To analyze the polynomial [tex]\( x^4 + 9x^2 + 81 \)[/tex], let's break down the expression step by step.
1. Understand the Polynomial:
The given polynomial is [tex]\( x^4 + 9x^2 + 81 \)[/tex]. This is a fourth-degree polynomial as the highest degree of [tex]\( x \)[/tex] is 4.
2. Identify the Structure:
The polynomial can be written in terms of powers of [tex]\( x \)[/tex]. Specifically, we have terms involving [tex]\( x^4 \)[/tex], [tex]\( x^2 \)[/tex], and a constant term (81). Notably, the polynomial can be rewritten as:
[tex]\[
x^4 + 9x^2 + 81
\][/tex]
3. Attempt to Factorize:
Let's see if the polynomial can be factorized. A common technique is to look for patterns or use algebraic identities. Here, let's consider the expression as a quadratic in terms of [tex]\( y \)[/tex], where [tex]\( y = x^2 \)[/tex]:
[tex]\[
y^2 + 9y + 81
\][/tex]
Rewriting yields:
[tex]\[
(x^2)^2 + 9(x^2) + 81
\][/tex]
4. Check for Roots or Factoring Possibilities:
We need to see if there exist any easy-to-find factors or roots using standard methods. However, this polynomial does not factor easily into simpler polynomials with integer or rational coefficients.
5. Convert back to Original Variable:
Converting back to the original variable [tex]\( x \)[/tex], we reaffirm that the polynomial remains:
[tex]\[
x^4 + 9x^2 + 81
\][/tex]
6. Conclusion:
The polynomial given, [tex]\( x^4 + 9x^2 + 81 \)[/tex], stands in its simplest form for regular factorization and may require advanced techniques beyond typical high school algebra to factor further.
Thus, the polynomial [tex]\( x^4 + 9x^2 + 81 \)[/tex] is analyzed and remains in its given form.
