The function of imaginary roots of the equation $(x-1)(x-2)(3x+1)=32$ is

Expand the given equation to form the polynomial x^3 - 8x^2 + 6x + 42 = 0. Because the coefficients are all real, any imaginary roots must occur in conjugate pairs, meaning the number of imaginary roots must be an even number. By testing simple integer values, you find that x = 3 is a root since 27 - 72 + 18 + 42 = 15. Dividing the cubic polynomial by (x - 3) leaves the quadratic x^2 - 5x - 14 = 0, which factors into (x - 7)(x + 2) = 0 to yield the real roots 7 and -2. Since all three roots are real, the number of imaginary roots is 0.

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