Find an​ nth-degree polynomial function with real coefficients satisfying the given conditions.

Question

n - 4; -1,4, and 2+3i are zeros; f(1)= -180

what is f(x)?

Answer 1

 Complex zeroes always come in pairs, so the complement to 2+3i is 2-3i.

f(x)=a(x-2-3i)(x-2+3i)(x+1)(x-4) where a is a constant.

f(x)=a(x²-4x+13)(x²-3x-4)=

a(x⁴-3x³ -4x²

       -4x³+12x²+16x

               +13x²-39x-52)=

a(x⁴-7x³+21x²-23x-52).

f(1)=a(1-7+21-23-52)=-180.

f(1)=-60a=-180, so a=3 and:

f(x)=3x⁴-21x³+63x²-69x-156.

Answer 2

n=4;

-1,3, and 4+4i are zeros;

f(1)=-100

Answer 3

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