Find an nthdegree polynomial function with real coefficients

Find an nth-degree polynomial function with real coefficients satisfying the given conditions n = 4: i and 4 i are zeros: f(- 1) = 68 f(x) = (Type an expression using x as the variable. Simplify your answer.)

Solution

n=4 , f(-1) = 68

Given zeroes of polynomial are i and 4i

All four zeroes will be i , -i , 4i , -4i

So, f(x) will be a factor of (x-i)(x+i)(x-4i)(x+4i)

Now , f(x) = c(x+i)(x-i)(x+4i)(x-4i) , where c is constant

f(x) = c(x²+1)(x²+16)

f(-1) = c(2)(17) = 68

=> c=68/34 = 2

Therefore , f(x) = 2(x²+1)(x²+16)

Or f(x) = 2(x⁴+17x²+16) = 2x⁴ + 34x² +32