Final answer:
To factor the polynomial P(x) = x^3 - 3x^2 + 4x - 12 with P(3)=0, we use the factor x - 3. The other factors must be complex conjugates because the original polynomial has real coefficients. The correct factors are P(x) = (x - 3)(x + 2i)(x - 2i).
Step-by-step explanation:
The student asked to write the polynomial P(x) = x^3 - 3x^2 + 4x - 12 as three linear factors, given that P(3) = 0. Since we know P(3) = 0, x - 3 is a factor of P(x). To find the other two factors, long division or synthetic division can be used to divide P(x) by (x - 3), or alternatively, the fact that the polynomial must equal zero can be utilized to infer the remaining factors if the polynomial can be factored easily.
The remaining factors will have to multiply to give a quadratic equation because we started with a third-degree polynomial and factored out one linear term. Since the coefficients in the original polynomial are all real numbers, and since complex roots occur in conjugate pairs, if x = 3 is one root and two others are complex, they must be complex conjugates of each other. With that knowledge, we can expect the other factors to take the form (x + ai) and (x - ai) where i is the imaginary unit and a is some real number.
Comparing with the given options, the correct answer is option A: P(x) = (x - 3)(x + 2i)(x - 2i). The other options either have a wrong sign in the linear terms or form a different quadratic term when multiplied out.