Problem 26 - College Algebra and Calculus: An Applied Approach

Question

Write the rational expression in simplest form.\(\frac{x^{2}-16}{4-x}\)

Step-by-Step Solution

Answer

The simplified form of the expression is \(- (x+4)\)

Step 1: Factorize the numerator

The numerator \(x^{2}-16\) is in the form \(a^2-b^2\), which represents difference of squares. The formula for the difference of squares \(a^{2} - b^{2}\) is \((a-b)(a+b)\). Here \(a = x\) and \(b = 4\), so, the factorized form is \((x-4)(x+4)\). \ Therefore, the expression becomes \(\frac{(x-4)(x+4)}{4-x}\)

Step 2: Rewrite the denominator

The denominator \(4-x\) is same as \(-(x-4)\) by the associative property of addition. \ Therefore, the expression becomes \(\frac{(x-4)(x+4)}{-(x-4)}\)

Step 3: Cancel out common factors

We can cancel out the common factor \((x-4)\) in the numerator and denominator. \ Now, the simplified form is \(\frac{x+4}{-1}\) or \(- (x+4)\).

Key Concepts

Difference of Squares

Sometimes, you'll come across a quadratic expression that can be written as a difference of two perfect squares. The classic formula to keep in mind is \(a^2 - b^2 = (a-b)(a+b)\). This is known as the difference of squares. It's a powerful tool in algebra that helps simplify expressions quickly.

In the expression \(x^2 - 16\), we can see that it matches the form \(a^2 - b^2\). Here, \(a = x\) and \(b = 4\), because \(4^2 = 16\). Knowing this, we apply the formula to factorize the expression: \((x-4)(x+4)\).

Applying the difference of squares can make complex expressions easier to handle, and is often the first step in many algebraic simplifications. Look for this pattern whenever you see a difference between two terms that appear to be squared.

Factoring

Factoring is a crucial algebraic technique used to simplify expressions or solve equations. It's the process of breaking down an expression into products of simpler expressions.

When you factor an expression, you're essentially dividing it into components that are multiplied together to give the original expression. For example, in our step- by-step solution, we factorized \(x^2 - 16\) into \((x-4)(x+4)\). This shows us the underlying structure, revealing how each piece works together.

Recognizing different types of factoring shortcuts, like difference of squares, is a part of building algebraic fluency. Master this skill to make tackling larger problems easier. Practice breaking down a variety of expressions to become comfortable with "seeing" these hidden factors.

Simplifying Algebraic Fractions

The goal of simplifying algebraic fractions is to make them as clean and simple as possible. This often involves canceling out common factors between the numerator and the denominator.

Let's look at the expression \(\frac{(x-4)(x+4)}{4-x}\). By rewriting the denominator \(4-x\) as \(-(x-4)\), you can see immediately there is a common factor.

Once identified, the next step is to cancel those common terms, in this case, \((x-4)\). Careful here: what remains isn't just the uncancelled part, but anything that might have been affected by signs or addition/subtraction rules earlier. Thus, the simplified form after canceling is \(- (x+4)\).

Getting comfortable with these processes will make your algebra life much easier and problems more approachable, since you're stripping them of unnecessary complexity.