Toán Lớp 8: CMR với mọi số nguyên m thì (2m-1)^2-(2m-1) luôn chia hết cho 8 Cho x^2+y^2+z^2=10.Tính gái trị biểu thức P=(xy+yz+xz)^2+(x^2-yz)^2+(y^

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Toán Lớp 8:  CMR với mọi số nguyên m thì (2m-1)^2-(2m-1) luôn chia hết cho 8 Cho x^2+y^2+z^2=10.Tính gái trị biểu thức P=(xy+yz+xz)^2+(x^2-yz)^2+(y^2-yz)^2+(z^2-xy)^2

maitrucloan · Answer

a)     (    2    m    &minus;    1    )       &sup2;       &minus;    (    2    m    &minus;    1    )    =    (    2    m    &minus;    1    )    (    2    m    &minus;    1    &minus;    1    )     (2m&minus;1)&sup2;&minus;(2m&minus;1)=(2m&minus;1)(2m&minus;1&minus;1)          =    (    2    m    &minus;    1    )    (    2    m    &minus;    2    )    =    2    (    2    m    &minus;    1    )    (    m    &minus;    1    )     =(2m&minus;1)(2m&minus;2)=2(2m&minus;1)(m&minus;1)     Ta c&oacute; 2(2m-1)(m-1) chia hết cho 8 &hArr; (2m-1)(m-1) chia hết cho 4   M&agrave; (2m-1);(m-1) = 1 &rArr; m = 4k+1 (k &isin; Z)   Vậy        (    2    m    &minus;    1    )       &sup2;       (    2    m    &minus;    1    )     (2m&minus;1)&sup2;(2m&minus;1)     chia hết cho 8 &hArr; m = 4k+1   b) P     =    (    x    y    +    y    z    +    x    z    )       &sup2;       +    (    x       &sup2;       &minus;    y    z    )       &sup2;       +    (    y       &sup2;       &minus;    y    z    )       &sup2;       +    (    z       &sup2;       &minus;    x    y    )       &sup2;        =(xy+yz+xz)&sup2;+(x&sup2;&minus;yz)&sup2;+(y&sup2;&minus;yz)&sup2;+(z&sup2;&minus;xy)&sup2;           =    x       &sup2;       y       &sup2;       +    y       &sup2;       z       &sup2;       +    x       &sup2;       z       &sup2;       +    2    x    y       &sup2;       z    +    2    x       &sup2;       y    z    +    2    x    y    z       &sup2;        =x&sup2;y&sup2;+y&sup2;z&sup2;+x&sup2;z&sup2;+2xy&sup2;z+2x&sup2;yz+2xyz&sup2;     +          x        4         x4     -     2    x       &sup2;       y    z    +    y       &sup2;       z       &sup2;        2x&sup2;yz+y&sup2;z&sup2;     +          y        4         y4     -     2    x    y       &sup2;       z    +    x       &sup2;       z       &sup2;        2xy&sup2;z+x&sup2;z&sup2;     +          z        4         z4     -     2    x    y    z       &sup2;       +    x       &sup2;       z       &sup2;        2xyz&sup2;+x&sup2;z&sup2;          =     =            x        4         x4     +          y        4         y4     +          z        4         z4     +        2      x        2          y        2         2x2y2     +        2      y        2          z        2         2y2z2     +        2      z        2          x        2         2z2x2          =     =          (      x        2        +      y        2        +      z        2          )      2       (x2+y2+z2)2           =     =            10        2        =    100     102=100

quynhthanhnghi · Answer

a)$(2m-1)&sup2; - (2m - 1) = (2m-1)(2m-1-1)  $   $= (2m-1)(2m-2) = 2(2m-1)(m-1)$   Ta c&oacute; 2(2m-1)(m-1) chia hết cho 8 &hArr; (2m-1)(m-1) chia hết cho 4   M&agrave; (2m-1);(m-1) = 1 &rArr; m = 4k+1 (k &isin; Z)   Vậy $(2m-1)&sup2;(2m-1)$ chia hết cho 8 &hArr; m = 4k+1   b) P$=(xy+yz+xz)&sup2;+(x&sup2;-yz)&sup2;+(y&sup2;-yz)&sup2;+(z&sup2;-xy)&sup2;$    $= x&sup2;y&sup2; + y&sup2;z&sup2; + x&sup2;z&sup2; + 2xy&sup2;z + 2x&sup2;yz + 2xyz&sup2; $ + $x^{4}$ -$2x&sup2;yz + y&sup2;z&sup2;$ + $y^{4}$ -$2xy&sup2;z + x&sup2;z&sup2;$ + $z^{4}$ -$2xyz&sup2; + x&sup2;z&sup2;$   $=$ $x^{4}$ + $y^{4}$ + $z^{4}$ + $2x^{2}y^{2}$ + $2y^{2}z^{2}$ + $2z^{2}x^{2}$   $=$ $(x^{2}+y^{2}+z^{2})^2$    $=$ $10^{2}=100$