Fatoração

por DaviMol16 Sex 21 Nov 2014, 14:53

Se a+b+c=0 e a²+b²+c²=1, calcule A= a⁴+b⁴+c⁴

por PedroCunha Sex 21 Nov 2014, 17:10

Olá.

Por Polinômios Simétricos:

\\ S_1 = \sigma_1 = 0, S_2 = 1, A = S_4 = ? \\\\ a+b+c = 0 \therefore (a+b+c)^2 = 0 \therefore a^2+b^2+c^2 + 2(ab+bc+ac) =0 \\\\ \rightarrow ab+bc+ac = \sigma_2 = -\frac{1}{2} \\\\ S_4 = S_3 \cdot \sigma_1 - S_2 \cdot \sigma_2 + S_1 \cdot \sigma_3 \therefore S_4 = S_3 \cdot 0 - 1 \cdot \left(-\frac{1}{2} \right) + 0 \cdot \sigma_3 \Leftrightarrow \boxed{\boxed{S_4 = A = \frac{1}{2} }}

por Luck Sex 21 Nov 2014, 17:57

Outra forma:
a+b+c = 0 , a²+b² +c² = 1 ∴ ab + bc + ac = -1/2

(a²)² + (b²)² + (c²)² = (a²+b²+c²)² - 2[(ab)²+ (ac)² + (bc)²]

(ab)² + (ac)² + (bc)² = (ab + ac+bc)² - 2(a²bc + ab²c + +abc²)
(ab)² +(ac)² +(bc)² = (ab+ac+bc)² - 2abc(a+b+c)
(ab)² + (ac)² + (bc)² = (1/4)

a^4 + b^4 + c^4 = 1² - 2(1/4)
a^4 + b^4 + c^4 = 1/2