1)(x+y)^3-(x-y)^3-2y = (x + y - x + y) * ((x + y)² + (x + y)(x - y) + (x - y)²) - 2y =
= 2y * (x² + 2xy + y² + x² - y² + x² - 2xy + y²) - 2y = 2y * (3x² + y²) - 2y = 2y(3x² + y² - 1)
2) (m+n)^3+(m-n)^3-2m = m³ + 3m²n + 3mn² + n³ + m³ - 3m²n + 3mn² - n³ - 2m =
= 2m³ + 6mn² - 2m = 2m( m² + 3n²) - 2m = 2m * (m² + 3n² - 1)
3) (a-b)^3-(c+d)^3-a+b+c+d = (a - b - c -d) ((a-b)² + (a - b)(c + d) + (c + d)²)- a + b + c + d =
= (a - b - c -d) ((a-b)² + (a - b)(c + d) + (c + d)²)- (a - b - c - d) = (a - b - c - d) ((a+ b)((a +b + c + d) + + (c + d)² - 1 )
№1
а) а4 –18а2 + 81=(a2-9)^2=(a2-9)(a2-9);
б) а5 + а3 – а2 –1=a2(a3-1)+a3-1=(a3-1)(a2+1);
в) а5 +3а4 – 4а3 – 12а2=;a4(a+3)-3a2(a+3)=(a+3)(a4-3a2);
г) а4 + 2а3 – 2а –1=(a2+1)(a2-1)+2a(a2-1)=(a2-1)(a2+1+2a;
№ 2
а) 9х3 – 18х2 = х – 2
9x2(x-2)-(x-2)=0
(x-2)(9x2-1)=0
(x-2)(3x-1)(3x+1)=0
x-2=0 ; 3x-1=0 ; 3x+1=0
x=2 ; x=1/3 ; x=-1/3
б) у3 – у2 = у – 1;
y2(y-1)-(y-1)=0
(y-1)(y2-1)=0
(y-1)(y-1)(y+1)=0
y-1=0 y+1=0
y=1 y=-1
в) х3 – 3х2 – 3х + 1 = 0;
х3+1 – 3х(x +1) = 0;
(x+1)(x2-x+1)-3x(x+1)=0
(x+1)(x2-x+1-3x)=0
(x+1)(x2-4x+1)=0
x+1=0 x2-4x+1=0
x=-1 D=16-4*1=12
x=-1 x=2±√3
г) х4 – 2х3 + 2х – 1 = 0
x4-1 -2x(x2-1)=0
(x2-1)(x2+1)-2x(x2-1)=0
(x2-1)(x2+1-2x)=0
x2-1=0 x2-2x+1=0
(x-1)(x+1)=0 D=4-4*1=0
x=±1 x=-b/2a=1
д) х5 + 5х3 – 6х2 = 0
x2(x3+5x-6)=0
x2=0 | x3+5x-6=0
x=0 | x3-1+5x-5=0
x=0 | (x-1)(x2+x+1)+5(x-1)=0
x=0 | (x-1)(x2+x+6)=0
x=0 | x-1=0 | x2+x+6=0
x=0 | x=1 | D=1-4*6=-23 - Нет корней.
=(cosx+sinx)/cosx :(sinx+cosx)=1/cosx