Problem 36 - Elementary Algebra

Question

Divide. (Assume all denominators are nonzero.) $$ 24 a 2 b 3(a-2 b) \div 12 a b(a-2 b) 5 $$

Step-by-Step Solution

Answer

\( \frac{6a}{5} \)

Step 1: Understand the Expression

The given expression to be divided is: \( \frac{24a^2 b 3(a-2b)}{12ab(a-2b)5} \). We need to simplify this expression by dividing the numerator by the denominator.

Step 2: Rewrite the Expression

Rewrite the expression: \( \frac{24a^2 b \cdot 3(a-2b)}{12ab \cdot (a-2b) \cdot 5} \). This will help us identify common factors easily.

Step 3: Cancel Common Factors

First, notice the \((a-2b)\) term in both the numerator and the denominator, so we can cancel it out. Then, divide both the numerator and the denominator by \(a\): \[\frac{24a^2 b \cdot 3}{12ab \cdot 5}\] becomes \[\frac{24a b \cdot 3}{12b \cdot 5}. \] This simplifies further by canceling \(b\).

Step 4: Simplify After Canceling

Now simplify: \( \frac{24a \cdot 3}{12 \cdot 5} \). Cancel out any further common factors: 24 and 12 have a greatest common divisor of 12, and 3 and 5 remain unchanged: \[\frac{2a \cdot 3}{5}. \]

Step 5: Final Simplification

After simplifying, the expression becomes \( \frac{6a}{5} \). Thus, the result of the division operation is this simplified form.

Key Concepts

Simplification Techniques

When tackling algebraic expressions, simplification makes them easier to work with. It involves reducing the expression to its simplest form by canceling out terms. In our exercise, simplification played a crucial role in streamlining the division of two polynomials.

• **Canceling Terms:** Look for terms that appear both in the numerator and denominator, like \((a-2b)\) found here. Canceling these terms reduces complexity.
• **Reducing Fractions:** Divide both numerator and denominator by their common factors, just like with normal numbers. Here, we divided by 12 and by \(b\).

This step-by-step canceling and reduction make complex expressions more manageable and easier to solve.

Factoring in Algebra

Factoring involves expressing a polynomial as a product of its factors. This technique helps simplify expressions and solve equations. It’s essential when you need to simplify or divide polynomials.
In our given solution, recognizing \((a-2b)\) as a common factor allowed us to cancel it out.
Factors can also help simplify expressions by breaking them down into smaller, more manageable pieces.

• **Identify Common Factors:** Always begin by spotting any repeating terms or expressions in the polynomials.
• **Use Factoring to Simplify:** Finding common factors, like \((a-2b)\), and removing them simplifies the division process and aids in finding the simplest form of your expression.

Factoring is like breaking down a big puzzle into smaller pieces that are easier to see and work with.

Division of Polynomials

Dividing polynomials is similar to dividing numbers, but with variables and terms. This process typically involves simplifying expressions first. Understanding how to handle each part of the expression is key.
In our problem, after canceling common terms and factors, we were left with a simpler expression: \(\frac{24a \cdot 3}{12 \cdot 5}\). This form was easier to manage and allowed straightforward division.

• **Handling Variables and Coefficients:** Treat each part of the polynomial separately, dividing coefficients and reducing powers of variables as needed.
• **Cancel Before Dividing:** Aim to simplify as much as possible by canceling first, which reduces the complexity of your division task.

Think of polynomial division as a systematic approach: simplify, divide, and verify. It turns a potentially daunting task into a structured process.