Let's break down the given expression step-by-step to determine if [tex]\(\frac{x^3 - 1}{x^2 - 1}\)[/tex] simplifies to [tex]\(x\)[/tex].
### Step 1: Factor the Numerator
The numerator is [tex]\(x^3 - 1\)[/tex]. This can be factored using the difference of cubes formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
Here, [tex]\(a = x\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ x^3 - 1 = (x - 1)(x^2 + x + 1) \][/tex]
So, the factored form of the numerator is:
[tex]\[ x^3 - 1 = (x - 1)(x^2 + x + 1) \][/tex]
### Step 2: Factor the Denominator
The denominator is [tex]\(x^2 - 1\)[/tex]. This can be factored using the difference of squares formula:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Here, [tex]\(a = x\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ x^2 - 1 = (x - 1)(x + 1) \][/tex]
So, the factored form of the denominator is:
[tex]\[ x^2 - 1 = (x - 1)(x + 1) \][/tex]
### Step 3: Simplify the Fraction
Now that we have factored both the numerator and the denominator, we can write the original expression as:
[tex]\[ \frac{x^3 - 1}{x^2 - 1} = \frac{(x - 1)(x^2 + x + 1)}{(x - 1)(x + 1)} \][/tex]
We can cancel the common factor of [tex]\((x - 1)\)[/tex] in the numerator and the denominator:
[tex]\[ \frac{(x - 1)(x^2 + x + 1)}{(x - 1)(x + 1)} = \frac{x^2 + x + 1}{x + 1} \][/tex]
### Step 4: Compare the Simplified Expression to [tex]\(x\)[/tex]
After canceling the common factor, the simplified expression is:
[tex]\[ \frac{x^2 + x + 1}{x + 1} \][/tex]
It is evident that [tex]\(\frac{x^2 + x + 1}{x + 1}\)[/tex] is not simply [tex]\(x\)[/tex]. Therefore, the expression [tex]\(\frac{x^3 - 1}{x^2 - 1}\)[/tex] does not simplify to [tex]\(x\)[/tex].
### Conclusion
Thus, the correct answer is:
No, the expression [tex]\(\frac{x^3 - 1}{x^2 - 1}\)[/tex] does not simplify to [tex]\(x\)[/tex].
