Answer:
[tex]f(x) = {x}^{3} - 9 {x}^{2} + 27x - 27[/tex]
Step-by-step explanation:
We want to create a third degree polynomial function with one zero at three.
In other words, we want to find a polynomial function with roots x=3 , multiplicity, 3.
Since x=3 is a solution, x-3 is the only factor that repeats thrice.
[tex]f(x) = {(x - 3)}^{3} [/tex]
We expand to get:
[tex]f(x) = x( {x}^{2} - 6x + 9) - 3( {x}^{2} - 6x + 9)[/tex]
[tex]f(x) = {x}^{3} - 6 {x}^{2} + 9x- 3 {x}^{2} + 18x - 27[/tex]
This simplifies to:
[tex]f(x) = {x}^{3} - 9 {x}^{2} + 27x - 27[/tex]
See attachment for graph.
