Вправи: 417 (1)

417. Спростіть вираз:

1) (1+x)-1x+2 + (2+x)-1x+3 + (3+x)-1x+4 = 1x+1(x+2) + 1x+2(x+3) + 1x+3(x+4) = x+3x+4+ x+1x+4+ (x+1)(x+2)x+1x+2x+3(x+4) = x2+ 4x+3x+12+ x2+ 4x+x+4+x2+ 2x+x+2x+1x+2x+3(x+4) = 3x2+ 15x+18x+1x+2x+3(x+4) = 3(x2+ 5x+6) x+1x+2x+3(x+4) = 3(x2+ 2x+3x+6)x+1x+2x+3(x+4) = 3(xx+2+ 3x+2)x+1x+2x+3(x+4) = 3x+2(x+3)x+1x+2x+3(x+4) = 3x+1(x+4).

2) (a-1+ b-1a-1- b-1 – a-1- b-1a-1+ b-1) : ((a² + b²)(a² – b²)^–1 – (a² – b²)(a² + b²)^–1) = (1a+ 1b1a- 1b – 1a- 1b1a- 1b) : (a2+ b2a2- b2 – a2- b2a2+ b2) = (b+aabb-aab – b-aabb+aab) : (a2+ b2)2- (a2- b2)2(a2- b2)(a2+ b2) = (b+a•abab •(b-a) – b-a•abb+a•ab) : a4+ 2a2b2+ b4- (a4- 2a2b2+ b4)a2- b2(a2+ b2) = (b+ab-a – b-ab+a) : a4+ 2a2b2+ b4- a4+ 2a2b2- b4a2- b2(a2+ b2) = b2+ 2ab+ a2- (b2- 2ab+ a2)b-a(b+a) : 4a2b2a2- b2(a2+ b2) = b2+ 2ab+ a2- b2+ 2ab- a2b-a(b+a) • a-ba+b(a2+ b2)4a2b2 = –4ab •a-ba+b(a2+ b2)a-ba+b•4a2b2 = –a2+ b2ab.

3) (ab-1+ 1)2ab-1- a-1b • a3b-3- 1a2b-2+ ab-1+ 1 : a2b-2+ 1ab-1+ a-1b = (ab+ 1)2ab- ba • a3b3 - 1a2b2+ abb+ 1 • ab+ baa2b2+ 1 = (a+bb)2a2- b2ab • a3- b3b3a2+ ab+ b2b2 • a2+ b2aba2+ b2b2 = (a+b)2 •abb2a+b(a-b) • a-ba2+ ab+ b2• b2b3(a2+ ab+ b2) • (a2+ b2) • b2ab(a2+ b2) = a+b• b4b5 = a+bb.