Certainly! Let's break down and examine the given mathematical expression step-by-step.
The expression provided is:
[tex]\[
\frac{1}{1+x^{11}+x^4} + \frac{1}{1+x^{16}+x^n} + \frac{1}{1+x+x^6}
\][/tex]
### 1. First Term: [tex]\(\frac{1}{1 + x^{11} + x^4}\)[/tex]
- This term represents a fraction where the numerator is 1.
- The denominator is composed of the constant [tex]\(1\)[/tex], added to [tex]\(x\)[/tex] raised to the power of [tex]\(11\)[/tex] and [tex]\(x\)[/tex] raised to the power of [tex]\(4\)[/tex] respectively.
- This term essentially shows how we handle a polynomial in the denominator with specific powers of [tex]\(x\)[/tex].
### 2. Second Term: [tex]\(\frac{1}{1 + x^{16} + x^n}\)[/tex]
- Similarly to the first term, the numerator is [tex]\(1\)[/tex].
- The denominator includes the constant [tex]\(1\)[/tex], added to [tex]\(x\)[/tex] raised to the power of [tex]\(16\)[/tex] and [tex]\(x\)[/tex] raised to the power of [tex]\(n\)[/tex], where [tex]\(n\)[/tex] can be any integer.
### 3. Third Term: [tex]\(\frac{1}{1 + x + x^6}\)[/tex]
- Again, the numerator here is [tex]\(1\)[/tex].
- The denominator consists of the constant [tex]\(1\)[/tex], added to [tex]\(x\)[/tex] and [tex]\(x\)[/tex] raised to the power of [tex]\(6\)[/tex].
### Combining Everything:
The final expression is a sum of these three fractions where each fraction is of the form [tex]\(\frac{1}{\text{some polynomial in x}}\)[/tex]. Without specific values for [tex]\(x\)[/tex] and [tex]\(n\)[/tex], we cannot further simplify these terms into numerical values. So, the expression in its simplest form is represented as follows:
[tex]\[
\frac{1}{1 + x^{11} + x^4} + \frac{1}{1 + x^{16} + x^n} + \frac{1}{1 + x + x^6}
\][/tex]
In summary:
1. The first term, [tex]\(\frac{1}{1 + x^{11} + x^4}\)[/tex], deals with [tex]\(x^{11}\)[/tex] and [tex]\(x^4\)[/tex] in the denominator.
2. The second term, [tex]\(\frac{1}{1 + x^{16} + x^n}\)[/tex], involves [tex]\(x^{16}\)[/tex] and [tex]\(x^n\)[/tex].
3. The third term, [tex]\(\frac{1}{1 + x + x^6}\)[/tex], includes [tex]\(x\)[/tex] and [tex]\(x^6\)[/tex].
Thus, when you sum these fractions together, the resultant expression yields the following result:
[tex]\[
\left( \frac{1}{1 + x^{11} + x^4}, \frac{1}{1 + x^{16} + x^n}, \frac{1}{1 + x + x^6} \right)
\][/tex]
This remains the final expression, reflective of the input variable values.
