По теореме Виета x1 + x2 = 6/2 = 3; x1*x2 = 3/2 = 1,5.
a) x1*x2^5 + x1^5*x2 = x1*x2*(x1^4 + x2^4) =
= 1,5*(x1^4 + 2x1^2*x2^2 + x2^4 - 2x1^2*x2^2) =
= 1,5*((x1^2 + x2^2)^2 - 2*(1,5)^2) =
= 1,5*((x1^2 + 2x1*x2 + x2^2 - 2x1*x2)^2) - 2*2,25) =
= 1,5*( [ (x1+x2)^2 - 2*1,5 ]^2 - 4,5) = 1,5*((3^2 - 3)^2 - 4,5) =
= 1,5*(6^2 - 4,5) = 1,5*(36 - 4,5) = 1,5*31,5 = 47,25
b) x1^4 + x2^4 = x1^4 + 2x1^2*x2^2 + x2^4 - 2x1^2*x2^2 =
(x1^2 + x2^2)^2 - 2*(1,5)^2 = (x1^2 + 2x1*x2 + x2^2 - 2x1*x2)^2) - 2*2,25 =
[ (x1+x2)^2 - 2*1,5 ]^2 - 4,5 = (3^2 - 3)^2 - 4,5 = 36 - 4,5 = 31,5
c)
![\frac{x1}{x2^2} + \frac{x2}{x1^2} = \frac{x1^3+x2^3}{x1^2*x2^2} = \frac{(x1+x2)(x1^2-x1*x2+x2^2)}{(x1*x2)^2} =](https://tex.z-dn.net/?f=+%5Cfrac%7Bx1%7D%7Bx2%5E2%7D+%2B+%5Cfrac%7Bx2%7D%7Bx1%5E2%7D+%3D+%5Cfrac%7Bx1%5E3%2Bx2%5E3%7D%7Bx1%5E2%2Ax2%5E2%7D+%3D+%5Cfrac%7B%28x1%2Bx2%29%28x1%5E2-x1%2Ax2%2Bx2%5E2%29%7D%7B%28x1%2Ax2%29%5E2%7D+%3D)
![= \frac{3(x1^2+2x1x2+x2^2-3x1x2)}{(1,5)^2} = \frac{1,5*2((x1+x2)^2-3*1,5)}{(1,5)^2} = \frac{2(3^2-4,5)}{1,5} = \frac{2*4,5}{1,5}=6](https://tex.z-dn.net/?f=%3D+%5Cfrac%7B3%28x1%5E2%2B2x1x2%2Bx2%5E2-3x1x2%29%7D%7B%281%2C5%29%5E2%7D+%3D+%5Cfrac%7B1%2C5%2A2%28%28x1%2Bx2%29%5E2-3%2A1%2C5%29%7D%7B%281%2C5%29%5E2%7D+%3D+%5Cfrac%7B2%283%5E2-4%2C5%29%7D%7B1%2C5%7D+%3D+%5Cfrac%7B2%2A4%2C5%7D%7B1%2C5%7D%3D6+)