Problem 49 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{2 x^{2}-8}{4 x-8} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x + 2}{2}\).

Step 1: Factor Numerator

Start by factoring out the greatest common factor from the numerator. The expression in the numerator is \(2x^2 - 8\). Both terms have 2 as a common factor, so factor it out: \(2(x^2 - 4)\).

Step 2: Factor Denominator

Check if the denominator \(4x - 8\) can be factored. The common factor here is 4, so factor it out to get \(4(x - 2)\).

Step 3: Recognize a Difference of Squares

Notice that \(x^2 - 4\) in the numerator is a difference of squares, which can be factored further as \((x + 2)(x - 2)\).

Step 4: Substitute the Factors Back

Substitute the factors back into the expression: \(\frac{2(x + 2)(x - 2)}{4(x - 2)}\).

Step 5: Simplify the Expression

Since \(x - 2\) is common in both the numerator and the denominator, they cancel each other out. You are left with the expression \(\frac{2(x + 2)}{4}\).

Step 6: Simplify Further

Divide both the numerator and the denominator by their common factor, which is 2. This gives you \(\frac{x + 2}{2}\).

Key Concepts

Factoring Expressions

Factoring expressions is like finding the building blocks of a mathematical phrase. You break down a bigger term into simpler parts that multiply together to yield the original expression. In algebra, this step is crucial because it helps in simplifying complex expressions.

For instance, consider the expression given in the exercise, the numerator is originally \(2x^2 - 8\). By factoring it, which means finding terms that multiply to give the original expression, we extract the greatest common factor (GCF). In this particular case, 2 is the GCF. Upon factoring it out, we transform the expression to \(2(x^2 - 4)\).

This process makes complicated mathematical problems easier to manage and solve. By reducing the expression to simpler forms, it opens doors to recognize and apply further simplification techniques. Identifying common factors is usually the start of the factoring process.

Difference of Squares

A difference of squares refers to a specific algebraic form that allows for easy factoring. It's one of the special types of polynomials and appears as two squared numbers separated by a minus sign: \(a^2 - b^2\).

The interesting part is, this can be easily broken down into two binomials: \((a + b)(a - b)\).

In our exercise, the term \(x^2 - 4\) appears. It fits the form of a difference of squares since 4 is the square of 2. Thus, \(x^2 - 4\) is equivalent to \((x + 2)(x - 2)\). Recognizing this form is critical because it transforms an irreducible expression into simple factors that are manageable.

Applying this strategy helps simplify expressions quickly and is a powerful technique in algebra for easy transformation of complex expressions.

Canceling Common Factors

Canceling common factors is a straightforward and satisfying step in algebra simplification. Once you have an expression fully factored, you compare the numerator and the denominator to identify and cancel out any identical terms.

In the exercise, once the expression was factored, the part \(x - 2\) appeared in both the numerator and the denominator. These identical factors can "cancel" each other out because dividing anything by itself equals 1.

For example:

• \(\frac{\text{factor}}{\text{same factor}} = 1\)

By canceling common factors, you simplify the expression to a more reduced form. Always inspect the entire expression for any terms that appear identically above and below the division line to keep simplification intact.

Expression Simplification

Expression simplification is the art of making expressions easier to read, interpret, and solve by applying different algebraic techniques. The goal is to rewrite the expression in the simplest form possible without changing its value.

After factoring and canceling common factors, we are left with a simpler expression in the exercise. From \(\frac{2(x + 2)}{4}\), it further simplifies by dividing both the numerator and the denominator by their greatest common factor, which in this case is 2. This leads us to \(\frac{x + 2}{2}\).

Simplifying expressions helps reduce the number of terms, making them easier to handle and solve. It's a fundamental algebra skill that highlights the power of understanding structure and relationships within mathematical expressions.