Giải:
Từ \(a+b+c+d=0\Leftrightarrow a+c=-\left(b+d\right)\)
\(\Leftrightarrow\left(a+c\right)^3=-\left(b+d\right)^3\)
\(\Leftrightarrow a^3+c^3+3ac\left(a+c\right)=-\left[b^3+d^3+3bd\left(b+d\right)\right]\)
\(\Leftrightarrow VT=a^3+b^3+c^3+d^3=-3bd\left(b+d\right)-3ac\left(a+c\right)\)
\(=-3bd\left(b+d\right)+3ac\left(b+d\right)=3\left(ac-bd\right)\left(b+d\right)=VP\) (Đpcm)
1, Cho a, b, c thỏa mãn :
\(\left\{{}\begin{matrix}\left(a+b\right)\left(b+c\right)\left(c+a\right)=abc\\\left(a^3+b^3\right)\left(b^3+c^3\right)\left(c^3+a^3\right)=a^3b^3c^3\end{matrix}\right.\\ CMR:abc=0\)
2, a, CMR nếu x + y + z = 0 thì :
\(2\left(x^5+y^5+z^5\right)=5xyz\left(x^2+y^2+z^2\right)\)
b, Cho a, b,c, d thỏa mãn : a + b + c + d = 0
CMR : \(a^3+b^3+c^3+d^3=3\left(ab-cd\right)\left(c+d\right)\)
Mọi người giải giúp mk, đc bài nào hay bài ấy nhé!
1,Cho \(a^2+b^2+c^2+3=2\left(a+b+c\right)\) .Cmr: \(a=b=c=1\)
2,Cho \(\left(a+b+c\right)^2=3\left(ab+ac+bc\right)\) .Cmr: \(a=b=c\)
3,Cho \(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=\left(a+b-2c\right)^2+\left(b+c-2a\right)^2+\left(c+a-2b\right)^2\) .Cmr: \(a=b=c\)
4,Cho a,b,c,d là các số khác 0 và:
\(\left(a+b+c+d\right)\left(a-b-c+d\right)=\left(a-b+c-d\right)\left(a+b-c-d\right)\) .Cmr: \(\dfrac{a}{c}=\dfrac{b}{d}\)
5,Cho \(x^2-y^2-z^2=0\) .Cmr: \(\left(5x-3y+4z\right)\left(5x-3y-4z\right)=\left(3x-5y\right)^2\)
HELP ME!mik cần gấp lắm rồi!Thank trước nhé!
Rút gọn
a) \(A=\left(3x+1\right)^2-2\left(3x+1\right)\left(3x+5\right)+\left(5x+5\right)^2\)
b) \(B=\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{18}+1\right)\left(3^{32}+1\right)\)
c) \(C=\left(a+b-c\right)^2+\left(a-b+c\right)^2-2\left(b-c\right)^2\)
d) \(D=\left(a+b+c\right)^2+\left(a-b-c\right)^2+\left(b-c-a\right)^2+\left(c-b-a\right)^2\)
e)\(E=\left(a+b+c+d\right)^2+\left(a+b-c-d\right)^2+\left(a+c-b-d\right)^2+\left(a+d-b-c\right)^2\)
1. Cho x + y = 1; \(x^2+y^2=13\). Tính \(x^3+y^3\)
2. Cho a+b+c+d=0. CMR: \(a^3+b^3+c^3+d^3=3\left(ac-bd\right)\left(b+d\right)\)
3. Cho x-y= -1; Tính GTBT: P = \(2\left(x^3-y^3\right)+3\left(x^2+y^2\right)\)
Chứng minh rằng :
a) \(\left(a+b\right)\left(a^2-ab+b^2\right)+\left(a-b\right)\left(a^2+ab+b^2\right)=2a^3\)
b) \(a^3+b^3=\left(a+b\right)\left[\left(a-b\right)^2+ab\right]\)
c) \(\left(a^2+b^2\right)\left(c^2+d^2\right)=\left(ac+bd\right)^2+\left(ad-bc\right)^2\)
Bài tập về hằng đẳng thức
CMR:
a) \(\left(a+b\right)\left(a^2-b+b^2\right)+\left(a-b\right)\left(a^2+ab+b^2\right)=2a^3\)
b) \(a^3+b^3=\left(a+b\right)\left[\left(a-b\right)^2+ab\right]\)
c) \(\left(a^2+b^2\right)\left(c^2+d^2\right)=\left(ac+bd\right)^2+\left(ad-bc\right)^2\)
cho a,b,c là 3 độ dài tam giác. cmr:
a) a3(b2-c2)+b3(c2-b2)+c3(a2-b2)<0 với a<b<c
b) \(a\left(b-c\right)^2+b\left(c-a\right)^2+c\left(a+b\right)^2>a^3+b^3+c^3\)
Bài 1: Tìm x
a) \(\left(5-2x\right)^2-16=0\)
b) \(x^2-4x=29\)
c) \(\left(x-3\right)^3-\left(x-3\right).\left(x^2+3x+9\right)+9.\left(x+1\right)^2=15\)
d) \(2.\left(x-5\right).\left(x+5\right)-\left(x+2\right).\left(2x-3\right)+x.\left(x^2-8\right)=\left(x+1\right).\left(x^2-x+1\right)\)
Bài 2: Rút gọn
a) \(\left(x^2+x+1\right).\left(x^2-x+1\right).\left(x^2-1\right)\)
b) \(\left(a+b-c\right)^2+\left(a-b+c\right)^2-2.\left(b-c\right)^2\)
c) \(\left(a+b+c\right)^2-\left(a+b\right)^2-\left(a+c\right)^2-\left(b+c\right)^2\)
d) \(\left(a+b+c\right)^2+\left(a-b-c\right)^2+\left(b-c-a\right)^2+\left(c-a-b\right)^2\)
b) Cho \(x-y=1\), tính \(B=x^3-y^3-3xy\)
c) Cho \(x-y=2\), tính \(C=2\left(x^3-y^3\right)-3\left(x+y\right)^2\)
d) Cho \(a+b=1\), tính \(D=a^3+b^3+3ab\left(a^2+b^2\right)+6a^2b^2\left(a+b\right)\)
