Final answer:
The value of alpha3 - beta3 for the zeros of the quadratic polynomial x2 - 2 is 0, as alpha and beta are opposites and cancel out when applying the difference of cubes formula.
Step-by-step explanation:
If alpha and beta are the zeros of the quadratic polynomial x2 - 2, we are asked to find the value of alpha3 - beta3. First, let us find the roots using the quadratic formula, which for a polynomial ax2 + bx + c = 0, is given by x = (-b ± √(b2 - 4ac)) / (2a). In this case, a = 1 and c = -2, while b = 0 since the term with x is missing.
This yields the roots alpha and beta as √2 and -√2. We then substitute these values into alpha3 - beta3 and simplify using the difference of cubes formula, which is a3 - b3 = (a - b)(a2 + ab + b2). Substituting alpha for a and beta for b, we get (alpha - beta)((alpha)2 + (alpha)(beta) + (beta)2).
From the equation, we can deduce that alpha + beta = -1. Now, we can use this information to find alpha^3 - beta^3.
Using the identity for the difference of cubes, alpha^3 - beta^3 = (alpha - beta)(alpha^2 + alpha * beta + beta^2).
Substituting the values we have, we get alpha^3 - beta^3 = (-1)(alpha^2 + alpha * beta + beta^2).
Simplifying further, we find the value of alpha3 - beta3 equals 0, because alpha and beta are opposites and cancel each other out.