Problem 18 - Precalculus Mathematics for Calculus

Question

Simplify the rational expression. $$\frac{x^{2}-x-12}{x^{2}+5 x+6}$$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{x-4}{x+2}\) for \(x \neq -3\).

Step 1: Identify expression and find factors

The problem asks us to simplify the rational expression \(\frac{x^{2}-x-12}{x^{2}+5x+6}\). First, we will identify and then factor both the numerator \(x^{2}-x-12\) and the denominator \(x^{2}+5x+6\).

Step 2: Factor the numerator

The numerator \(x^{2} - x - 12\) can be factored by finding two numbers that multiply to \(-12\) and add up to \(-1\). These numbers are \(-4\) and \(3\). Therefore, the factorization is \((x - 4)(x + 3)\).

Step 3: Factor the denominator

The denominator \(x^{2} + 5x + 6\) can be factored by finding two numbers that multiply to \(6\) and add up to \(5\). These numbers are \(2\) and \(3\). Therefore, the factorization is \((x + 2)(x + 3)\).

Step 4: Cancel out common factors

After factorization, the expression becomes \(\frac{(x - 4)(x + 3)}{(x + 2)(x + 3)}\). The common factor \((x + 3)\) in both the numerator and the denominator can be cancelled out, leaving us with \(\frac{x - 4}{x + 2}\).

Step 5: Write the simplified expression

The simplified form of the rational expression is \(\frac{x - 4}{x + 2}\). Note that this simplification is valid as long as \(x eq -3\) to avoid division by zero from the original expression.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra that helps to simplify expressions. In this exercise, we need to break down both the numerator and denominator into simpler expressions or "factors."

To factor a quadratic expression like \(x^2 - x - 12\), we look for two numbers that multiply to the constant term, which is \(-12\), and add up to the linear coefficient, which is \(-1\).

• For \(x^2 - x - 12\), the numbers are \(-4\) and \(3\), leading to the factorization into \((x-4)(x+3)\).
• Similarly, to factor \(x^2 + 5x + 6\), we find two numbers that multiply to \(6\) and add up to \(5\). Those numbers are \(2\) and \(3\), giving us \((x+2)(x+3)\) as the factors.

Once the polynomials are factored into their simplest components, we can proceed to simplify the expression by cancelling common terms.

Simplification

Simplification involves reducing expressions to their simplest form. In the context of rational expressions, this often occurs after factoring both the numerator and the denominator.

Through simplification, you turn a complicated expression into something more manageable. After factoring both parts of the original rational expression, see if there are any common factors in the numerator and denominator that can be eliminated:

• The expression \(\frac{(x - 4)(x + 3)}{(x + 2)(x + 3)}\) has a common factor of \((x + 3)\).

Removing this common factor simplifies the expression significantly, allowing easier interpretation and further calculations. Always remember that simplification doesn't change the value of the expression as long as you are not equating \(x\) to any value that makes a denominator zero.

Common Factors Cancellation

The cancellation of common factors is a primary method used in simplifying rational expressions, helping to make the expression simpler and clearer.

When the same factor appears in both the numerator and the denominator of a fraction, you can "cancel" this factor. This is akin to dividing both terms by the same factor. However, ensure that this factors' cancellation does not lead to division by zero in the initial expression.

• In this example, \( (x + 3) \) appears both in the numerator and the denominator, so it can be cancelled, simplifying the expression.
• Cancelling \( (x + 3) \) leaves us with \( \frac{x - 4}{x + 2} \).

The result of this cancellation is a simplified expression that still holds true unless \(x\) results in a zero in the original denominator. It is crucial to note any restrictions, such as \(x eq -3\), which ensures the original fraction would not lead to division by zero.