A given polynomial equation of 5 degree has three equal roots . Given equation is $x^5-10a^3x^2+b^4x+c^5=0$ which has 3 equal roots. What I know is that since its a 5th degree equation therefore it must have 5 roots out (of which 3 are equal).
Aim is to establish the relationship between the constants $a,b$ and $c$.
Options given are:
1) $6a^5+c^5=0$.
2) $b^4=15a^4.$
I have to find which one is correct out of the two, any help?

Let $$f(x) = x^5 - 10a^3x^2 + b^4x + c^5 = (x - m)^3g(x)$$ where $m$ is the repeated root, and $g(x)$ is some second-order polynomial. 
Then, differentiating and substituting $x = m$,
$$5x^4 - 20a^3x + b^4 = (x - m)^3g'(x) + 3(x - m)^2g(x)$$
$$5m^4 - 20a^3m + b^4 = 0$$
Differentiating another time, and similarly substituting $x = m$,
$$20x^3 - 20a^3  = (x - m)^3g''(x) + 3(x - m)^2g'(x) + 3(x - m)^2g'(x) + 6(x - m)g(x)$$
$$20m^3 - 20a^3 = 0$$
$$m = a$$
Substituting this back into our first result,
$$5a^4 - 20a^3\cdot a + b^4 = 0$$
$$b^4 = 15a^4$$

Note: The fact that $(x - m)$ appears in every term of $f'(x)$ and $f''(x)$ is in fact a usable basis for proving what Gerry Myerson mentioned in the comments - that $f(x), f'(x), f''(x)$ all have a common factor.
