B.2. An enthusiastic theoretical physics student is working on a numerical algorithm for finding roots to higher order polynomials.
(i) Show that if $(x - \alpha)^2$ is a factor of the polynomial $p(x)$, then $p'(\alpha) = 0$. [2]
(ii) Show a corresponding result if $(x - a)^4$ is a factor of $p(x)$. [4]
(iii) Hence, given that the polynomial $$x^6-7x^5+15x^4-40x^2+\kappa x+\lambda$$ has a factor of the form $(x - \alpha)^4$, find $\kappa$ and $\lambda$. [14]
Hint: Find $\alpha$.
The question, as mentioned above part (i) and (ii) seem simple enough,
but I am unsure where to start for part (iii). It provides a hint to try and find the constant a first but I am unsure how to do this. Any guidance would be massively appreciated. (The University don't provide their own answer schemes)