\(A=\dfrac{\left(a+b\right)\left(-x-y\right)-\left(a-y\right)\left(b-x\right)}{abxy\left(xy+ay+ab+by\right)}\)

\(=\dfrac{a\left(-x-y\right)+b\left(-x-y\right)-a\left(b-x\right)+y\left(b-x\right)}{abxy\left(xy+ay+ab+by\right)}\)

\(=\dfrac{-ax-ay-bx-by-ab+ax+by-xy}{abxy\left(xy+ay+ab+by\right)}\)

\(=\dfrac{-ay-bx-ab-xy}{abxy\left(xy+ay+ab+by\right)}\)

\(=\dfrac{-xy+ay+ab+by}{abxy\left(xy+ay+ab+by\right)}=\dfrac{-1}{abxy}\)

Với \(a=\dfrac{1}{3};b=-2;x=\dfrac{3}{2};y=1\)

\(\Rightarrow A=\dfrac{-1}{\dfrac{1}{3}.\left(-2\right).\dfrac{3}{2}.1}=-1\)

Nhân khai triển tử và mẫu của B, thấy ab + bc + ca thì thay bằng 1

a) Ta có : \(a^2+1=a^2+ab+bc+ac=a\left(a+b\right)+c\left(a+b\right)=\left(a+b\right)\left(a+c\right)\)

Tương tự : \(b^2+1=\left(b+a\right)\left(b+c\right)\) ; \(c^2+1=\left(c+a\right)\left(c+b\right)\)

Suy ra \(\left(1+a^2\right)\left(1+b^2\right)\left(1+c^2\right)=\left(a+b\right)^2\left(b+c\right)^2\left(c+a\right)^2\)

Vậy \(A=\frac{\left(a+b\right)^2\left(b+c\right)^2\left(c+a\right)^2}{\left(a+b\right)^2\left(b+c\right)^2\left(c+a\right)^2}=1\)

b) Ta có ; \(a^2+2bc-1=a^2+2bc-\left(ab+bc+ac\right)=a^2-ab+bc-ac=a\left(a-b\right)-c\left(a-b\right)\)

\(=\left(a-b\right)\left(a-c\right)\)

Tương tự : \(b^2+2ac-1=\left(a-b\right)\left(c-b\right)\) ; \(c^2+2ab-1=\left(a-c\right)\left(b-c\right)\)

Suy ra \(\left(a^2+2bc-1\right)\left(b^2+2ac-1\right)\left(c^2+2ab-1\right)=\left(a-b\right)^2.\left(c-a\right)^2.\left[-\left(b-c\right)^2\right]\)

Vậy : \(B=\frac{-\left(a-b\right)^2\left(b-c\right)^2\left(c-a\right)^2}{\left(a-b\right)^2\left(b-c\right)^2\left(c-a\right)}=-1\)

a) phương trình \(x^3-3x^2+1\) có 3 nghiệm thực phân biệt là a,b,c(đề bài). Áp dụng Định lí Vi-ét cho đa thức bậc 3 ta có:\(\left\{{}\begin{matrix}a+b+c=3\\ab+bc+ac=0\\a.b.c=-1\end{matrix}\right.\)

ta có

      a+b+c=3

<=>\(\left(a+b+c\right)^2=9\)

<=>\(a^2+b^2+c^2+2ab+2bc+2ac=9\)

<=>\(a^2+b^2+c^2=9\)

<=>\(\left(a^2+b^2+c^2\right)^2=81\)

<=>\(a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+a^2c^2\right)=81\)(1)

ta có ab+bc+ac=0

   <=>\(\left(ab+bc+ac\right)^2=0\)

   <=>\(a^2b^2+b^2c^2+a^2c^2+2abc\left(a+b+c\right)=0\)

   <=>\(a^2b^2+b^2c^2+a^2c^2-2.1.3=0\)

   <=>\(a^2b^2+b^2c^2+a^2c^2=6\)(2)

