Problem 60 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{x^{2}-x-12}{8+2 x-x^{2}}$$

Step-by-Step Solution

Answer

The simplified form is \(-\frac{x+3}{x+2}\).

Step 1: Factor the Numerator

The numerator is \( x^2 - x - 12 \). To factor this, we look for two numbers that multiply to \(-12\) and add to \(-1\). These numbers are \( -4 \) and \( 3 \). Thus, we can factor the numerator as \((x - 4)(x + 3)\).

Step 2: Rearrange and Factor the Denominator

The denominator is \( 8 + 2x - x^2 \). Rearrange this as \(-x^2 + 2x + 8\). To make factoring easier, rewrite it as \(-1(x^2 - 2x - 8)\). We look for two numbers that multiply to \(-8\) and add to \(-2\). These numbers are \(-4\) and \(2\). Thus, the denominator factors as \(-1(x - 4)(x + 2)\).

Step 3: Simplify the Fraction

The fraction \( \frac{(x - 4)(x + 3)}{-1(x - 4)(x + 2)} \) shows a common factor of \( x - 4 \) in both the numerator and the denominator. Cancel this common factor, which simplifies the fraction to \( \frac{x + 3}{-(x + 2)} \).

Step 4: Simplify Further if Possible

The fraction \( \frac{x + 3}{-(x + 2)} \) can be rewritten by distributing the negative sign in the denominator: \( -(1) \times \frac{x+3}{x+2} = -\frac{x+3}{x+2} \). Therefore, the final simplified form is \( -\frac{x+3}{x+2} \).

Key Concepts

Factoring Polynomials

When it comes to simplifying algebraic fractions, the first important step is often factoring the polynomials present in the numerator and the denominator. Let's start with the numerator of the given fraction, which is \(x^2 - x - 12\). To factor this polynomial, we need to find two numbers that multiply to \(-12\) (the constant term) and add up to \(-1\) (the coefficient of \(x\)). In this case, those numbers are \(-4\) and \(3\). This is because \(-4 \times 3 = -12\) and \(-4 + 3 = -1\). Thus, the numerator factors to \((x - 4)(x + 3)\). Moving to the denominator, the expression is initially \(8 + 2x - x^2\). Rearranging this to \(-x^2 + 2x + 8\) helps to see it better for factoring. To make it easier, factor out \(-1\) first, which gives us \(-1 \times (x^2 - 2x - 8)\). Now, we need numbers that multiply to \(-8\) and add to \(-2\). The numbers \(-4\) and \(2\) work, because \(-4 \times 2 = -8\) and \(-4 + 2 = -2\). This means the denominator factors to \(-1(x - 4)(x + 2)\).
Factoring simplifies our expressions for the next steps!

Canceling Common Factors

After factoring both the numerator and the denominator, the next crucial step is to cancel any common factors. This is a key part of simplifying algebraic fractions. Let's look back at our factored fraction: \[ \frac{(x - 4)(x + 3)}{-1(x - 4)(x + 2)} \] Here, the factor \((x - 4)\) appears in both the numerator and the denominator. Since these terms are identical, they can be "canceled" out. Canceling involves removing this common factor from both parts of the fraction, as long as it is not zero (since division by zero is undefined).Once we cancel \((x - 4)\), our fraction becomes:\[ \frac{x + 3}{-(x + 2)} \] Now, we have a simplified expression that still represents the original problem, but in a much simpler form, making it easier to understand and work with in further calculations.

Simplification Steps

Simplifying an algebraic fraction often involves several structured steps, and understanding these steps makes the process straightforward and efficient. After canceling common factors, such as \((x - 4)\) in our example, we then look to present the fraction in its cleanest form.From the fraction \(\frac{x + 3}{-(x + 2)}\), observe that the negative sign can be distributed in various ways. One option is to factor it out of the entire fraction, resulting in:\[-1 \times \frac{x+3}{x+2} = -\frac{x+3}{x+2}\]This step of dealing with the negative sign is essential. It ensures that the fraction's structure follows the standard conventions, usually preferred in mathematical notation.
Finally, if there are no further like terms or common factors, you are left with the simplest possible version of the original expression: \(-\frac{x + 3}{x + 2}\).Remember, each step in the process is about making the expression easier to work with, and ultimately leading to a solution that provides clear insight into the relationships between the components of the fraction.