Prove that, if \(\alpha\) is a repeated root of \(f(x)=0\), where \(f(x)\) is a polynomial, then \(\alpha\) is a root of the equation \(f'(x)=0\).

If \(f\) has a repeated root at \(\alpha\), how can we write \(f\)? Can we now differentiate?

Given that the equation \(4x^4+x^2+3x+1=0\) has a repeated root, find its value.

Can we apply the first part of the question, together with the Factor Theorem?