Answer:
The correct answer is
[tex]f(x)=(x-2)^{2} (x+1)(x-3)[/tex]
Step-by-step explanation:
Every polynomial function can be factored based on its roots, then expressed in factored form.
The roots of a polynomial are the values of the variable for which the polynomial function takes the value of zero. The roots of a polynomial can be real or complex. In the graph of the polynomial function, the real roots are identified as the intersections with the x-axis (those values in which the function is zero). A polynomial function of degree "n" will have at most n roots.
Then
Polynomial function: [tex]f(x)=a_{n} x^{n} +a_{n-1} x^{n-1} + ......... + a_{2}x^{2} +a_{1} x^{1} +a_{0} x^{0[/tex]
Factorized function: [tex]f(x)=a_{n} *(x-x_{1} )(x-x_{2} ).........(x-x_{n-1})(x-x_{n})[/tex]
where [tex]x_{1} ,x_{2} ,x_{n-1} ,x_{n}[/tex] are the roots of the polynomial function.
On the other hand, the number of times that the root repeats itself is called the multiplicity order of a root. If the order of multiplicity of the root is PAR, the graph of the function touches the
x axis but does not traverse it, REBOUNDS. If the order of multiplicity of the root is ODD, the graph of the function crosses the x axis, SHORT.
The multiplicity is expressed as an exponent in each root, as shown in the example below, where k represents the multiplicity: [tex](x-x_{1} )^{k}[/tex]
In this case, when looking at the graph, you can see that there are 3 roots in: -1, 2 and 3.
As you can also see in the graph, at -1 and 3 the root crosses the "x" axis. This indicates that the multiplicity is odd. On the other hand, in 2 the root bounces (does not cross the x axis), indicating that the multiplicity is par.
Then, the correct answer is
[tex]f(x)=(x-2)^{2} (x+1)(x-3)[/tex]
