(a) Find the roots, $P_n(t)=0$, of the Legendre polynomials $P_2, P_3$ and $P_4$. (b) Prove that for $0 \leq j \leq k$, the polynomial $R_{j, k}(t)$ defined in (4.62) has roots of order $k-j$ at $t= \pm 1$, and $j$ additional simple roots lying between -1 and 1 . (c) Conclude that all $k$ roots of the Legendre polynomial $P_k(t)$ are real and simple, and that they lie in the interval $-1<t<1$.