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

Question

Part I: Write the factors (in the form x - a) that are associated with the roots (a) given in the problem. (2 points)

Part II: Multiply the 2 factors with complex terms to produce a quadratic expression. (4 points)

Part III: Multiply the quadratic expression you just found by the 1 remaining factor to find the resulting cubic polynomial. (4 points)

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Answer 1

Here the roots of the polynomial function 3 , 3+i and 3-i

Part I: 

The factors of the polynomial function  (x - 3), [x-(3+i)] and [x - (3 - i)]

 

Now product of these factors gives the polynomial function

Part II:

[x-(3+i)] [x - (3 - i)] (x-3)

[x(x) - x(3-i) -x(3+i) + (3+i)(3-i)] (x-3)

[x² -3x + xi - 3x - xi + (3² - i²)]  (x-3)    [ (a+b)(a-b) = (a²-b³) ]

[x² -6x + 9 - (-1)] (x - 3)

(x² -6x + 10) (x - 3)

Part III:

(x² -6x +10) (x - 3)

x²(x) - x²(3) - 6x(x) + 6x(3) + 10(x) -10(3)

x³ -3x² - 6x² +18x +10x -30

x³ -9x² + 28x - 30

The polynomial function is   x³ -9x² + 28x - 30