Very difficult!!!!!!!!!!!!!!!!

If a,b,c,d are the roots of the equation x⁴+P₁x²+P₂x²+P₃x+P₄=0 then (1+a²)(1+b²)(1+c²)(1+d²)=

2 Answers

Please give a short/simple solution!!!!

341

Hari Shankar ·2008-12-25 17:34:30

We have x⁴+P₁x³+P₂x²+P₁x+P₄ = (x-a)(x-b)(x-c)(x-d) = (a-x)(b-x)(c-x)(d-x)

Put x = i,

we get (1-P₂+P₄)+i(P₃-P₁) = (a-i)(b-i)(c-i)(d-i)

Taking the square of mudulus on both sides we get

(1-P₂+P₄)²+(P₃-P₁)² = (a²+1)(b²+1)(c²+1)(d²+1)

√ √ || {} [] () → ~ ^ x x x → → ln log lg lim lim cos sin tan cot arcsin arccos arctan

∫ ∫ ∫ ∫ ∫ ∬ ∬ ∬ ∬ ∬ ∭ ∭ ∭ ∭ ∭ ⨌ ⨌ ⨌ ⨌ ⨌

∑ ∑ ∑ ∑ ∑ ∮ ∮ ∮ ∮ ∮ ∏ ∏ ∏ ∏ ∏ ∐ ∐ ∐ ∐ ∐

+ - × ÷ ± ∓ = ∪ ∩ ≠ ∼ ≈ ∅ ∀ ∃ ∈ ∋ ∝ ≤ ≥ ≃ ≅ ≡ < > ≪ ≫ ⊂ ∉ ⊃ ⊆ ⊇ ⇒ ⇐ ⇔ ⇒ ⇐ ⇔ ⇑ ⇓ ⇕ → ← ↔ → ← ↔ ↑ ↓ ↕ ↖ ↗ ↘ ↙ ↪ ↩ ⇀ ↼ ⇁ ↽

° ∂ ∞ α β γ δ θ λ μ ν ξ π ρ σ φ ω ε ƒ κ τ υ Γ Δ Λ Σ Π Ω