Problem 61 - Intermediate Algebra

Question

Perform each division. \(\frac{5 a^{3}-10 a}{25 a^{3}}\)

Step-by-Step Solution

Answer

The division simplifies to \(\frac{a^2 - 2}{5a^2}\).

Step 1: Identify the Common Factors in the Numerator

First, look at the numerator, which is \(5a^3 - 10a\). We can factor out the greatest common factor (GCF), which is \(5a\). This simplifies the numerator to \(5a(a^2 - 2)\).

Step 2: Rewrite the Division Problem

Now rewrite the division expression with the factored numerator: \(\frac{5a(a^2 - 2)}{25a^3}\). This makes it easier to simplify.

Step 3: Simplify the Fraction

Look at \(\frac{5a(a^2 - 2)}{25a^3}\) and simplify it by canceling out common factors. Both the numerator and the denominator have a factor of \(5a\). Divide both by \(5a\) to get \(\frac{a^2 - 2}{5a^2}\).

Step 4: Final Simplification Check

Check if \(\frac{a^2 - 2}{5a^2}\) can be simplified further. There are no more common factors between the numerator \(a^2 - 2\) and the denominator \(5a^2\), so this is the simplest form.

Key Concepts

Understanding the Greatest Common Factor (GCF)

The Greatest Common Factor, or GCF, is a crucial concept in dealing with algebraic fractions and expressions. It refers to the largest factor that two or more numbers or terms share in common. Understanding how to find the GCF is important because it helps simplify expressions by eliminating shared factors.

Consider the algebraic expression in the numerator of our problem: \(5a^3 - 10a\). To identify the GCF, examine each term:

• Both terms have a numerical factor of 5.
• Both terms include the variable 'a'.

Therefore, the GCF is \(5a\). By factoring out \(5a\) from the expression, we get \(5a(a^2 - 2)\). This step is essential because it leads to the simplification of the fraction in the next processes.

Simplifying Complex Fractions

Once you've identified and factored out the GCF, the next step is to simplify the fraction itself. This process involves canceling out common factors in both the numerator and the denominator.

In our expression, once we factor the numerator, it becomes \(\frac{5a(a^2 - 2)}{25a^3}\). Next, we need to look for any common factors shared between both the numerator and the denominator.

• In this case, the common factor is \(5a\).

Dividing both numerator and denominator by \(5a\), the expression simplifies to \(\frac{a^2 - 2}{5a^2}\). Being able to identify and cancel these common factors is key to solving many algebraic fraction problems efficiently.

Factoring Polynomials for Simplification

Factoring polynomials is a strategic method to break down complex expressions into simpler ones. It has a wide range of applications, particularly in simplifying algebraic fractions.

In the context of the given expression \(5a^3 - 10a\), identifying potential common factors allows us to simplify the polynomial expression efficiently. Here’s how it works:

• Look for the GCF or any pattern that allows you to rewrite the polynomial.
• In our case, factoring out \(5a\) gives \(5a(a^2 - 2)\).

Factorization often reveals simpler expressions that are easier to work with, particularly when substituting into algebraic fractions. The factorization should always aim to simplify the structure of the polynomial, making it more workable for further operations such as simplifications or solving equations.