Which shows the expression x^2-1/x^2-x in simplest form
The correct Answer and Explanation is:
To simplify the expression (\frac{x^2 – 1}{x^2 – x}), we will start by factoring both the numerator and the denominator.
Step 1: Factor the Numerator and Denominator
- Numerator: (x^2 – 1) The expression (x^2 – 1) is a difference of squares, which can be factored using the formula (a^2 – b^2 = (a – b)(a + b)). Here, we can let (a = x) and (b = 1): [
x^2 – 1 = (x – 1)(x + 1)
] - Denominator: (x^2 – x) The expression (x^2 – x) can be factored by taking out the common factor (x): [
x^2 – x = x(x – 1)
]
Step 2: Rewrite the Expression
Now, substituting the factored forms back into the original expression gives us:
[
\frac{x^2 – 1}{x^2 – x} = \frac{(x – 1)(x + 1)}{x(x – 1)}
]
Step 3: Simplify the Expression
Next, we notice that ((x – 1)) appears in both the numerator and the denominator. Since we cannot divide by zero, we must specify that (x \neq 1) to avoid any undefined expressions. Therefore, we can cancel ((x – 1)):
[
\frac{(x – 1)(x + 1)}{x(x – 1)} = \frac{x + 1}{x} \quad \text{for } x \neq 1
]
Final Expression
The simplified expression is:
[
\frac{x + 1}{x} \quad \text{for } x \neq 1
]
Conclusion
Thus, the expression (\frac{x^2 – 1}{x^2 – x}) simplifies to (\frac{x + 1}{x}) as long as (x \neq 1). This method of factoring is essential in algebra as it allows us to simplify complex rational expressions, making them easier to work with in various mathematical contexts. Understanding these techniques is crucial for solving equations, simplifying expressions, and analyzing functions effectively.