Question

Create a polynomial function in factored form that crosses through the x-axis at -2 and touches the x-axis and turns around at 4.​

Answer 1

Given:

A polynomial crosses through the x-axis at -2 and touches the x-axis and turns around at 4.​

To find:

The polynomial function in factored form.

Solution:

If the graph of a polynomial intersect the x-axis at $x=a$, then $(x-a)$ is a factor of the polynomial.

If the graph of a polynomial touches the x-axis at $x=b$, then $(x-b)$ is a factor of the polynomial with multiplicity 2. In other words $(x-b)^2$ is the factor of the polynomial.

It is given that the polynomial crosses through the x-axis at -2. So, $(x+2)$ is a factor of required polynomial.

It is given that the polynomial touches the x-axis and turns around at 4. So, $(x-4)^2$ is a factor of required polynomial.

Now, the required polynomial is:

$P(x)=(x+2)(x-4)^2$

Therefore, the required polynomial is $P(x)=(x+2)(x-4)^2$.