OOPS I got the zeros wrong in the last post...please show work.

Find a polynomial of the specified degree that has the given zeros

Degree 4: zeros -2,0,2,4

(x+2)(x)(x-2)(x-4) = 0 in factored form.

That can be multiplied out to give:

x*(x^2 -4)(x-4) = 0
[x^3 -4x^2 -4x +16]x = 0
x^4 -4x^3 -4x^2 +16x = 0

To find a polynomial of degree 4 with the given zeros (-2, 0, 2, 4), we can use the fact that a polynomial with zeros a, b, c, and d can be written in factored form as (x - a)(x - b)(x - c)(x - d).

Using this formula, we can write the polynomial as:

P(x) = (x - (-2))(x - 0)(x - 2)(x - 4)

Simplifying this equation gives us:

P(x) = (x + 2)(x - 0)(x - 2)(x - 4)

Now, we can expand this expression to find the polynomial:

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

Multiplying the factors:

P(x) = (x^2 + 2x)(x^2 - 2x - 4x + 8)

Simplifying:

P(x) = (x^2 + 2x)(x^2 - 6x + 8)

P(x) = x^4 - 4x^3 - 4x^2 + 16x

Therefore, a polynomial of degree 4 with zeros -2, 0, 2, and 4 is P(x) = x^4 - 4x^3 - 4x^2 + 16x.

To find a polynomial that has the given zeros, you can use the fact that the product of the factors corresponding to the zeros gives the polynomial.

In this case, the given zeros are -2, 0, 2, and 4. So, the factors corresponding to these zeros would be (x+2), x, (x-2), and (x-4), respectively.

To find the polynomial of degree 4, we need to multiply these factors together.

(x+2) * x * (x-2) * (x-4) = (x^2 + 2x)(x^2-4x-8) = (x^4 - 2x^3 - 4x^2 + 8x^2 - 16x - 8x + 16)

Simplifying the above expression, we get:

x^4 - 2x^3 + 4x^2 + 16x - 8x - 16

Therefore, a polynomial of degree 4 with the given zeros -2, 0, 2, and 4 is:

f(x) = x^4 - 2x^3 + 4x^2 + 16x - 16