Discussion
What is the H.C.F. of \(2x^{2} - 3x - 9\) and \(2x^{3} - 54\)?
Answer: B
The H.C.F. is \(x - 3\).
Factorise both expressions:
\[ \begin{aligned} 2x^2 - 3x - 9 &= (2x + 3)(x - 3) \\ 2x^3 - 54 &= 2(x^3 - 27) \\ &= 2(x - 3)(x^2 + 3x + 9) \end{aligned} \]
The only common factor is \(x - 3\). \(2x + 3\) divides only the first expression, and \(x + 3\) and \(2x - 3\) divide neither.
Check by substitution: at \(x = 3\) the first expression is \(18 - 9 - 9 = 0\) and the second is \(54 - 54 = 0\), so both vanish and share the factor. At \(x = -3/2\) the second expression is \(-27/4 - 54\), not zero, so \(2x + 3\) is not common.
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