PLEASE HELP ASAP!! Write a polynomial f(x) that satisfies the following conditions. Polynomial of lowest degree with zeros of -4 (multiplicity of 1), 2 (multiplicity of 3), and with f(0)=64

See below.

Step-by-step explanation:

So, we have the zeros -4 with a multiplicity of 1, zeros 2 with a multiplicity of 3, and f(0)=64.

Recall that if something is a zero, then the equation must contain (x - n), where n is that something. In other words, for a polynomial with a zero of -4 with a multiplicity of 1, then (x+4)^1 must be a factor.

Therefore, (x-2)^3 (multiplicity of 3) must also be a factor.

Lastly, f(0)=64 tells that when x=0, f(x)=64. Don't simply add 64 (like what I did, horribly wrong). Instead, to keep the zeros constant, we need to multiply like this:

In other words, we will have:

, where n is some value.

Let's determine n first. We know that f(0)=64, thus:

Now, let's expand:

Expand:

This is the simplest it can get.

correct answers below: