Hello, umbraculum!

For #2, you need some "equation theory" . . .

The theory applies to all polynomials, but I'll explain it for this cubic only.2) Given that a, b, and c are the roots of the cubic: .P(x) .= .x³ - 2x² - 23x + k,

. . find the exact value of: .a² + b² + c².

Given a cubic equation: .x³ + px² + qx + r .= .0 .with roots a, b, c:

. . a + b + c .= .-p, . ab + bc + ac .= .q, . abc .= .-r

We have: .(1)a + b + c .= .2, .(2)ab + bc + ac .= .-23, .(3)abc .= .-k

Square equation(1): .(a + b + c)² .= .2²

And we have: .a² + b² + c² + 2ab + 2bc + 2ac .= .4

Factor: .a² + b² + c² + 2(ab + bc + ac) .= .4

. . . . . . . . . . . . . . . . . . \_________/

. . . . . . . . . . . . . . . . . . . .This is -23

So we have: .a² + b² + c² + 2(-23) .= .4 . → .a² + b² + c² .= .50