Bonjour ;
Exercice 1 :
1)
a) 123²-122²-121²+120² = (123²-122²)-(121²-120²)
= (123-122)(123+122) - (121-120)(121+120) = 1 * 245 - 1 * 241
= 245 - 241 = 4
b) 12²-11²-10²+9² = (12²-11²) - (10²-9²) = (12-11)(12+11) - (10-9)(10+9)
= 1 * 23 - 1 * 19 = 23 - 19 = 4
c) 45²-44²-43²+42² = (45²-44²) - (43²-42²) = (45-44)(45+44) - (43-42)(43+42)
= 1 * 89 - 1 * 85 = 89 - 85 = 4
2) ∀ x ∈ R : (x+3)² - (x+2)² - (x+1)² + x² = 4
3) ∀ x ∈ R : (x+3)² - (x+2)² - (x+1)² + x² = ((x+3)² - (x+2)²) - ((x+1)² - x²)
= ((x+3)-(x+2)) ((x+3)+(x+2)) - ((x+1)-x) ((x+1)+x)
= 1 * (2x+5) - 1 * (2x+1) = 2x + 5 - 2x - 1 = 4
