Question

COROANA!!!
Demonstrati ca:
(x²-x+1)=x^4-2x³+3x²-2x+1
oricare ar fi numarul real x

Answer 1

(x²-x+1)²=$x^{4}$ -2x³+3x²-2x+1

(a+b+c)²=a²+b²+c²+2ab+2bc+2ac

(x²-x+1)²= $x^{4}$ +x²+1² +2x²-2x³-2x=$x^{4}$ -2x³+3x²-2x+1

$x^{4}$ -2x³+3x²-2x+1=$x^{4}$ -2x³+3x²-2x+1 , ∀ x∈R  

Answer 2

x^4 - 2x³ +3x² -2x + 1 = (x^4 - x³ +x²) -(x³ -x² +x) + ( x² -x + 1) =
= x²(x² -x + 1) - x(x² - x+ 1) + (x² -x + 1) = (x²- x + 1)(x² - x + 1) = (x² - x + 1)²
sau (x² - x + 1)² = [x(x-1) + 1]² = x²(x² - 2x + 1) + 2x(x-1) + 1 =
= x^4 - 2x³ +x² + 2x² - 2x + 1 = x^4 - 2x³ + 3x² - 2x + 1