Determine the Equation

Question

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. If n=3, 4 and 2i are zeros, and f(1)=-30

Answer 1

Since we know that n = 3, then there are 3 roots. One is 4, one is 2i and the third is -2i according to conjugate roots.

So the factors are ( x - 4 ) (x - 2i ) and ( x + 2i)

Putting these together gives:

f (x) = a ( x - 4 ) (x - 2i ) ( x + 2i) Multiplying the binomials.

f (x) = a (x - 4) (x² + 4) Since f (1) = - 30 then

-30 = a (1 - 4) (1² + 4)

-30 = a ( -3) ( 5)

-30 = -15 a

a = 2

So f (x) = 2 (x - 4) (x² + 4) = 2 ( x³ - 4x² + 4x - 16) = 2x³ - 8x² + 8x - 32

Answer: f (x) = 2x³ - 8x² + 8x - 32