Problem 16 - Algebra 1: Concepts and Skills

Question

Write the product in simplest form. $$\frac{3 x}{x^{2}-2 x-24} \cdot \frac{x-6}{6 x^{2}}$$

Step-by-Step Solution

Answer

The simplified form of the given product is \(\frac{1}{2(x+4)x}\).

Step 1: Factorize the polynomials

Factorize the polynomial expressions. The polynomial \(x^{2}-2 x-24\) can be factored into \((x-6)(x+4)\). The expression thus turns into: \(\frac{3 x}{(x-6)(x+4)} \cdot \frac{x-6}{6 x^{2}}\)

Step 2: Cross out common terms

Now we can see that \(x-6\) is common in the numerator and denominator. Also, the term \(x\) is common. Cross them out, yielding the following expression: \(\frac{3}{x+4} \cdot \frac{1}{6 x}\)

Step 3: Simplify the expression

Lastly, multiply the remaining terms in the numerator and the denominator, which results in a simplified expression:\(\frac{3}{6(x+4)x} = \frac{1}{2(x+4)x}\)

Key Concepts

Factoring Polynomials

Factoring polynomials is like breaking expressions down into parts that are easier to manage. Imagine polynomials are like a complex puzzle. Factoring helps us see the smaller pieces, which makes it easier to solve or simplify larger expressions. It's essential when working with rational expressions because it can help reveal common factors that can be canceled. Let's look at the polynomial from our example: \(x^2 - 2x - 24\). Here, the goal is to find two numbers that multiply to \(-24\) (the constant term) and add up to \(-2\) (the coefficient of \(x\)). These two numbers are \(-6\) and \(4\), which means we can rewrite the polynomial as \((x - 6)(x + 4)\).

• Check: \((-6) \times 4 = -24\) and \((-6) + 4 = -2\).
• This factorization unveils the structure of the expression.

Breaking polynomials down like this is a crucial step before simplifying rational expressions.

Multiplying Rational Expressions

Multiplying rational expressions involves working with fractions that contain polynomials. It's very much like multiplying numerical fractions. Consider two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\). When multiplied, they become \(\frac{a \, \cdot \, c}{b \, \cdot \, d}\). Rational expressions work the same way:

• You multiply the numerators together.
• You multiply the denominators together.

In our example, we started with:\(\frac{3x}{(x-6)(x+4)} \cdot \frac{x-6}{6x^2}\).Here, the numerators \(3x\) and \(x-6\) are multiplied, and the denominators \((x-6)(x+4)\) and \(6x^2\) are multiplied as well. This step sets the stage for the crucial operation of canceling common factors.

Canceling Common Factors

The final touch in simplifying rational expressions is canceling common factors. This step looks for factors that appear in both the numerator and denominator, similar to reducing fractions.From our example, the expression is \(\frac{3x}{(x-6)(x+4)} \cdot \frac{x-6}{6x^2}\). We can spot that both \(x - 6\) and \(x\) appear in the numerator and the denominator. By canceling them, we simplify the expression.

• Identify common terms: \(x-6\) and \(x\).
• Cancel out these common terms.

Now, we have \(\frac{3}{(x+4) \, \cdot \, 6x}\). Reducing doesn't just make the expression simpler; it also makes it easier to work with in future steps or problems. Always check for these opportunities when working with rational expressions.