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.

Answer Details

If (x + 3) is a factor of the polynomial, then we can write the polynomial as: \[(x+3)(ax^2+bx+c)\] Expanding the above equation, we get: \[ax^3 + (b+3a)x^2 + (c+3b)x + 3c = x^3 + 3x^2 + nx - 12\] Equating the coefficients of corresponding powers of x, we get: \[a = 1, b + 3a = 3, c + 3b = n, 3c = -12\] From the last equation, we have: \[c = -4\] Using this value of c in the third equation, we have: \[n = c + 3b = -4 + 3b\] From the second equation, we have: \[b + 3a = 3\] \[b + 3(1) = 3\] \[b = 0\] Hence, we have: \[n = -4 + 3b = -4\] Therefore, the value of n is -4.