Does the expression x^3-1/x^2-1 simplify to x?
No, because x3 – 1 can be factored as x(x2 – x + 1) and x2 – 1 can be factored as x(x – 1), so only x can be canceled.
Yes, because x3(x – 1) can be factored as x2(x – 1) and x2 – 1 can be factored as x(x – 1), so (x – 1) can be canceled.
No, because the –1 in the numerator and denominator is not a common factor and cannot be canceled.
Yes, because –1 in the numerator and denominator is a common factor and can be canceled.
The Correct Answer and Explanation is :
The expression (\frac{x^3 – 1}{x^2 – 1}) does not simplify to (x). Let’s analyze this step by step.
- Factoring the Expressions:
- The numerator (x^3 – 1) can be factored using the difference of cubes formula, which states (a^3 – b^3 = (a – b)(a^2 + ab + b^2)). Here, (a = x) and (b = 1):
[
x^3 – 1 = (x – 1)(x^2 + x + 1)
] - The denominator (x^2 – 1) can be factored using the difference of squares formula, which states (a^2 – b^2 = (a – b)(a + b)). Here, (a = x) and (b = 1):
[
x^2 – 1 = (x – 1)(x + 1)
]
- Rewriting the Expression:
Substituting the factored forms back into the original expression gives:
[
\frac{x^3 – 1}{x^2 – 1} = \frac{(x – 1)(x^2 + x + 1)}{(x – 1)(x + 1)}
] - Canceling Common Factors:
We can cancel the common factor (x – 1) from the numerator and denominator, assuming (x \neq 1):
[
\frac{x^2 + x + 1}{x + 1}
] - Final Simplification:
After canceling, the expression simplifies to:
[
\frac{x^2 + x + 1}{x + 1}
]
This does not simplify to (x). Instead, we can analyze this expression further, but it won’t yield (x) directly. - Conclusion:
Therefore, the statement that (\frac{x^3 – 1}{x^2 – 1} = x) is incorrect. The common factor (x – 1) can be canceled, but it does not lead to the conclusion that the entire expression simplifies to (x). Instead, it simplifies to (\frac{x^2 + x + 1}{x + 1}), which is a distinct rational expression.
In summary, the correct answer is No, the expression does not simplify to (x), as it simplifies to (\frac{x^2 + x + 1}{x + 1}).