Certainly! Let's break down and understand the given rational function step-by-step.
The rational function we are given is:
[tex]\[
\frac{2x^2 - 3}{x^3 + 3x^2 + 3x + 1}
\][/tex]
### Step 1: Examine the numerator and the denominator
1. Numerator: [tex]\(2x^2 - 3\)[/tex]
- This is a quadratic polynomial, specifically a second-degree polynomial. It consists of two terms, [tex]\(2x^2\)[/tex] and [tex]\(-3\)[/tex].
2. Denominator: [tex]\(x^3 + 3x^2 + 3x + 1\)[/tex]
- This is a cubic polynomial, specifically a third-degree polynomial. It has four terms: [tex]\(x^3\)[/tex], [tex]\(3x^2\)[/tex], [tex]\(3x\)[/tex], and [tex]\(1\)[/tex].
### Step 2: Identifying properties of the rational function
- Domain: The domain of the rational function includes all real numbers except where the denominator is equal to zero, because division by zero is undefined. To find the points where the denominator is zero, we would solve:
[tex]\[
x^3 + 3x^2 + 3x + 1 = 0
\][/tex]
- Simplification: There may not be any further simplification possible unless we find common factors between the numerator and the denominator.
### Step 3: Factorize if possible (optional detail)
To completely understand the behavior of a rational function, sometimes we try to factorize the polynomials if possible.
However, it is not necessary for expressing the given function as is. For now, we will assume it is in its simplest form based on the expression provided:
### Final Expression
Hence, the given expression for the rational function we are working with is:
[tex]\[
\frac{2x^2 - 3}{x^3 + 3x^2 + 3x + 1}
\][/tex]
This concludes the detailed step-by-step understanding and writing of the rational function.
