Is it possible to have a such polynomials?

Question

An exercise asks me to write an example of such polynomials, if they exist:

1. an irreducible polynomial of degree 5 in $\mathbb{R}[x]$.
2. a polynomial of degree 5 in $\mathbb{R}[x]$ that has no roots
3. a polynomial of degree 5 such that it is a product of five irreducible polynomials in $\mathbb{R}[x]$ and such that it has exactly one root in $\mathbb{R}$

My hypothetical answers:

1. It does not exist because a polynomial in $\mathbb{R}[x]$ is irreducible if and only if degree is 1 - OR - degree is 2 and discriminant is $< 0$
2. It does not exists because a polynomial of degree 5 can be divided in three polynomial of degree 2, 2, 1 where the one with degree 1 has always a root.
3. I don't really know.

Are my answers correct? If not, could you please correct them? Thank you.

Answer 1

1. This is correct.
2. It won't always be $2,2,1$. For example, it could be $2,1,1,1$ or $1,1,1,1,1$. What's important here is that the degree is odd, so there will always be a polynomial of degree $1$ and thus there will always be a root.
3. Your initial answer of a bunch of $x-1$ and $x+1$ multiplied together was almost correct. The thing was is that $x+1$ has a root of $-1$. Instead, you should've left out $x+1$ entirely and only used $x-1$, giving us $f(x)=(x-1)^5$.