Problem 14 - Algebra and Trigonometry

Question

\(7-20=\) Simplify the rational expression. $$ \frac{x^{2}-x-2}{x^{2}-1} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{x - 2}{x - 1} \), where \(x \neq \pm 1\).

Step 1: Identify the Problem

We are given the rational expression \( \frac{x^2 - x - 2}{x^2 - 1} \). Our task is to simplify this expression.

Step 2: Factor the Numerator

The numerator is \( x^2 - x - 2 \). We are looking for two numbers that multiply to \(-2\) (the constant term) and add to \(-1\) (the coefficient of \(x\)). These numbers are \(-2\) and \(1\). Thus, the numerator factors as \((x - 2)(x + 1)\).

Step 3: Factor the Denominator

The denominator is \( x^2 - 1 \), which is a difference of squares. We can factor it as \((x - 1)(x + 1)\).

Step 4: Simplify the Expression

The factored form of the rational expression is \( \frac{(x - 2)(x + 1)}{(x - 1)(x + 1)} \). Since \((x + 1)\) appears in both the numerator and the denominator, we can cancel it out, assuming \(x eq -1\). The simplified expression is \( \frac{x - 2}{x - 1} \).

Step 5: State the Domain Restrictions

The original expression has restrictions where the denominator cannot be zero. Setting \(x^2 - 1 = 0\) gives \(x = 1\) or \(x = -1\). Thus, \(x eq 1\) and \(x eq -1\) should be stated as domain restrictions.

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill that helps simplify expressions. To factor a polynomial, we break it down into simpler components, known as factors, that, when multiplied together, give the original polynomial. Here's how you can factor a simple trinomial like \(x^2 - x - 2\):

• Look for two numbers that multiply to give you the constant term, which is \(-2\) in this case.
• At the same time, these numbers must add up to the coefficient of the \(x\) term, which is \(-1\).

For the expression \(x^2 - x - 2\), the numbers \(-2\) and \(1\) fit both conditions:

• \(-2 \times 1 = -2\)
• \(-2 + 1 = -1\)

Thus, the expression can be factored into \((x - 2)(x + 1)\). Factoring helps in simplifying calculations by revealing common factors that can be canceled in rational expressions.

Difference of Squares

The difference of squares is a special pattern in polynomials where two squares are subtracted from each other, written as \(a^2 - b^2\). This can be factored using the formula:\[ a^2 - b^2 = (a + b)(a - b) \]In our exercise, the denominator \(x^2 - 1\) is a perfect example of the difference of squares:

• Here, \(a^2\) is \(x^2\) and \(b^2\) is \(1^2\).
• The factors are determined as \(a + b\) and \(a - b\), which results in \((x + 1)(x - 1)\).

Applying the difference of squares factorization simplifies the expression and allows us to see if any terms cancel between the numerator and the denominator, crucial for simplifying rational expressions further.

Domain Restrictions

Domain restrictions refer to the values that variables in an expression cannot take. They're essential because dividing by zero is undefined in mathematics. For rational expressions like \( \frac{x^2 - x - 2}{x^2 - 1} \), we look for values of \(x\) that make the denominator zero.Here's how to find these restrictions:

• Set the denominator \(x^2 - 1\) equal to zero: \(x^2 - 1 = 0\).
• Solving \(x^2 = 1\) gives \(x = 1\) and \(x = -1\).
• These are the values where the expression is undefined, so they are domain restrictions.

Thus, when simplifying or working with the expression, we must remember that \(xeq 1\) and \(xeq -1\) to avoid division by zero. Always state these restrictions to ensure clarity and correctness in your calculations.