Answer:

Answer in factored form:

Answer in standard form:

Step-by-step explanation:

I don't see your choices but I can still give you a polynomial fitting your criteria. I will give the answer in both factored form and standard form.

The following results are by factor theorem:

So if x=-2 is a zero then x+2 is a factor.

If x=7 is a zero then x-7 is a factor.

If x=5 is a zero then x-5 is a factor. It says we have this factor twice. I know this because it says with multiplicity 2.

So let's put this together. The factored form of the polynomial is

A(x+2)(x-7)(x-5)(x-5)

or

Now A can be any number satisfying a polynomial with zeros -2 and 7 with multiplicity 1, and 5 with multiplicity 5.

However, it does say we are looking for a polynomial function with leading coefficient 1 which means A=1.

Now the factored form is easy.

The standard form requires more work (multiplying to be exact).

I'm going to multiply (x+2)(x-7) using foil.

First: x(x)=x^2

Outer: x(-7)=-7x

Inner: 2(x)=2x

Last: 2(-7)=-14

--------------------Adding.

I'm going to multiply using formula .

.

So now we have to multiply these products.

That is we need to do:

I'm going to distribute every term in the first ( ) to

every term in the second ( ).

------------------------------------------ Distributing:

-------------------------------------------Adding like terms: