Problem 20 - Algebra 1: Concepts and Skills

Question

Write the product in simplest form. $$\frac{3 x^{2}-6 x}{2 x+1} \cdot \frac{4 x+2}{x-2}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(6x\).

Step 1: Factorize the numerators and denominators

To simplify the expression, first both the numerators and denominators should be factorized. The numerators can be factored as \(3x(x - 2)\) and \(2(2x + 1)\), while the denominators are already in simplest form (apart from \(2x + 1\) which is also a factor). The expression becomes \(\frac{3x(x - 2)}{2x + 1} \cdot \frac{2(2x + 1)}{x - 2}\).

Step 2: Simplify the expression by cancelling out common factors

Now, cancel out common factors occurring in both the numerator and the denominator across the whole expression. Here, \(x - 2\) and \(2x + 1\) are common factors in the numerator and denominator and can hence be cancelled out. This results in the expression: \(3x * 2 = 6x\).

Step 3: Write the final answer

After performing all these simplifications, we are left with \(6x\) as the final product.

Key Concepts

Factoring Polynomials

In algebra, factoring polynomials is a fundamental process. It involves breaking down a complex polynomial into simpler factors that, when multiplied together, produce the original polynomial. Let's see how this can be crucial in simplifying rational expressions.

Polynomials can often contain terms that are common and can be extracted as factors. For example, in the expression \(3x^2 - 6x\), both terms share a common factor of \(3x\). Hence, it can be rewritten as \(3x(x - 2)\). Identifying these common factors is key to reducing complexity and simplifying expressions.

Factoring involves looking for these common terms:

• Identify common numerical coefficients, like finding 3 in both 3 and 6.
• Look for common variables, such as \(x\) in \(x^2\) and \(x\).

By breaking down polynomials into factors, you transform a seemingly complex expression into a set of simpler components. This makes it easier to cancel terms later, especially when dealing with rational expressions in algebraic fractions.

Simplifying Expressions

Simplifying expressions in algebra is all about reducing them to their most basic form. This means removing any unnecessary parts by cancelling out common factors.

Let's consider our given exercise, where we had the expression: \( \frac{3x(x - 2)}{2x + 1} \cdot \frac{2(2x + 1)}{x - 2} \). The goal is to cancel out identical components in the numerators and denominators across the fractions, like \(x-2\) and \(2x+1\).

• Identify and cancel common factors between the numerators and denominators.
• Ensure that the remaining terms are multiplied, as seen with \(3x\) and \(2\) becoming \(6x\).

This cleanup leads to a much more manageable and understandable expression. Simplifying not only makes calculations easier, but it also makes it simpler to understand the behavior and value of the expression.

Algebraic Fractions

Algebraic fractions function similarly to numerical fractions, but they include variables. These fractions can seem intimidating at first due to their mixture of numbers and symbols. However, simplifying them follows similar rules to simplifying numerical fractions.

In our example, we encountered fractions like \[ \frac{3x^2-6x}{2x+1} \text{ and } \frac{4x+2}{x-2} \]. Here’s the approach:

• Factorize the numerators and denominators to reveal any common factors.
• Cancel out these common factors to simplify the fraction.
• Multiply the remaining elements to find the simplified product, much like multiplying simple fractions.

Understanding algebraic fractions is important, as they often appear in various algebraic equations and functions. By practicing these techniques, students can demystify algebraic fractions and become comfortable working with them in any context.