Respuesta :
ANSWER
f(x) = x³ - 4x² + 25x - 100
EXPLANATION
The degree of the polynomial is 3, so there are 3 factors,
[tex]f(x)=(x-x_1)(x-x_2)(x-x_3)_{}[/tex]Two of the zeros are 4 and 5i. The second given zero is complex, so its complex conjugate must be a zero too. Therefore, the third zero is -5i,
[tex]f(x)=(x-4)(x-5i)(x+5i)[/tex]Let's verify that f(1) is -78,
[tex]f(1)=(1-4)(1-5i)(1+5i)[/tex]The product of a complex number and its complex conjugate is the sum of the squares of the real and imaginary parts because it has the form of a difference of two squares,
[tex]\begin{gathered} f(1)=(-3)(1^2-(5i)^2) \\ f(1)=(-3)(1-(-25)) \\ f(1)=(-3)(1+25) \end{gathered}[/tex]Solve the product,
[tex]f(1)=-3\cdot26=-78[/tex]Thus, the zero we found is correct. Now we have to write the function in standard form, by multiplying the factors,
[tex]\begin{gathered} f(x)=(x-4)(x-5i)(x+5i)=f(x)=(x-4)(x^2-5ix+5ix-5i\cdot5i) \\ f(x)=(x-4)(x^2+25) \end{gathered}[/tex]Multiply these two factors,
[tex]f(x)=x^3-4x^2+25x-100[/tex]Hence, the function that satisfies the given conditions is f(x) = x³ - 4x² + 25x - 100
