If (x + 3) is a factor of the polynomial \(x^{3} + 3x^{2} + nx - 12\), where n is a constant, find the value of n.
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Correct Answer: Option D
Explanation:
\(x + 3 = 0 \implies x = -3\)
Using remainder theorem, if x + 3 is a factor, f(-3) = 0.
\(f(-3) = (-3)^{3} + 3(-3)^{2} + n(-3) - 12 = 0\)
\(-27 + 27 - 3n - 12 = 0 \implies -3n = 12\)
\(n = -4\)
\(x + 3 = 0 \implies x = -3\)
Using remainder theorem, if x + 3 is a factor, f(-3) = 0.
\(f(-3) = (-3)^{3} + 3(-3)^{2} + n(-3) - 12 = 0\)
\(-27 + 27 - 3n - 12 = 0 \implies -3n = 12\)
\(n = -4\)