Вправи: 1085 (1-2)

1085. Спростіть вираз:

1) 1x- 1y+z1x+ 1y+z • (1 + y2+ z2- x22yz) : x-y-zxyz = x(x-y-z)2.

1. 1x- 1y+z1x+ 1y+z = y+z-xx(y+z) : y+z+xx(y+z) = y+z-xy+z+x;

2. 1 + y2+ z2- x22yz = y2+ 2yz+ z2- x22yz = (y+ z)2- x22yz;

3. y+z-xy+z-xy+z+xxyzy+z+x • 2yz(x-y-z) = (x-y-z)2x2(x-y-z) = x-y-zx2.

2) m-n2m-n- m2+ n2+ m2m2+ mn- n24n4+ 4mn2+ m2 : (2n2+ m) • (n² + n + mn + m) = n+1n-2m.

1. m-n2m-n – m2+ n2+ m2m2+ mn- n2 = 2m3+ m2n-mn2- 2m2n-mn2+ n3- (2m3+ 2mn2+ 2m2- m2n- n3- nm)2m-n(2m2+ mn- n2) = 2m3+ m2n- mn2-2m2n- mn2+n3 -2m3-2mn2-2m2+ m2n+ n3+ nm(2m- n)(2m2+ mn-n2) = 2n3-4mn2+ mn-2m2(2m- n)(2m2+ mn-n2) = 2n2n-2m+ m(n-2m)(2m- n)(2m2+ mn-n2) = (n-2m)(2n2+ m)(2m- n)(2m2+ mn-n2) = –2n2+ m2m2+ mn-n2;

2. - 2n2+ m2m2+ mn-n2(2n2+m)22n2+ m • (n(n + 1) + m(n + 1)) = –(n+1)(n+ m)2m2+ mn-n2 = (n+1)(n+ m)n2- mn-2m2.

n² – mn – 2m² = 0 – розв'яжемо відносно n.

D = (–m)² – 4 • (–2m²) = 9m²;

n₁ = m+3m2 = 2m; n₂ = m-3m2 = –m.

Маємо (n+1)(n+ m)(n-2m)(n+ m) = n+1n-2m.

3) 4x+1y+1z : 1x+1y – 4y(xyz+ x+ z) = 4.

1. 4x+ 1y+1z = 4x+ 1yz+1z = 4x+ zyz+1 = 4xyz+ x+ zyz+1 = 4(yz+1)xyz+ x+ z;

2. 4(yz+1)xyz+ x+ z : 1xy+1y = 4(yz+1)xyz+ x+ z : yxy+1 = 4(yz+1)(xy+1)y(xyz+ x+ z);

3. 4(yz+1)(xy+1)y(xyz+ x+ z) – 4y(xyz+ x+ z) = 4 • xy2z+ yz+ xy+1-1y(xyz+ x+ z) = 4 • y(xyz+ x+ z)y(xyz+ x+ z) = 4.

4) ((ab- a)^–2 – (a+ b)2-4aba2- ab)² • a4 a2b2- b4 = a- ba+ b.

1. (a+ b)2-4aba2- ab = a2+ 2ab+ b2-4aba2- ab = a2- 2ab+b2 a2- ab = (a- b)2a(a- b) = a- ba;

2. ((ab- a)^–2 – a- ba)² = ((a- b)2a2 – a- ba)² = (a- ba • (a- ba – 1))² = (a- b)2a2 • (a- b- aa)² = (a- b)2a2 • b2a2 = b2(a- b)2a4;

3. b2(a- b)2a4 • a4a2b2- b4 = b2a4(a- b)2 a4b2(a- b)(a+ b) = a- ba+ b.

5) p-6- 644+2p-1+ p-2 • p24-4p-1+p-2 – 4p2(2p+1)1-2p = 1 + 2p.

1. (p-3)2- 82 4+2p-1+ p-2 • p24-4p-1+ p-2 = (p-3 -8)(p-3 + 8)p2(4+2p-1+ p-2)(4-4p-1+ p-2) = (p-1 -2)(p-2 +2p-1+4)(p-1 +2)(p-2 -2p-1+4)p2(4+2p-1+ p-2)(p-1 -2)2 = p2(p-1 +2)(p-2 -2p-1+4)1p- 2 = pp-1 +2p2(p-2 -2p-1+4)1-2p = (1+2p)(1-2p+4p2)1-2p = 1+8p31-2p;

2. 1+8p31-2p – 4p2(2p+1)1-2p = 1+8p3-8p3-4p21-2p = 1-4p21-2p = (1-2p)(1+2p)1-2p = 1 + 2p.

6) x-1-y-1 x-3- y-3 : x2y2(x+ y)2-3xy • (x2-y2 xy)^–1 = –xy(x+ y)2.

1. x-1-y-1x-3- y-3 = 1x-1y1x3+ 1y3 = y- xxyy3+x3x3y3 = y- xx3y3xy(y3 + x3) = y- xx2y2y3+ x3;

2. x2y2(x+ y)2-3xy = x2y2x2+ 2xy+ y2-3xy = x2y2x2-xy+ y2;

3. (y- x)x2y2y3+ x3 : x2y2x2- xy+ y2 • (x2-y2xy)^–1 = (y- x)x2y2(x2 - xy+ y2)y+ xx2 - xy+ y2 • x2y2 • xy(x- y)(x+ y) = –xy(x+ y)2.