Simplifying the Expression: (x^4  1)/(x1)
This expression involves a polynomial division, where the numerator is a fourthdegree polynomial and the denominator is a linear polynomial. We can simplify this expression using various methods.
Method 1: Polynomial Long Division
This method involves dividing the numerator by the denominator using long division, similar to the way we divide numbers. Here's how it works:

Set up the division:
_______ x1  x^4 + 0x^3 + 0x^2 + 0x  1

Divide the leading terms: x^4 / x = x^3 Write x^3 above the line.

Multiply the quotient (x^3) by the divisor (x1): x^3 * (x1) = x^4  x^3 Write this result below the dividend.

Subtract:
_______ x1  x^4 + 0x^3 + 0x^2 + 0x  1 x^4  x^3  x^3 + 0x^2

Bring down the next term:
_______ x1  x^4 + 0x^3 + 0x^2 + 0x  1 x^4  x^3  x^3 + 0x^2 + 0x

Repeat steps 25: x^3 / x = x^2 x^2 * (x1) = x^3  x^2
_______ x1  x^4 + 0x^3 + 0x^2 + 0x  1 x^4  x^3  x^3 + 0x^2 + 0x x^3  x^2  x^2 + 0x

Continue the process until the degree of the remainder is less than the degree of the divisor:
_______ x1  x^4 + 0x^3 + 0x^2 + 0x  1 x^4  x^3  x^3 + 0x^2 + 0x x^3  x^2  x^2 + 0x  1 x^2  x  x  1 x  1  0

The simplified expression is: (x^4  1)/(x1) = x^3 + x^2 + x + 1
Method 2: Factoring
We can also simplify the expression by factoring both the numerator and the denominator.

Factor the numerator: x^4  1 is a difference of squares: x^4  1 = (x^2 + 1)(x^2  1) Furthermore, (x^2  1) is another difference of squares: (x^2 + 1)(x^2  1) = (x^2 + 1)(x + 1)(x  1)

Cancel the common factor: (x^4  1)/(x1) = [(x^2 + 1)(x + 1)(x  1)]/(x1) = (x^2 + 1)(x + 1)
Conclusion
Both methods lead to the same simplified expression. However, factoring is generally faster and more efficient for simpler expressions like this one. The simplified expression (x^2 + 1)(x + 1) is equivalent to x^3 + x^2 + x + 1, but it might be more convenient to leave it in factored form depending on the context.