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The problem asks for the remainder when dividing the polynomial \( x^{81} + x^{49} + x^{25} + x^9 + x \) by \( x^3 - x \).

To solve this, we will use **polynomial division**. However, note that the divisor \( x^3 - x \) can be factored as \( x(x^2 - 1) = x(x - 1)(x + 1) \). This factorization might help simplify the process, but we will proceed directly with polynomial division first to find the remainder.

The key is that the remainder when dividing by a polynomial of degree 3 will be a polynomial of degree less than 3, i.e., it will be of the form \( ax^2 + bx + c \).

### Process of Polynomial Division:
Let's divide \( f(x) = x^{81} + x^{49} + x^{25} + x^9 + x \) by \( g(x) = x^3 - x \). Since this is a large polynomial, a key observation is that each term involving high powers of \( x \) will reduce under the division by \( x^3 - x \).

Given the periodic behavior of powers of \( x \) modulo \( x^3 - x \), for example:
- \( x^3 \equiv x \) (mod \( x^3 - x \))
- \( x^4 \equiv x^2 \) (mod \( x^3 - x \))
- \( x^5 \equiv x \cdot x \equiv x \) (mod \( x^3 - x \))

Using this property, we can reduce each term in the original polynomial by its remainder when divided by \( x^3 - x \).

Let's reduce each term in the given polynomial:
- \( x^{81} \equiv x \) (mod \( x^3 - x \)) since \( 81 \mod 3 = 0 \)
- \( x^{49} \equiv x \) (mod \( x^3 - x \)) since \( 49 \mod 3 = 1 \)
- \( x^{25} \equiv x \) (mod \( x^3 - x \)) since \( 25 \mod 3 = 1 \)
- \( x^9 \equiv x \) (mod \( x^3 - x \)) since \( 9 \mod 3 = 0 \)
- \( x \equiv x \) (mod \( x^3 - x \))

### Adding the Reduced Terms:
Now, summing up the results of the reductions:
\[
x + x + x + x + x = 5x
\]

Thus, the remainder when dividing \( x^{81} + x^{49} + x^{25} + x^9 + x \) by \( x^3 - x \) is:

\[
\boxed{5x}
\]

Would you like further clarification on any step, or details on polynomial division techniques?

### Related Questions:
1. How can you apply the factorization of \( x^3 - x \) in solving this problem?
2. What are the rules for finding remainders of polynomials with large powers?
3. Can you explain how powers of \( x \) modulo \( x^3 - x \) behave periodically?
4. How do you determine the degree of the remainder when dividing by a polynomial?
5. Why does the remainder have to be a polynomial of degree less than 3?

**Tip**: When working with large powers of variables in polynomial division, look for periodic patterns or modular behavior to simplify the process.

Find the remainder when you divide x^81 + x^49 + x^25 + x^9 + x by x^3 - x.

This detailed guide explores how to find the remainder when dividing x^81 + x^49 + x^25 + x^9 + x by x^3 - x using polynomial division techniques and modular arithmetic. Learn how to reduce high powers of x mod x^3 - x for a faster solution.

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