Вправи: 677(1-3)

677. Спростіть вираз:

1) (x\x-yx+ y + y\x+yx- y + 2xyx-y) : (–y+x y- yx) = x-xy+ xy+ y+2xy x- y • (–y-yx y+x) = (x+y) 2x- y • (–y(y- x)y+ x) = (x+ y)2x- y • y(x- y)x+ y = x+ y• yx+y = y.

2) (x\x+yx- y – y\x-yx+ y) • xy- yyy = x+xy- xy+ y x-y • y(x- y)y = x+ yx- y • y(x- y)y = x+ yy.

3) (x+1+x2-1x+1-x2-1 – x)² = ((x+1+ x2-1 )2 (x+1)2-(x2-1) – x)² = ((x+1)2+2x+1x2-1+ x2-1x2+ 2x+1- x2+1 – x)² = (x2+ 2x+1+2x+2x2-1+ x2-12x+2 – x)² = (2x2+2x+2x+2x2-12x+2 – x\2x+3)² = (2x2+2x+2x+2x2-1- 2x2-2x2x+2)² = ((2x+2)x2-12x+2)² = (x2-1)² = x² – 1.

4) xx+ yyx+ y • (x – y)^–1 + 3xy-3yx- y – 3y(y+ x)(y- x) = xx+ yy(x+ y)(x- y) + 3xy-3yx- y + 3yx- y = xx+ yy(x+ y)(x- y) + 3xy-3y+3yx- y = xx+ yy(x+ y)(x- y) + 3xyx- y = (xx+ yy)(x- y)(x- y)(x- y) + 3xyx- y = x2- xxy + yxy-y2 (x- y)2 + 3xyx- y = x2- y2-xy(x- y)(x- y)2 + 3xyx- y = x- yx+ y- xy(x- y)(x- y)2 + 3xyx- y = (x- y)(x+ y- xy)x- y2 + 3xyx- y= x+ y- xyx- y + 3xyx- y = x+ y- xy+3xyx- y = x+ y+2xyx- y = (x+ y)2(x+ y)(x- y) = x+ yx-y.