If p(x) = kx^2 - kx - 12, for all values of x, what is the value of k?

Obs.: although the above version of the question stem is a bit "wordy", we believe it´s better (mathematically speaking).

\(p\left( x \right) = k{x^2} - kx - 12\,\,\,\,\,\left( * \right)\)

\(? = k\)

\(\left( 1 \right)\,\,\,0\,\, = \,\,p\left( 3 \right)\,\,\mathop = \limits^{\left( * \right)} \,\,k{\left( 3 \right)^2} - k\left( 3 \right) - 12\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,k\,\,{\rm{unique}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}.\)

\(\left( 2 \right)\,\,\,0 = p\left( { - 2} \right)\,\,\mathop = \limits^{\left( * \right)} \,\,k{\left( { - 2} \right)^2} - k\left( { - 2} \right) - 12\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,k\,\,{\rm{unique}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}{\rm{.}}\)


The correct answer is therefore (D).

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.