down the numerator into \( (x - 4)(x + 4) \).
Identifying and applying difference of squares saves time and effort when tackling algebraic simplifications.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. The main goal is to simplify expressions and solve equations.
In the context of the exercise, we focused on the quadratic polynomial \( x^2 - 16 \). By recognizing it as a difference of squares, we were able to factor it efficiently into \((x - 4)(x + 4)\).

• Identify the type of polynomial: Recognize whether it can be expressed as a product of binomials you're familiar with.
• Factor systematically: Use techniques like grouping, trial and error, or special formulas such as the difference of squares.

Factoring is crucial in algebra, as it simplifies algebraic fractions and solves polynomial equations effectively.
For complex expressions, breaking them into simpler components makes handling them much easier, as demonstrated in factoring \( x^2 - 16 \).

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. Simplifying rational expressions involves reducing them to their simplest form.
In our exercise, we begin with \( \frac{x^2 - 16}{x - 4} \). After factoring the numerator through the difference of squares method, we rewrite it as \( \frac{(x - 4)(x + 4)}{x - 4} \).

• Simplify by canceling common factors: Identify common terms in the numerator and denominator, and cancel them out.
• Verify the domain: Ensure that you don't cancel factors that would change the domain, such as dividing by zero.

In this example, \( x - 4 \) was a common factor, allowing us to cancel it and simplify the expression to \( x + 4 \).
Understanding rational expressions helps in streamlining complex algebraic operations, making them more manageable and efficient.