Write an equation to a polynomial function that has the following properties:

Fourth degree equation
Lead coefficient is -2
Two negative real roots and one positive real root
The positive real root has multiplicity of 2

(x-2)(x-2) (-2x-1)(x+1) = 0

roots
2
2
-1/2
-1

To construct a polynomial function with the given properties, we can start by considering the equation in factored form. Since the polynomial has two negative real roots, we know that the factors of the equation will be of the form (x - a)(x - b)(x - c)(x - d), where a, b, c, and d are the roots.

We also know that one of the roots has a multiplicity of 2, which means we will have a repeated factor. Let's assume that the positive real root is represented by (x - e) and its multiplicity is 2.

Therefore, the equation in factored form will be:

f(x) = -2(x - a)(x - b)(x - c)(x - e)^2

Now let's determine the values of a, b, c, and e.

Since we want two negative real roots, we will consider a and b as negative numbers. Let's assume a = -p and b = -q, where p and q are positive real numbers.

Since the positive real root has a multiplicity of 2, we will consider e as a positive number. Let's assume e = r, where r is a positive real number.

Therefore, the final equation in factored form is:

f(x) = -2(x + p)(x + q)(x + c)(x - r)^2

This equation satisfies the given properties of a fourth-degree polynomial with a lead coefficient of -2, two negative real roots, and one positive real root with a multiplicity of 2.

To create a polynomial function with the given properties, we can start by considering the roots and their multiplicities.

From the given information, we know that there are two negative real roots. Let's call them "a" and "b." Since they are both negative, the polynomial factors will include (x - a) and (x - b).

We also know there is one positive real root with a multiplicity of 2. Let's call it "c." Because it has a multiplicity of 2, the corresponding factor will be (x - c)^2.

Now, if we consider the lead coefficient, which is -2, the equation can be written as:

f(x) = -2(x - a)(x - b)(x - c)^2

So, the equation of the polynomial function that satisfies the given properties is:

f(x) = -2(x - a)(x - b)(x - c)^2