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 the function and verify the real zeros and the given function value. n = 3; 3 and 5 i are zeros; f(2)= -29 f(x)= (Type an expression using x as the variable. Simplify your answer.)

Solution

degree = 3

zeros = 3 , 5i

complex zeros occur in pair , so if 5i is one zero other would be -5i

f(x) = a (x - x1 )(x-x2)(x-x3)

where, x1,x2,x3 are the zeros

plugging the values we get

f(x) = a ( x-3)(x-5i)(x+5i)

f(x) = a ( x^3 -3x^2 + 25x - 75 )

given f(2) = -29

-29 = a( 2^3 - 3(2)^2 + 25(2) - 75 )

-29 = a( -29 )

a = 1

hence, f(x) = x^3 -3x^2 + 25x - 75