Answer:
[tex]f(x) = {x}^{3} -12x - 16 [/tex]
Step-by-step explanation:
We want to for a polynomial function whose real zeros are -2 with multiplicity 2 and 4 with multiplicity 1.
If -2 is a zero of a polynomial, then by the factor theorem, x+2 is a factor.
Since -2 has multiplicity 2, (x+2)² is a factor.
Also 4 is a zero which means x-4 is a factor.
We write the polynomial in factored form as:
[tex]f(x) = {(x + 2)}^{2} (x - 4)[/tex]
We expand to get:
[tex]f(x) = ({x}^{2} + 4x + 4)(x - 4)[/tex]
We expand further to get:
[tex]f(x)=x({x}^{2} + 4x + 4) - 4({x}^{2} + 2x + 4)[/tex]
[tex]f(x) = {x}^{3} + 4 {x}^2 + 4x - 4 {x}^{2} + 8x - 16 [/tex]
[tex]f(x) = {x}^{3} - 12x - 16 [/tex]
