Final answer:
The third degree polynomial is first found from the given roots. A scalar is then found using the given point on the polynomial. The expanded form of the polynomial is -x³+2x²-9x+18.
Step-by-step explanation:
This question pertains to the domain of polynomials, specifically
third degree polynomials. Given the roots x=2 and x=3i, we need to first understand that complex roots always come in conjugate pairs. Therefore, if x=3i is a root, its conjugate x=-3i is also a root. Hence, the polynomial factorized form becomes (x-2)(x-3i)(x+3i). Expanding this out, we get x³-2x²+9x-18 as the unreduced form of the polynomial. However, the problem also states that this polynomial passes through the point (3,-9), so substituting x=3 in the polynomial should result in y=-9. If it doesn’t, it indicates that we have to multiply the polynomial with a scalar. Assume the scalar to be 'a' in which case -9 =a(27-18*3+9*3-18). Solving this gives a = -1. Hence the polynomial in expanded form is -x³+2x²-9x+18.
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