Question

Write the equation of the graph shown below in factored form.

f(x) = (x − 2)2(x + 1)(x − 3)

f(x) = (x + 2)2(x − 1)(x + 3)

f(x) = (x + 2)2(x + 1)(x + 3)

f(x) = (x − 2)2(x − 1)(x − 3)

Answer 1

3x

Explanation:

Answer 2

The correct answer is

`f(x)=(x-2)^(2) (x+1)(x-3)`

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:
`f(x)=a_(n) x^(n) +a_(n-1) x^(n-1) + ......... + a_(2)x^(2) +a_(1) x^(1) +a_(0) x^{0`

Factorized function:
`f(x)=a_(n) *(x-x_(1) )(x-x_(2) ).........(x-x_(n-1))(x-x_(n))`

where
`x_(1) ,x_(2) ,x_(n-1) ,x_(n)` 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:
`(x-x_(1) )^(k)`

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

`f(x)=(x-2)^(2) (x+1)(x-3)`