What is obtained by taking out the common factor from \(27a^3-9a^2\)?

Correct answer: \(9a^2(3a-1)\)

The direct answer is D: \(9a^2(3a-1)\). To factor an expression, first find the greatest common factor of every term. The numerical coefficients 27 and 9 have greatest common factor 9. The common variable part is \(a^2\), because the smaller power among \(a^3\) and \(a^2\) is \(a^2\). Thus the greatest common factor is \(9a^2\). Dividing each term by it gives \(27a^3\div9a^2=3a\) and \(-9a^2\div9a^2=-1\). Therefore the expression is \(9a^2(3a-1)\). A is not the greatest factor and does not reproduce the original expression. B gives \(27a^3-27a^2\), so its second coefficient is wrong. C simplifies to \(27a^3-9a^3\), also incorrect. D expands to \(27a^3-9a^2\). Memory cue: take the numerical HCF and the lowest common power of each variable.