K
Khách

Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.

11 tháng 8 2016

Đặt x = a+b , y = b+c , z = c+a

Khi đó : \(\left(a+b\right)^3+\left(b+c\right)^3+\left(c+a\right)^3-3\left(a+b\right)\left(b+c\right)\left(c+a\right)\)

\(=x^3+y^3+z^3-3xyz=\left(x+y\right)^3+z^3-3xy\left(x+y\right)-3xyz\)

\(=\left(x+y+z\right)\left(x^2+y^2+z^2+2xy-yz-xz\right)-3xy\left(x+y+z\right)\)

\(=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)\)

\(=\frac{x+y+z}{2}\left[\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\right]\)

Thay x,y,z bởi a,b,c vào và rút gọn :)

10 tháng 8 2016
2a^3 + 2b^3 + 2 c^3 -6abc
10 tháng 8 2016

Đặt\(a+b=x\)

\(b+c=y\)

\(c+a=z\)

\(\Rightarrow x^3+y^3+z^3-3xyz=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)\)

\(=\frac{1}{2}\left(x+y+z\right)\left[\left(x-y\right)^2+\left(y-z\right)^2+\left(x-z\right)^2\right]\)

\(=\left(a+b+c\right)\left[\left(a-c\right)^2+\left(a-b\right)^2+\left(b-c\right)^2\right]\)

6 tháng 8 2016

(a+b)3+(b+c)+(c+a)3-3(a+b)(b+c)(c+a)

=(a+b+b+c+c+a)[ (a+b)2+(b+c)2+(c+a)2-(a+b)(b+c)-(b+c)(c+a)-(c+a)(a+b)]

=2(a+b+c)[(a+b)2+(b+c)2+(c+a)-(a+b)(b+c)-(b+c)(c+a)-(c+a)(a+b)]

18 tháng 8 2016

Áp dụng hđt a3+b3+c3=3abc ta có

(a+b)3+(b+c)3+(c+a)3=3(a+b)(b+c)(c+a)

→đề bài có giá trị =0

 

 

26 tháng 11 2021

\(B=\left(\dfrac{a-b}{a^2+ab}-\dfrac{a}{b^2+ab}\right):\left(\dfrac{b^3}{a^3-ab^2}+\dfrac{1}{a+b}\right)\)

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

    \(=\dfrac{b\left(a-b\right)-a^2}{ab\left(a+b\right)}:\dfrac{b^3+a\left(a-b\right)}{a\left(a-b\right)\left(a+b\right)}\)

    \(=\dfrac{ab-b^2-a^2}{ab\left(a+b\right)}\cdot\dfrac{a\left(a-b\right)\left(a+b\right)}{a^2-ab+b^3}\)

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

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

Đề lỗi rồi chứ mình ko rút gọn đc nữa

20 tháng 8 2018

Ta có A=\(\dfrac{a^3\left(b-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}=\dfrac{a^3\left(b-c\right)+b^3c-c^3b-a\left(b^3-c^3\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}=\dfrac{a^2\left(b-c\right)+bc\left(b^2-c^2\right)-a\left(b-c\right)\left(b^2+bc+c^2\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)

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

10 tháng 9 2018

Sao lại bằng -(a + b + c) vậy bạn?

\(\left(a+b+c\right)^3=\left[\left(a+b\right)+c\right]^3=\left(a+b\right)^3+c^3+3\left(a+b\right)c\left(a+b+c\right)\)

\(=a^3+b^3+c^3+3ab\left(a+b\right)+3\left(a+b\right)c\left(a+b+c\right)\)

\(=a^3+b^3+c^3+3\left(a+b\right)\left(ab+ca+bc+c^2\right)\)

\(=a^3+b^3+c^3+3\left(a+b\right)\left(b+c\right)\left(c+a\right)\)

\(=a^3+b^3+c^3+3\left(abc+c^2a+b^2c+bc^2+a^2b+ca^2+ab^2+abc\right)\)

\(=a^3+b^3+c^3+3\left[ab\left(a+b+c\right)+bc\left(a+b+c\right)+ca\left(a+b+c\right)\right]\)

\(=a^3+b^3+c^3+3\left(a+b+c\right)\left(ab+bc+ca\right)\)

\(\Rightarrow\)\(a^3+b^3+c^3=\left(a+b+c\right)^3-3\left(a+b+c\right)\left(ab+bc+ca\right)\)

Lại có: \(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=\left(a-b+b-c+c-a\right)^2\)

\(-2\left[\left(a-b\right)\left(b-c\right)+\left(b-c\right)\left(c-a\right)+\left(c-a\right)\left(a-b\right)\right]\)

\(=-2\left(ab-ca-b^2+bc+bc-ab-c^2+ca+ca-bc-a^2+ab\right)\)

\(=2\left(a^2+b^2+c^2-ab-bc-ca\right)=2\left(a+b+c\right)^2-6\left(ab+bc+ca\right)\)

\(\Rightarrow\)\(P=\frac{\left(a+b+c\right)^3-3\left(a+b+c\right)\left(ab+bc+ca\right)}{2\left(a+b+c\right)^2-6\left(ab+bc+ca\right)}\)

\(=\frac{\left(a+b+c\right)\left[\left(a+b+c\right)^2-3\left(ab+bc+ca\right)\right]}{2\left[\left(a+b+c\right)^2-3\left(ab+bc+ca\right)\right]}=\frac{a+b+c}{2}\)