Problem 52 - 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 the Common Denominator

Both fractions have the same denominator: \(x^3 - 8x\). This simplifies the process, as we don't need to adjust for any differences in the denominators.

Step 2: Combine the Numerators

Since the denominators are the same, we can simply combine the numerators: \(-2x + 3x = x\). Thus, the expression simplifies to: \(\frac{x}{x^3 - 8x}\).

Step 3: Factor the Denominator

To simplify the expression further, factor the common denominator \(x^3 - 8x\). This can be factored as: \(x(x^2 - 8)\).

Step 4: Simplify the Fraction

Now, rewrite the expression using the factored form of the denominator: \(\frac{x}{x(x^2 - 8)}\). Cancel the common factor of \(x\) from the numerator and the denominator to get \(\frac{1}{x^2 - 8}\).

Key Concepts

Common Denominator

Whenever you're dealing with rational expressions, finding a common denominator is crucial. A common denominator is the same bottom part, or denominator, for two or more fractions. When the fractions you're working with already have the same denominator, you're in luck! This simplifies the operations immensely because you can directly combine the numerators.

Here's a quick reminder on why this matters:

• It allows you to add or subtract fractions with ease.
• The operation becomes simpler, as there’s no need to adjust or change the denominators.

For our original exercise, both fractions share the denominator \(x^3 - 8x\). Therefore, in this scenario, we could immediately focus on combining the numerators, skipping any additional work to modify the denominators.

Factoring Polynomials

Factoring is like tearing down a wall into bricks. It involves breaking down a complex expression into simpler, multipliable components. This makes it easier to work with polynomials, especially when you're trying to simplify rational expressions.

In our problem, the denominator \(x^3 - 8x\) needed simplification. A common technique is factoring out the greatest common factor first. Here, you notice that \(x\) is common in both terms, so you pull out \(x\) as a factor:

• \(x^3 - 8x = x(x^2 - 8)\)

By factoring, you've transformed a tricky polynomial into a more manageable component product. This makes it much easier for you to work through and simplify the expression further.

Remember, factoring is useful not just for simplifying, but also for solving equations and finding roots of polynomials. It's a versatile tool in algebra!

Simplifying Fractions

Once you've dealt with the components, simplifying the fraction is the next step. This means reducing the fraction to its simplest form by canceling out common factors in the numerator and the denominator. Simplification is all about making expressions as straightforward as possible.

In the original exercise, after factoring the denominator, you ended up with the expression \(\frac{x}{x(x^2 - 8)}\). Here, both the numerator and the denominator contain \(x\) as a factor:

• Cancel out the common \(x\) to simplify the expression to \(\frac{1}{x^2 - 8}\).

This reduces the fraction significantly, simplifying any further operations with it. Simplifying fractions keeps expressions neat and is crucial for solving algebraic problems efficiently.