4. Let r be a root of x² + x + 1 = 0. Show the following:

(a) x³ - a³ = (x - a)(x - ar)(x - ar²).

(b) x³ + y³ + z³ - 3xyz = (x + y + z)(x + ry + r²z)(x + r²y + rz).

5. Let a, b, and c be the roots of x³ + 3x + 3 = 0. Find (a + 1)(b + 1)(c + 1).

6. Given that (x - a)(x - b) = x² - px + q, express each of the following in terms of p and q:

(a) a + b.

(b) ab.

(c) a² + 2ab + b².

(d) a² + ab + b².

(e) ab(a² + ab + b²).

(f) a³b³.

(g) The coefficients of the expansion of (x - a²)(x - ab)(x - b²).

7. Let (x - r)(x - s) = x² - px + q, and let (x - r³)(x - r²s)(x - rs²)(x - s³) = x⁴ - ax³ + bx² - cx + d. Express a, b, c, and d in terms of p and q.

8. Let (x - a)(x - b) = x² - ex + f, (x - c)(x - d) = x² - gx + h, and (x - ac)(x - ad)(x - bc)(x - bd) = x⁴ - px³ + qx² - rx + s. Find p, q, r, and s in terms of e, f, g, and h.

9. Let (x - a)(x - b)(x - c) = x³ - 3x + 1. Find each of the following:

(a) 2(a + b + c).

(b) (a + b)(a + c) + (a + b)(b + c) + (a + c)(b + c).

(c) (a + b)(a + c)(b + c).

(d) the equation y³ - py² + qy - r = 0 whose roots are a + b, a + c, and b + c.