Đáp án :
`a)A=2015`
`b)a^7+b^7=1168`
Giải thích các bước giải :
`a)`
`+)x^2/(x+y)+y^2/(y+z)+z^2/(z+x)=2015 (I)`
`+)A=y^2/(x+y)+z^2/(y+z)+x^2/(z+x) (II)`
Lầy `(I)` trừ `(II),` ta được :
`2015-A=(x^2/(x+y)+y^2/(y+z)+z^2/(z+x))-(y^2/(x+y)+z^2/(y+z)+x^2/(z+x))`
`<=>2015-A=x^2/(x+y)+y^2/(y+z)+z^2/(z+x)-y^2/(x+y)-z^2/(y+z)-x^2/(z+x)`
`<=>2015-A=(x^2-y^2)/(x+y)+(y^2-z^2)/(y+z)+(z^2-x^2)/(z+x)`
`<=>2015-A=((x-y)(x+y))/(x+y)+((y-z)(y+z))/(y+z)+((z-x)(z+x))/(z+x)`
`<=>2015-A=(x-y)+(y-z)+(z-x)`
`<=>2015-A=x-y+y-z+z-x`
`<=>2015-A=(x-x)+(y-y)+(z-z)`
`<=>2015-A=0`
`<=>A=2015`
Vậy `A=2015`
`b)a+b=2; ab=-2`
`+)a^2+b^2=(a^2+2ab+b^2)-2ab=(a+b)^2-2ab=2^2-2×(-2)=4+4=8`
`+)a^3+b^3=(a+b)(a^2-ab+b^2)=2[8-(-2)]=2×10=20`
`+)a^4+b^4=(a^4+2a^2b^2+b^4)-2a^2b^2=(a^2+b^2)^2-2(ab)^2=8^2-2(-2)^2=64-8=56`
`+)a^6+b^6=(a^6+2a^3b^3+b^6)-2a^3b^3=(a^3+b^3)^2-2(ab)^3=20^2-2(-2)^3=400+16=416`
`+)a^7+b^7=(a+b)(a^6-a^5b+a^4b^2-a^3b^3+a^2b^4-ab^5+b^6)`
`<=>a^7+b^7=(a+b)[(a^6+b^6)-(a^5b-ab^5)+(a^4b^2+a^2b^4)-3a^3b^3]`
`<=>a^7+b^7=(a+b)[(a^6+b^6)-ab(a^4-b^4)+a^2b^2(a^2+b^2)-3a^3b^3]`
`<=>a^7+b^7=2[416-(-2)56+(-2)^2×8-3(-2)^3]`
`<=>a^7+b^7=2[416+112+32+24]`
`<=>a^7+b^7=2×584`
`<=>a^7+b^7=1168`
Vậy `a^7+b^7=1168`
~Chúc bạn học tốt !!!~
