Determine the polynomials have multiple roots

Question

Solve the below:

Q: Determine whether the polynomials have multiple roots.

Let F be a field and let f(x) =a_nxⁿ+a_n-1x^n-1+...+a0 ∈ F[x]. The derivative, D(f(x)), of f(x) is defined by Dx(f(x))=na_nx^n-1+(n-1)a_n-1x^n-2+....+a1 where, as usual, na=a+a+.....+a(n times). Note that D(f(x)) is again a polynomial with coefficients in F.

The polynomial f(x) is said to have a multiple root if there is some field E containing F and some a∈E such that (x - a)² divides f(x) in E[x}. Use the criterion to determine whether the following polynomials have multiple roots:

(a)x³-3x-2∈ Q[x]

(b)x³+3x+2∈ Q[x]

(c)x⁶-4x⁴+6x³+4x²-12x+9∈ Q[x]

(d) Show for any prime p and any a ∈ Fp that the polynomial x^p-a has a multiple root.