Thay (2) vào (1) ta có \(a^4+b^4+c^4+2.6=81\)

                                <=>\(a^4+b^4+c^4=69\)

b) \(\dfrac{a+1}{\left(b+c\right)\left(1-a\right)+1}=\dfrac{a+1}{\left(3-a\right)\left(1-a\right)+1}=\dfrac{a+1}{3+a^2-4a+1}=\dfrac{a+1}{a^2-4a+4}=\dfrac{a+1}{\left(a-2\right)^2}\)

cmtt =>\(B=\dfrac{a+1}{\left(a-2\right)^2}+\dfrac{b+1}{\left(b-2\right)^2}+\dfrac{c+1}{\left(c-2\right)^2}\)=\(\dfrac{1}{a-2}+\dfrac{1}{b-2}+\dfrac{1}{c-2}+3\left[\dfrac{1}{\left(a-2\right)^2}+\dfrac{1}{\left(b-2\right)^2}+\dfrac{1}{\left(c-2\right)^2}\right]\)=\(\dfrac{3\left[\left(a-2\right)\left(b-2\right)\right]^2+3\left[\left(b-2\right)\left(c-a\right)\right]^2+3\left[\left(c-2\right)\left(a-2\right)\right]^2}{\left[\left(a-2\right)\left(b-2\right)\left(c-2\right)\right]^2}\)

đặt t=(a-2)(b-2);u=(b-2)(c-2);v=(c-2)(a-2)     =>t+u+v=0

B thành \(\dfrac{3\left(t^2+u^2+v^2\right)}{t.u.v}\) bạn biến đổi để xuất hiện t+u+v

=>B=\(\dfrac{3\left(t+u+v\right)^2-6\left(t.u+u.v+t.v\right)}{t.u.v}=\dfrac{-6.\left(a-2\right)\left(b-2\right)\left(c-2\right)\left(a-2+b-2+c-2\right)}{t.u.v}=\dfrac{18}{\left(a-2\right)\left(b-2\right)\left(c-2\right)}\)

(a-2)(b-2)(c-2)= abc-2(ab+bc+ac)+4(a+b+c)-8=12-9=3

Vậy B=3

c) ta có \(\dfrac{a^3}{a^2+2bc}=\dfrac{a^3}{a^2-2ac-2ab}=\dfrac{a^2}{a-2c-2b}=\dfrac{a^2}{3a-2\left(a+b+c\right)}=\dfrac{a^2}{3\left(a-2\right)}\)

cmtt =>C=\(\dfrac{a^2}{3\left(a-2\right)}+\dfrac{b^2}{3\left(b-2\right)}+\dfrac{c^2}{3\left(c-2\right)}=\dfrac{a^2\left(b-2\right)\left(c-2\right)+b^2\left(a-2\right)\left(c-2\right)+c^2\left(a-2\right)\left(b-2\right)}{3\left(a-2\right)\left(b-2\right)\left(c-2\right)}\)

bạn nhân vô thì ra C=\(\dfrac{4a^2-2a\left(ab+ac\right)-a+4b^2-2b\left(bc+ab\right)-b+4c^2-2c\left(ac+bc\right)-c}{3\left(a-2\right)\left(b-2\right)\left(c-2\right)}=\dfrac{ }{ }4\dfrac{ }{ }=\dfrac{4\left(a^2+b^2+c^2\right)-\left(a+b+c\right)+6abc}{3\left(a-2\right)\left(b-2\right)\left(c-2\right)}=\dfrac{4.9-3-6}{3.3}=\dfrac{27}{9}=3\)

a + b + c = 0

<=> (a + b + c)^2 = 0

<=> a^2 + b^2 + c^2 + 2(ab + bc + ca)  = 0

<=> a^2 + b^2 + c^2 = 0

<=> a = b = c = 0

=> Q = - 1 + 1 + 1 = 1

bai nay de

dễ thì lm giúp đi