Вправи: 105 (1-3)

105. Спростіть вираз:

1) a2+ 1a2- 2a+1 + a+1a-1 = a2+ 1(a-1)2 + a+ 1\a-1a-1 = a2+ 1+a-1(a+1)(a-1)2 = a2+ 1+ a2- 1(a-1)2 = 2a2(a-1)2;

2) a2+ b2a2- b2 – a-ba+b = a2+ b2a-b(a+b) – a- b\a-ba+b = a2+ b2- (a-b)2a-b(a+b) = a2+ b2- (a2- 2ab+ b2)a-b(a+b) = a2+ b2- a2+ 2ab- b2(a-b)(a+b) = 2aba2- b2;

3) c+7c-7 + 28c49- c2 = c+7c-7 – 28cc2- 49 = c+ 7\c+7c-7 – 28cc-7(c+7) = (c+7)2- 28cc-7(c+7) = c2+ 14c+49-28cc-7(c+7) = c2- 14c+49c-7(c+7) = (c-7)2c-7(c+7) = c-7c+7;

4) 5a+32a2+ 6a + 6-3aa2- 9 = 5a+ 3\a-32a(a+3) + 6-3a\2aa-3(a+3) = a-35a+3+ 2a(6-3a)2aa-3(a+3) = 5a2+ 3a-15a-9+12a-6a22aa-3(a+3) = -a2- 92aa-3(a+3) = -a2- 92aa-3(a+3) = –a2+ 92a3- 18a;

5) aa2- 4a+4 – a+4a2- 4 = a\a+2(a-2)2 – a+ 4\a-2a-2(a+2) = aa+2- (a+4)(a-2)(a-2)2(a+2) = a2+ 2a-(a2- 2a+4a-8)(a-2)2(a+2) = a2+ 2a- a2- 2a+8(a-2)2(a+2) = 8(a-2)2(a+2);

6) 2pp-5 – 5p+5 + 2p225- p2 = 2p\p+5p-5 – 5\1p-5p+5 – 2p2p2- 25 = 2pp+5- 5p-5- 2p2p-5(p+5) = 2p2+ 10p-5p+25-2p2p2- 25 = 5p+25p-5(p+5) = 5(p +5)p-5(p+5) = 5p-5;

7) 1y – y+816- y2 – 2y-4 = 1\y2-16y + y+ 8\yy2- 16 – 2\y(y+4)y-4 – y2- 16+yy+8- 2y(y+4)yy-4(y+4) = y2- 16+ y2+ 8y-2y2- 8yyy-4(y+4) = -16y(y2- 16) = 16y(16- y2) = 1616y- y3;

8) 2b-14b+2 + 4b4b2- 1 + 2b+13-6b = 2b- 1\3(2b-1)2(2b+1) + 4b\62b-1(2b+1) – 2b+ 1\2(2b+1)3(2b-1) = 3(2b-1)2+ 24b-2(2b+1)262b-1(2b+1) = 34b2- 4b+1+ 24b-2(4b2+ 4b+1)62b-1(2b+1) = 12b2- 12b+3+24b-8b2- 8b-262b-1(2b+1) = 4b2+ 4b+16(4b2- 1) = (2b+1)262b-1(2b+1) = 2b+112b-6.