Problem 51 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(-x-2\).

Step 1: Identify the Expression to Factor

The expression in the numerator is \(4 - x^2\). Recognize that it can be rewritten in a form that allows factoring as a difference of squares.

Step 2: Factor the Difference of Squares

Recall that \(a^2 - b^2 = (a-b)(a+b)\). In this case, \(4 - x^2 = (2^2) - (x^2)\) can be factored as \((2-x)(2+x)\).

Step 3: Simplify the Expression

The expression \(\frac{4-x^2}{x-2}\) becomes \(\frac{(2-x)(2+x)}{x-2}\). Recognize that \(2-x = -(x-2)\), so the expression simplifies to \(\frac{-(x-2)(2+x)}{x-2}\).

Step 4: Cancel Out Common Terms

The terms \((x-2)\) in the numerator and denominator cancel each other, leaving \(-(2+x)\).

Step 5: Simplify the Expression Further

Distribute the negative sign in \(-(2+x)\) to get \(-2-x\). The expression is now fully simplified.

Key Concepts

Difference of Squares

When you come across an expression like \(4 - x^2\), you might notice it is a special type of polynomial known as a "difference of squares." The difference of squares is an expression of the form \(a^2 - b^2\). Recognizing these can help you simplify expressions easily. Why is this useful? Because it has a neat factorization: \(a^2 - b^2 = (a-b)(a+b)\).

In the original exercise, \(4 - x^2\) becomes \((2^2) - (x^2)\). Here, \(a\) is 2, and \(b\) is \(x\). You factor this as \((2-x)(2+x)\). This step shows how powerful recognizing the difference of squares can be, simplifying a complex expression into a product of two simpler binomials.

So, remember, whenever you see a subtraction between two squared terms, check if it's a difference of squares. Factor it out to see if it helps reduce complexity!

Simplification of Expressions

Once you've factored the difference of squares, the next step is simplification. This involves manipulating the expression to reduce it to a simpler form. In our example, after factoring, the expression became \(\frac{(2-x)(2+x)}{x-2}\).

Simplifying often means looking for opportunities to cancel out terms in both the numerator and denominator. In this case, notice that \(2-x\) is related to \(x-2\) by a negative factor. Specifically, \(2-x = -(x-2)\).

This knowledge allows you to rewrite the fraction as \(\frac{-(x-2)(2+x)}{x-2}\), and then cancel out the \((x-2)\) terms. This leaves a much cleaner \(-(2+x)\). Simplifying expressions this way is a handy algebra skill, making complex problems manageable.

It's like cleaning up a room—removing unnecessary items to focus on what's essential. Keep practicing simplification, and over time, you'll get faster and more efficient at it!

Polynomial Division

Polynomial division often involves breaking down complex expressions into simpler parts, very much like dividing numbers. However, unlike basic numerical division, you'll often be working with variables, which adds a layer of complexity.

Consider the exercise: you started with \(\frac{4-x^2}{x-2}\). After factoring and simplifying, we saw how to make use of polynomial division by effectively canceling terms. Although this example simplifies without direct long division, understanding that division allows reduction and simplification is crucial.

In more complex cases, dividing polynomials might require long division or synthetic division techniques. These methods help manage expressions that do not simplify directly. The goal of polynomial division, especially when factoring, is to give you a simpler, clear pathway through algebra problems.

Think of polynomial division as a toolkit helping you take apart expressions, step by step. Keep practicing, and you'll soon find it intuitive!