Вправи: 1086 (1-2)

1086. Доведіть тотожність:

1) (x2 -1y2)x • (y+ 1x)y-x(y2 -1x2)y • (x-1y)x-y = (x2y2-1y2)x • (xy+1x)y-x (x2y2-1x2)y • (xy-1y)x-y = (x2y2-1)xy2x • (xy+1)y-xxy-x(x2y2-1)yx2y • (xy-1)x-yxx-y = (xy-1)x(xy+1)x(xy+1)y-x • x2y • yx-y(xy-1)y(xy+1)y(xy-1)x-y • y2x • xy-x = (xy-1)x(xy+1)yx2y-(y-x)yx-y-2x(xy-1)x(xy+1)y = x^y+zy^x–y = xx+yyx+y = (xy)^x+y, що й треба було довести.

2) (1 + b2+c2 -a22bc) : ((1a + 1b+c) : (1a – 1b+ c)).

1. 1 + b2+ c2-a22bc = b2+ 2bc+c2 -a22bc = (b+ c)2-a2 2bc;

2. (1a + 1b+ c) : (1a – 1b+ c) = a+ b+ ca(b+ c) : b+ c- aa(b+ c) = a+ b+ cb+ c- a;

3. (b+ c- a)(b+ c+ a)2bc : a+ b+ cb+ c- a = (b+ c- a)(b+ c+ a)(b+ c- a)2bc(a+ b+ c) = (b+ c- a)22bc, що й треба було довести.

3) (x- y)2+ xy(x+ y)2- xy : x5+ y5+ x2y3+x3y2(x3 + y3+ x2y+ xy2)(x3 - y3) = x2 -2xy+y2+ xy x2+2xy+ y2-xy • (x2x+ y+y2x+ y)(x- y)(x2+xy+y2) x2x3+y3+ y2(x3+y3) = (x2-xy+y2)(x+ y)(x2+y2)(x- y)(x2+xy+y2) (x2+xy+ y2)(x2+y2)(x3+y3) = (x- y)(x3+y3)x3+y3 = x – y, що й треба було довести.

4) (2- yy-1 + 2(x-1)x-2) : (y(x-1)y-1 + x(2- y)x-2).

1. 2- yy-1 + 2(x-1)x-2 = 2x- xy-4+2y+2xy-2x-2y+2(y-1)(x-2) = xy-2(y-1)(x-2);

2. y(x-1)y-1 + x(2- y)x-2 = yx2-3x+2+ x(3y-2- y2)(y-1)(x-2) = yx2-3xy+2y+3xy-2x- xy2(y-1)(x-2) = yx2+2y-2x- xy2(y-1)(x-2) = xyx- y-2(x- y)(y-1)(x-2) = (x- y)(xy-2)(y-1)(x-2);

3. xy-2(y-1)(x-2) : (x- y)(xy-2)(y-1)(x-2) = (xy-2)(y-1)(x-2)(y-1)(x-2)(x- y)(xy-2) = 1x- y, що й треба було довести.