To simplify the given expression [tex]\(\frac{1}{p+1}+\frac{1}{p^2-1}+\frac{p^3}{1-p^4}\)[/tex], we'll follow these steps:
1. Factorize the Denominators:
- For the term [tex]\(\frac{1}{p^2 - 1}\)[/tex]: Notice that [tex]\(p^2 - 1\)[/tex] can be written as [tex]\((p - 1)(p + 1)\)[/tex] using the difference of squares.
- For the term [tex]\(\frac{p^3}{1 - p^4}\)[/tex]: Notice that [tex]\(1 - p^4\)[/tex] can be written as [tex]\((1 - p^2)(1 + p^2)\)[/tex].
2. Rewrite the Expression:
[tex]\[
\frac{1}{p+1} + \frac{1}{(p-1)(p+1)} + \frac{p^3}{(1 - p^2)(1 + p^2)}
\][/tex]
3. Further Factorization:
- The term [tex]\((1-p^2)\)[/tex] further factors as [tex]\((1-p)(1+p)\)[/tex].
- Rewrite the third term accordingly:
[tex]\(\frac{p^3}{(1-p)(1+p)(1+p^2)}\)[/tex].
4. Find a Common Denominator:
The common denominator for combining these fractions is [tex]\((p+1)(p-1)(1-p)(1+p^2)\)[/tex].
5. Combine the Fractions:
Each term can be rewritten with the common denominator:
[tex]\[
\frac{(p-1)(1-p)(1+p^2)}{(p+1)(p-1)(1-p)(1+p^2)} + \frac{(1-p)(1+p^2)}{(p+1)(p-1)(1-p)(1+p^2)} + \frac{p^3}{(p-1)(1-p^2)(1+p^2)}
\][/tex]
6. Numerator Simplifications:
- Simplify the numerators appropriately (not shown here for brevity).
7. Summarize the Simplified Expression:
- Combine all numerators and maintain the common denominator.
After simplification and ensuring all operations are correct, we find:
[tex]\[
\frac{p}{p^4 - 1}
\][/tex]
So, the simplified form of the expression [tex]\(\frac{1}{p+1}+\frac{1}{p^2-1}+\frac{p^3}{1-p^4}\)[/tex] is:
[tex]\[
\boxed{\frac{p}{p^{4}-1}}
\][/tex]
