Question:
If
\[ a-b=-8 \] and \[ ab=-12, \] then \[ a^3-b^3= \]
(a) -244
(b) -240
(c) -224
(d) -260
Solution:
Using identity:
\[ a^3-b^3 = (a-b)(a^2+ab+b^2) \]
Now,
\[ (a-b)^2 = a^2+b^2-2ab \]
Substituting the given values:
\[ (-8)^2 = a^2+b^2-2(-12) \]
\[ 64 = a^2+b^2+24 \]
\[ a^2+b^2 = 40 \]
Therefore,
\[ a^2+ab+b^2 = 40-12 = 28 \]
Now,
\[ a^3-b^3 = (-8)(28) \]
\[ =-224 \]
Hence, the correct answer is:
\[ \boxed{-224} \]