Problem 37 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \frac{2 x}{x^{2}-9}-\frac{4 x+2}{x^{2}-9}-\frac{4}{x^{2}-9} $$

Step-by-Step Solution

Answer

The simplified expression is \[-2\frac{(x+3)}{x^{2} - 9}\]

Step 1: Identify common denominator

Notice that all terms in the expression have the same denominator, \(x^{2}-9\). This means the fractions can be easily combined.

Step 2: Combine fractions

Combine fractions by subtracting or adding the numerators since the denominators are the same. So, the expression becomes \[\frac{2 x -(4 x+2)-4}{x^{2}-9}\].

Step 3: Simplify the numerator

Simplify the numerator of the fraction by doing the subtraction. It simplifies to \[\frac{2x - 4x - 2 - 4}{x^{2}-9} = \frac{-2x - 6}{x^{2}-9}\].

Step 4: Factor out common term

In the numerator, we can factor out -2, resulting in \[-2\frac{(x+3)}{x^{2} - 9}\].

Key Concepts

Common Denominator

Understanding the concept of a common denominator is fundamental in simplifying algebraic expressions involving fractions. It is the shared denominator among fractions that allows them to be combined more easily.

When all terms in an algebraic expression have the same denominator, we don't need to perform any additional steps like finding the least common denominator (LCD), which is essential when denominators differ. Instead, we can go straight to combining the numerators.

For example, in the exercise provided, the common denominator is \(x^{2}-9\). By recognizing this shared denominator across all terms, combining the numerators becomes a simple task. It is like combining apples with apples; the resulting numerator is the accumulation of all the changes (additions or subtractions) required, still over that same, unchanged common denominator.

Combine Like Terms

Once we have a common denominator in place, we can focus on the next critical step in simplifying algebraic expressions, which is to combine like terms. Like terms are terms that have the same variable raised to the same power.

For instance, if you have \(2x\) and \(4x\), these terms can be combined because they both contain the variable \(x\) to the first power. To simplify, you would subtract or add the coefficients (the numbers in front of the variables) and keep the variable part unchanged.

In our exercise, the expression in the numerator after subtraction is \(2x - 4x - 2 - 4\). There are two terms with \(x\), which are combined to \(2x - 4x\), resulting in \( - 2x\). The constants \( - 2\) and \( - 4\) are also combined, leading to \( - 6\). Combining like terms is a straightforward process once you identify the terms with common variables and focus on just altering their coefficients.

Factoring Algebraic Expressions

Factoring is a powerful tool in algebra that simplifies expressions and solves equations. When factoring algebraic expressions, we look for a common factor in the terms that we can pull out, simplifying the expression further.

In the numerator of our example, after combining like terms, we are left with the expression \( - 2x - 6\). We notice that both terms have a common factor of \( - 2\). Factoring out the \( - 2\) gives us \( - 2(x + 3)\), which is a more simplified form of the expression.

Factoring is like reverse distribution. Instead of applying a number to each term inside a parenthesis, we are doing the opposite—finding a number that can be taken out of each term to leave us with a more condensed version of our expression inside the parentheses. This technique is not only useful in simplifying expressions but also plays a crucial role in solving quadratic equations and finding zeros of a function.