Discussion
If \(a^{3} - b^{3}\) = 91 and a - b = 1, what is the value of (ab) ?
Answer: D
\(ab = \) 30.
Use the identity for the difference of cubes:
\[ \begin{aligned} a^{3} - b^{3} &= (a - b)^{3} + 3ab(a - b) \\ 91 &= 1 + 3ab \\ ab &= 30 \end{aligned} \]
Check with \(a = 6\), \(b = 5\): \(216 - 125 = 91\) and \(6 - 5 = 1\), with \(ab = 30\). 27 and 9 come from slips in the identity.
Another route: from \(a - b = 1\), \((a - b)^{2} = 1\), so \(a^{2} + b^{2} = 1 + 2ab\); then \(a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2}) = 1 + 3ab = 91\), giving the same result.
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