Problem 53 - Algebra A Combined Function

Question

Perform the indicated operations. $$ \frac{-2 x}{x^{3}-8 x}+\frac{3 x}{x^{3}-8 x} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{x^2 - 8}\).

Step 1: Identify Common Denominator

Observe that both fractions have the same denominator: \(x^3 - 8x\). Since the denominators are the same, we can proceed to add the numerators directly.

Step 2: Add the Numerators

Add the numerators of the fractions. So, \(-2x + 3x\) which simplifies to \(x\).

Step 3: Write the Resulting Fraction

Combine the result from Step 2 with the common denominator. This gives us the fraction \( \frac{x}{x^3 - 8x} \).

Step 4: Factor the Denominator

Notice that the denominator \(x^3 - 8x\) can be factored. Factor by taking out a common \(x\), resulting in \(x(x^2 - 8)\).

Step 5: Simplify the Fraction

Since \(x\) appears in both the numerator and as a factor in the denominator, we can simplify by cancelling \(x\) (assuming \(x eq 0\)). The simplified fraction is \(\frac{1}{x^2 - 8}\).

Key Concepts

Factoring Expressions

Factoring expressions is a crucial skill in algebra that simplifies complex problems. The goal is to write an expression as the product of its factors, which can reveal potential simplifications.
For example, in the expression \(x^3 - 8x\), observe that both terms share a common factor of \(x\). By factoring \(x\) out, the expression can be rewritten as \(x(x^2 - 8)\). This reveals the inner structure of the expression.
Recognizing patterns is key to effective factoring:

• Common Factors: Identify any number or variable that divides all terms in the expression evenly.
• Difference of Squares: Expressions like \(a^2 - b^2\) factor to \((a + b)(a - b)\).
• Trinomials: Try to express in the form \((x + a)(x + b)\), especially when it's a quadratic.

Continually practicing factoring can make recognizing these patterns intuitive and straightforward.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, ensuring they are as easy to work with as possible.
This often requires factoring the numerator and the denominator to cancel common factors. In the step involving \(\frac{x}{x(x^2 - 8)}\), the variable \(x\) appears in both places. Canceling common factors, like \(x\) (so long as \(x eq 0\)), greatly simplifies the fraction to \(\frac{1}{x^2 - 8}\).
Here's a quick guide to simplify fractions:

• Factor Completely: Always factor both the numerator and the denominator completely to identify all common factors.
• Cancel Common Factors: Divide both the numerator and the denominator by their greatest common factor (GCF) to simplify.
• Check Restrictions: Be mindful of values that make the denominator zero, as they are not included in the solution set.

By mastering these steps, you ensure fractions are as concise, tidy, and correct as they need to be.

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions with different denominators. It allows you to combine fractions, much like combining like terms in an algebraic expression.
In the original problem, both fractions, \(\frac{-2x}{x^3 - 8x}\) and \(\frac{3x}{x^3 - 8x}\), already share the same denominator \(x^3 - 8x\), making it unnecessary to find a new common denominator.
When fractions have different denominators, follow these steps:

• Identify the Least Common Denominator (LCD): The smallest expression that each denominator divides into evenly.
• Adjust the Fractions: Multiply the numerator and denominator of each fraction by whatever is needed to bring them to the common denominator.
• Combine the Fractions: Once adjusted, you can add or subtract the numerators, keeping the common denominator.

Using these steps ensures accurate and straightforward addition or subtraction of fractions, helping to keep your work clear.