Write a polynomial with rational coefficients having roots 3,3+i and 3-i

Part 1: write the factors (in the form x-a) that are associated with the roots (a) given in the problem.

Part 2: Multiply the 2 factors with complex terms to produce a quadratic expression

Part 3: Multiply the quadratic expression you just found by the 1 remaing factor to find the resulting cubic polynomial

Part 1: If the roots of a polynomial are a, b, and c, then the factors can be written in the form (x - a), (x - b), and (x - c). In this case, we have (x - 3), (x - [3 + i]), and (x - [3 - i]). The three factors are (x - 3) (x - 3 - i) (x - 3 + i).

Part 2: The two factors with complex terms are (x - 3 - i) (x - 3 + i), and multiplying these terms as a difference of two squares can give:

(x - 3) ^2 - i^2 = x^2 - 6x + 9 - (-1) = x^2 - 6x + 10

Part 3: We now multiply (x^2 - 6x + 10) by the remaining factor of (x - 3). This results in the cubic expression:

(x^2 - 6x + 10) (x - 3)

= x^3 - 3x^2 - 6x^2 + 18x + 10x - 30

= x^3 - 9x^2 + 28x - 30