10. Do Problem 9 with (x - a)(x - b)(x - c) = x³ + 3x - 1.
 

11. Let s₁ = a + b + c, and s₂ = ab + ac + bc, and s₃ = abc. Find numbers x, y, z, t, u, v, and w such that for all a, b, and c:

(a) a³ + b³ + c³ = xs₃ + ys₁s₂ + zs₁³.

(b) a⁴ + b⁴ + c⁴ = ts₁s₃ + us₂² + vs₁²s₂ + ws₁⁴.

12. Let (x - r)(x - s)(x - t) = x³ - ax² + bx - c. Express r⁵ + s⁵ + t⁵ in terms of a, b, and c.

13. Let (x -1)(x -2)(x - 3)...(x - n) = xⁿ - s₁x^n-1 + s₂x^n-2 - ... + (-1)ⁿs_n. Show the following:

(a)

(b) s_n = n!.

(c) 2s₂ = (1³ + 2³ + ... + n³) - (1² + 2² + ... + n²).