Find an nthdegree polynomial function with real coefficients

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph she function and verify the real zeros and the given function value, n = 3; -2 and 2 + 5 i are zeros; f(1) = 78 f(x) = (Type an expression using x as the variable, Simplify your answer.)

Solution

degree , n = 3

zeros x = -2 , 5+3i so another zero would be 5 -3i

There is only one real zero

f(x) = k(x+2)( x - 5 -3i)(x- 5+3i)

= k (x +2) (x^2 +3ix -5x +25 -15i -3ix +15i +9) = k (x +2)(x^2 -5x + 34)

f(1) = 78

78 = k(3)( 1 - 5 +34) = 3k(30)

k = 78/90 = 26/30 = 13/10 = 1.3

f(x) = 1.3(x +2)(x^2 -5x + 34)