problem regarding theory of equations

Question

given quadratic equation : ${x^2+bx+c=0}$

let the roots of the equation be ${u}$ and $v$.

let ${S_0 = u^0+v^0}$

let ${S_1 = u^1+v^1}$

let ${S_2 = u^2+v^2}$

show that : ${S_2+bS_1+S_0 = 0}$

Answer 1

See, ${S_0 = 1 +1=2}$ and ${S_1 = u+ v}$ but ${u +v}$ is the sum of roots and hence ${ S_1=u + v = -b}$. Now, ${ (u+v)^2 = u^2 + v^2 + 2uv}$ is equivalent to $(S_1)^2 = S_2 + 2\text{(product of roots)}$. And $\text{product of roots} = c$. Now put the appropriate values into the equations. You should get your solution. :))