Let's solve the given expression [tex]\( x^4 + 9x^2 + 81 \)[/tex].
We will walk through the steps to thoroughly understand the problem.
### Step 1: Identify the type of expression
The given expression [tex]\( x^4 + 9x^2 + 81 \)[/tex] is a polynomial expression with the highest power of [tex]\( x \)[/tex] being 4.
### Step 2: Analyze the terms of the polynomial
The polynomial [tex]\( x^4 + 9x^2 + 81 \)[/tex] consists of three terms:
- [tex]\( x^4 \)[/tex], which is the fourth power of [tex]\( x \)[/tex]
- [tex]\( 9x^2 \)[/tex], which is the coefficient 9 multiplied by the square of [tex]\( x \)[/tex]
- [tex]\( 81 \)[/tex], which is a constant term
### Step 3: Simplification and factorization
To determine if this polynomial can be factored or simplified further, we will look at potential ways to express the polynomial in a simpler form. However, a quick inspection shows this expression doesn't readily factor into simpler polynomial terms over the real numbers.
### Step 4: Conclusion
Hence, the polynomial [tex]\( x^4 + 9x^2 + 81 \)[/tex] is left in its simplest form as it cannot be factored or simplified further over the real numbers.
### Summary
The expression [tex]\( x^4 + 9x^2 + 81 \)[/tex] represents a quartic polynomial and is already in its simplest form. No further simplification or factorization can be achieved.
Thus, the final result of the given expression remains:
[tex]\[ x^4 + 9x^2 + 81 \][/tex]
