For all real and positve numbers p and q, the following identity holds:
(p^1/3 - q^1/3)(p^2/3 +(p^1/3)(q^1/3) + q^2/3) = (p - q)
Does this identity still hold for negative p and/or q?
To determine if the given identity holds for negative values of p and/or q, we can substitute negative values into the equation and check if both sides are still equal.
Let's start by considering negative values for p and q separately.
1. Case: Negative p and positive q
Substituting p = -1 and q = 1 into the given identity:
((-1)^(1/3) - 1^(1/3)) * ((-1)^(2/3) + ((-1)^(1/3))(1^(1/3)) + 1^(2/3)) = (-1 - 1) = -2 ≠ 0
2. Case: Positive p and negative q
Substituting p = 1 and q = -1 into the given identity:
((1)^(1/3) - (-1)^(1/3)) * ((1)^(2/3) + ((1)^(1/3))((-1)^(1/3)) + (-1)^(2/3)) = (1 - (-1)) = 2 ≠ 0
3. Case: Negative p and negative q
Substituting p = -1 and q = -1 into the given identity:
((-1)^(1/3) - (-1)^(1/3)) * ((-1)^(2/3) + ((-1)^(1/3))((-1)^(1/3)) + (-1)^(2/3)) = (0) * (0) = 0
From the above calculations, we can conclude that the given identity only holds for negative p and q when p and q are equal. In all other cases, i.e., when p and q have different signs (either one positive and another negative), the identity does not hold.