Certainly! Let’s break down the function [tex]\(\frac{x^2 - 1}{x^2 + x + 1}\)[/tex] step-by-step.
### Step 1: Simplify the Numerator
The numerator [tex]\(x^2 - 1\)[/tex] is a difference of squares, which can be factored as:
[tex]\[ x^2 - 1 = (x + 1)(x - 1). \][/tex]
### Step 2: Analyze the Denominator
The denominator [tex]\(x^2 + x + 1\)[/tex] does not factor nicely with real numbers. However, we need to keep it in the quadratic form.
### Step 3: Substitute Factored Form Into the Function
Now, substitute the factored numerator back into the original function:
[tex]\[ \frac{x^2 - 1}{x^2 + x + 1} = \frac{(x + 1)(x - 1)}{x^2 + x + 1}. \][/tex]
### Step 4: Simplify the Function if Possible
Check if any terms can be canceled out. In this case, there are no common factors between the numerator and the denominator that can be canceled.
### Conclusion
Thus, the simplified form of the function is:
[tex]\[ \frac{(x + 1)(x - 1)}{x^2 + x + 1} \][/tex]
There are no further simplifications possible, so the final form remains:
[tex]\[ \frac{x^2 - 1}{x^2 + x + 1}. \][/tex]
This is the simplest form of the given expression.
