Question

Create a 3rd degree polynomial function with one zero at three. Sketch a graph

Answer 1

Answer:

$f(x) = {x}^{3} - 9 {x}^{2} + 27x - 27$

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.

$f(x) = {(x - 3)}^{3}$

We expand to get:

$f(x) = x( {x}^{2} - 6x + 9) - 3( {x}^{2} - 6x + 9)$

$f(x) = {x}^{3} - 6 {x}^{2} + 9x- 3 {x}^{2} + 18x - 27$

This simplifies to:

$f(x) = {x}^{3} - 9 {x}^{2} + 27x - 27$

See attachment for graph.