Problem 19 - Algebra and Trigonometry

Question

\(7-20=\) Simplify the rational expression. $$ \frac{2 x^{3}-x^{2}-6 x}{2 x^{2}-7 x+6} $$

Step-by-Step Solution

Answer

\(\frac{x(2x + 3)}{2x - 3}\)

Step 1: Factorize the Numerator

The numerator of the expression is \(2x^3 - x^2 - 6x\). We first factor out the common factor, which is \(x\). This gives us \(x(2x^2 - x - 6)\). Next, we need to factor the quadratic \(2x^2 - x - 6\). To do this, we look for two numbers that multiply to \(-12\) (the product of \(2\) and \(-6\)) and add to \(-1\) (the middle coefficient). The numbers \(-4\) and \(3\) work. So we write \(2x^2 - x - 6 = (2x^2 - 4x) + (3x - 6)\) and factor by grouping: \(2x(x - 2) + 3(x - 2) = (2x + 3)(x - 2)\). Thus, the numerator becomes \(x(2x + 3)(x - 2)\).

Step 2: Factorize the Denominator

The denominator is \(2x^2 - 7x + 6\). We look for two numbers that multiply to \(12\) (product of \(2\) and \(6\)) and add to \(-7\). These numbers are \(-3\) and \(-4\). So, we rewrite \(2x^2 - 7x + 6 = (2x^2 - 4x) + (-3x + 6)\). Factor by grouping: \(2x(x - 2) - 3(x - 2) = (2x - 3)(x - 2)\). Thus, the denominator becomes \((2x - 3)(x - 2)\).

Step 3: Simplify the Rational Expression

Now, we have \(\frac{x(2x + 3)(x - 2)}{(2x - 3)(x - 2)}\). Notice that \((x - 2)\) is a common factor in both the numerator and denominator, so we can cancel it out. This simplification gives us \(\frac{x(2x + 3)}{2x - 3}\).

Key Concepts

Factorization

Factorization is a crucial step in simplifying rational expressions. It involves breaking down expressions into simpler, more manageable parts known as factors. With rational expressions, you'll often factor both the numerator and the denominator to identify and cancel common factors. The goal is to simplify the expression as much as possible without changing its value.

Let's consider a basic example: to factor the expression \(2x^3 - x^2 - 6x\), we first look for a common factor in all the terms. Here, \(x\) is present in each term, so it's factored out, resulting in \(x(2x^2 - x - 6)\). By factoring expressions like polynomials further, we can simplify them for easier manipulation or solving.

The process involves identifying pairs of terms and finding common factors. For quadratic expressions like \(2x^2 - x - 6\), finding two numbers that both multiply to the product of the leading coefficient and the constant term (in this case, \(-12\)) and add to the linear coefficient (\(-1\)) helps in breaking it down further. Identifying these numbers allows us to group and factorize effectively.

Numerators and Denominators

In rational expressions, understanding the roles of numerators and denominators is essential. These are the top and bottom portions of a fraction, respectively.

**Why Factor Both?**

• Factoring both the numerator and the denominator helps in simplifying the expression. It'll highlight any shared factors that can be cancelled out.
• A common factor between the two suggests that the expression can be simplified without changing its value, which is the ultimate goal when working with rational expressions.

For example, in the exercise solution, the numerator \(x(2x + 3)(x - 2)\) and the denominator \((2x - 3)(x - 2)\) both contain \((x - 2)\) as a factor. Recognizing such commonalities allows us to simplify the expression, leading to a more streamlined result.

Simplification Steps

Simplifying rational expressions involves several systematic steps, and patience is key. Here's a breakdown of the key actions required:

**Step-by-Step Process**

• Identify common factors in both the numerator and the denominator.
• Factorize these expressions completely to reveal any common factors.
• Cancel out common factors. This step is crucial as it reduces the expression to its simplest form.
• Always ensure that your final expression cannot be simplified further. Double-check by re-evaluating both the numerator and the denominator.

In our example, once you arrive at \(\frac{x(2x + 3)(x - 2)}{(2x - 3)(x - 2)}\), noticing that \((x - 2)\) appears in both positions allows us to cancel it. This process leaves the simplified expression as \(\frac{x(2x + 3)}{2x - 3}\).

By following these simplification steps meticulously, you ensure that the rational expression is reduced cleanly and accurately, making it much easier to work with or to insert into other mathematical contexts.