![](https://rs.olm.vn/images/avt/0.png?1311)
Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
a+b+c+d=0
=>a+b=-(c+d)
=> (a+b)^3=-(c+d)^3
=> a^3+b^3+3ab(a+b)=-c^3-d^3-3cd(c+d)
=> a^3+b^3+c^3+d^3=-3ab(a+b)-3cd(c+d)
=> a^3+b^3+c^3+d^3=3ab(c+d)-3cd(c+d) ( vi a+b = - (c+d))
==> a^3 +b^^3+c^3+d^3==3(c+d)(ab-cd) (đpcm)
![](https://rs.olm.vn/images/avt/0.png?1311)
c, Ta có : a+b+c=0 ⇒ c=-(a+b)
⇒ a3+b3+c3= a3+b3-(a+b)3= x3+y3-(x3+3x2y+3xy2+y3)= x3+y3-x3-3x2y-3xy2-y3= -3x2y-3xy2= -3xy(x+y)= 3xyz(đpcm)
Câu a : Ta có :
\(x^3+x^2z+y^2z-xyz+y^3=0\)
\(\Leftrightarrow\left(x^3+y^3\right)+\left(x^2z-xyz+y^2z\right)=0\)
\(\Leftrightarrow\left(x+y\right)\left(x^2-xy+y^2\right)+z\left(x^2-xy+y^2\right)=0\)
\(\Leftrightarrow\left(x^2-xy+y^2\right)\left(x+y+z\right)=0\)
\(\Leftrightarrow x+y+z=0\)
Câu b : Khai triển VT ta có :
\(VT=\left(a+b+c\right)^3-a^3-b^3-c^3=a^3+b^3+c^3+3\left(a+b\right)\left(b+c\right)\left(c+a\right)-a^3-b^3-c^3=3\left(a+b\right)\left(b+c\right)\left(c+a\right)=VP\)
Câu c : Ta có :
\(a^3+b^3+c^3=3abc\)
\(\Leftrightarrow a^3+b^3+c^3-3abc=0\)
\(\Leftrightarrow\left(a+b\right)^3+c^3-3ab\left(a+b\right)-3abc=0\)
\(\Leftrightarrow\left(a+b+c\right)\left(a^2+2ab+b^2-bc-ca+c^2\right)-3ab\left(a+b+c\right)=0\)
\(\Leftrightarrow\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)=0\)
Luôn đúng vì \(a+b+c=0\)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
- Xét vế Trái (T): (a+b+c)3=a3+b3+c3+3ab2+3a2b+3ac2+3a2c+3bc2+3b2c+6abc
=a3+b3+c3+3(ab2+a2b+ac2+a2c+bc2+b2c+2abc)
- Xét vế Phải (P):
a3+b3+c3+3(a+b)(b+c)(a+c)=a3+b3+c3+3(a+b)(ab+bc+ac+c2)
=a3+b3+c3+3(ab2+a2b+ac2+a2c+bc2+b2c+2abc)
=>T=P
Vậy (a+b+c)3=a3+b3+c3+3(a+b)(b+c)(a+c)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có: \(VT=\left(a+b+c\right)^3-a^3-b^3-c^3\)
\(=\left[\left(a+b\right)+c\right]^3-a^3-b^3-c^3\)
\(=\left(a+b\right)^3+3\left(a+b\right)^2.c+3\left(a+b\right).c^2+c^3-a^3-b^3-c^3\)
\(=a^3+3a^2b+3ab^2+b^3+3.\left(a+b\right)^2.c+3\left(a+b\right).c^2-a^3-b^3\)
\(=3ab\left(a+b\right)+3\left(a+b\right)^2.c+3\left(a+b\right).c^2\)
\(=3\left(a+b\right)\left[\left(ab+ac\right)+\left(bc+c^2\right)\right]\)
\(=3\left(a+b\right)\left(a+c\right)\left(b+c\right)=VP\)
P/s: Bài nhà cô Yến à m? :">
![](https://rs.olm.vn/images/avt/0.png?1311)
Bài làm
a) x4+x3+2x2+x+1=(x4+x3+x2)+(x2+x+1)=x2(x2+x+1)+(x2+x+1)=(x2+x+1)(x2+1)
b)a3+b3+c3-3abc=a3+3ab(a+b)+b3+c3 -(3ab(a+b)+3abc)=(a+b)3+c3-3ab(a+b+c)
=(a+b+c)((a+b)2-(a+b)c+c2)-3ab(a+b+c)=(a+b+c)(a2+2ab+b2-ac-ab+c2-3ab)=(a+b+c)(a2+b2+c2-ab-ac-bc)
c)Đặt x-y=a;y-z=b;z-x=c
a+b+c=x-y-z+z-x=o
đưa về như bài b
d)nhóm 2 hạng tử đầu lại và 2hangj tử sau lại để 2 hạng tử sau ở trong ngoặc sau đó áp dụng hằng đẳng thức dề tính sau đó dặt nhân tử chung
e)x2(y-z)+y2(z-x)+z2(x-y)=x2(y-z)-y2((y-z)+(x-y))+z2(x-y)
=x2(y-z)-y2(y-z)-y2(x-y)+z2(x-y)=(y-z)(x2-y2)-(x-y)(y2-z2)=(y-z)(x2-2y2+xy+xz+yz)
Ta có: (a+b+c)3 = (a+b)3 + 3.(a+b)2.c + 3.(a+b).c2 + c3
= a3 + 3.a2.b +3.ab2 + b3 + 3.(a+b).c.(a+b+c)
= a3 + b3 + c3 + 3ab.(a+b) + 3.(a+b).c.(a+b+c)
= a3 + b3 + c3 + 3.(a+b).(ab+ac+bc+c2)
= a3 + b3 + c3 + 3(a+b)(b+c)(c+a) (ĐPCM)