Simplifying the Expression (p^3  6) / (p  1)
The expression (p^3  6) / (p  1) represents a rational function, meaning it's a fraction where both the numerator and denominator are polynomials. To simplify this expression, we can use polynomial long division.
Polynomial Long Division

Set up the division: Write the numerator (p^3  6) as the dividend and the denominator (p  1) as the divisor.
_________ p  1  p^3  6

Divide the leading terms: The leading term of the dividend (p^3) is divided by the leading term of the divisor (p). This gives us p^2.
p^2 ______ p  1  p^3  6 p^3  p^2 

Subtract and bring down the next term: Subtract the result (p^3  p^2) from the dividend. Bring down the next term (6).
p^2 ______ p  1  p^3  6 p^3  p^2  p^2  6

Repeat the process: Divide the leading term of the new dividend (p^2) by the leading term of the divisor (p). This gives us p.
p^2 + p ______ p  1  p^3  6 p^3  p^2  p^2  6 p^2  p   p  6

Continue until the degree of the remainder is less than the degree of the divisor: Repeat the process, dividing p by p, and bring down 6.
p^2 + p  1 ______ p  1  p^3  6 p^3  p^2  p^2  6 p^2  p   p  6  p + 1  7
Result
The quotient is p^2 + p  1, and the remainder is 7. Therefore, we can express the original expression as:
(p^3  6) / (p  1) = p^2 + p  1  7/(p  1)
This simplified form is equivalent to the original expression for all values of p except for p = 1 (where the denominator would be zero, making the expression undefined).