Verified answer

If α, β, γ are the roots of the equations

x3 + 4x + 8 = 0,

then find the values of

a. α+β+γ

b. αβ+βγ+γα

c. αβγ

d. α^3+β^3+γ^3

x³ + 4x + 8 = 0 → let: x = u + v

(u + v)³ + 4(u + v) + 8 = 0

[(u + v)²(u + v)] + 4(u + v) + 8 = 0

[(u² + 2uv + v²)(u + v)] + 4(u + v) + 8 = 0

[u³ + u²v + 2u²v + 2uv² + uv² + v³] + 4(u + v) + 8 = 0

[u³ + v³ + 3u²v + 3uv²] + 4(u + v) + 8 = 0

(u³ + v³) + (3u²v + 3uv²) + 4(u + v) + 8 = 0

(u³ + v³) + 3uv(u + v) + 4(u + v) + 8 = 0

(u³ + v³) + [(u + v)(3uv + 4)] + 8 = 0 → suppose that: (3uv + 4) = 0 ← equation (1)

(u³ + v³) + [(u + v) * 0] + 8 = 0

(u³ + v³) + 8 = 0 ← equation (2)

You get a system of 2 equations:

(1) : (3uv + 4) = 0

(1) : 3uv = - 4

(1) : uv = - 4/3

(1) : u³v³ = - 4³/3³

(2) : (u³ + v³) + 8 = 0

(2) : u³ + v³ = - 8

Let: U = u³

Let: V = v³

The system becomes:

(1) : UV = - 4³/3³ ← this is the product P

(2) : u³ + v³ = - 8 ← this is the sum S

You know that U and V are the solution of the equation:

x² - Sx + P = 0

x² + 8x - (4³/3³) = 0

Δ = 8² - 4[1 * - (4³/3³)]

Δ = 64 + (256/27)

Δ = 1984/27

Δ = (64/9) * (31/3)

x1 = [- 8 - (8/3).√(31/3)] / 2 = - 4 - (4/3).√(31/3) ← this is U

x2 = [- 8 + (8/3).√(31/3)] / 2 = - 4 + (4/3).√(31/3) ← this is V

U = u³ → u = U^(1/3) → u = [- 4 - (4/3).√(31/3)]^(1/3)

V = v³ → v = V^(1/3) → v = [- 4 + (4/3).√(31/3)]^(1/3)

Recall: x = u + v → x ≈ - 1.36465560765604 ← this is the only root

Why did you tell that there are 3 roots?