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\(a;\left(x+y\right)^5-x^5-y^5=x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5-x^5-y^5\)
\(=5x^4y+10x^3y^2+10x^2y^3+5xy^4\)
\(=5xy\left(x^3+2x^2y+2xy^2+y^3\right)\)
\(=5xy\left[\left(x^3+y^3\right)+\left(2x^2y+2xy^2\right)\right]\)
\(=5xy\left[\left(x+y\right)\left(x^2-xy+y^2\right)+2xy\left(x+y\right)\right]\)
\(=5xy\left(x+y\right)\left(x^2-xy+y^2+2xy\right)\)
\(=5xy\left(x+y\right)\left(x^2+xy+y^2\right)\left(dpcm\right)\)
\(\left(ab+bc+ca\right)^2=a^2b^2+b^2c^2+c^2a^2+2a^2bc+2ab^2c+2abc^2\)
\(=a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right)\)
\(=a^2b^2+b^2c^2+c^2a^2+2abc\cdot0=a^2b^2+b^2c^2+c^2a^2\left(dpcm\right)\)
a)\(x^3+y^3+z^3-3xyz\\ \left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz\\ =\left[\left(x+y\right)^3+z^3\right]-\left[3xyz+3xy\left(x+y\right)\right]\\=\left(x+y+z\right)\left[\left(x+y\right)^2-z\left(x+y\right)+z^2\right] \\ =\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2\right)-3xy\left(x+y+z\right)\\ =\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2-3xy\right)\\ =\left(x+y+z\right)\left(x^2+y^2+x^2-xy-xz-yz\right)\)
1. Ta có : x + y + z = 0 \(\Rightarrow\)( x + y + z )2 = 0 \(\Rightarrow\)x2 + y2 + z2 = - 2 ( xy + yz + xz )\(S=\frac{x^2+y^2+z^2}{\left(y-z\right)^2+\left(z-x\right)^2+\left(x-y\right)^2}=\frac{-2\left(xy+yz+xz\right)}{2\left(x^2+y^2+z^2\right)-2\left(yz+xz+xy\right)}\)
\(S=\frac{-2\left(xy+yz+xz\right)}{-4\left(xy+yz+xz\right)-2\left(yz+xz+xy\right)}=\frac{-2\left(xy+yz+xz\right)}{-6\left(xy+yz+xz\right)}=\frac{1}{3}\)
a. \(\left(x-y\right)\left(x^4+x^3y+x^2y^2+xy^3+y^4\right)\)
\(\Rightarrow x^5+x^4y+x^3y^2+x^2y^3+y^5-yx^4-x^3y^2-x^2y^3-xy^4-y^5=VP\)
\(\Rightarrow dpcm\)
b. \(\left(x+y\right)\left(x^4-x^3y+x^2y^2-xy^3+y^4\right)\)
\(\Rightarrow x^5-x^4y+x^3y^2-x^2y^3+xy^4+yx^4-x^3y^2-xy^4+y^5=VP\)
\(\Rightarrow dpcm\)
c.d làm tương tự
Bài làm
a) Biến đổi vế trái, ta được:
\(VT=\left(x-y\right)\left(x^4+x^3y+x^2y^2+xy^3+y^4\right)\)
\(=x^5+x^4y+x^3y^2+x^2y^3+xy^4-x^4y-x^3y^2-x^2y^3-xy^4-y^5\)
\(=\left(x^5-y^5\right)+\left(x^4y-x^4y\right)+\left(x^3y^2-x^3y^2\right)+\left(x^2y^3-x^2y^3\right)+\left(xy^4-xy^4\right)\)
\(=x^5-y^5=VP\left(đpcm\right)\)
b) Biến đổi vế trái, ta có:
\(VT=\left(x+y\right)\left(x^4-x^3y+x^2y^2-xy^3+y^4\right)\)
\(=x^5-x^4y+x^3y^2-x^2y^3+xy^4+x^4y-x^3y^2+x^2y^3-xy^4+y^5\)
\(=\left(x^5+y^5\right)+\left(-x^4y+x^4y\right)+\left(x^3y^2-x^3y^2\right)+\left(-x^2y^3+x^2y^3\right)+\left(xy^4-xy^4\right)\)
\(=x^5+y^5=VP\left(đpcm\right)\)
c) Biến đổi vế trái, ta có:
\(VT=\left(a+b\right)\left(a^3-a^2b+ab^2-b^3\right)\)
\(=a^4-a^3b+a^2b^2-ab^3+a^3b-a^2b^2+ab^3-b^4\)
\(=\left(a^4-b^4\right)+\left(-a^3b+a^3b\right)+\left(a^2b^2-a^2b^2\right)+\left(-ab^3+ab^3\right)\)
\(=a^4-b^4=VP\left(đpcm\right)\)
d) Đây là hằng đẳng thức, như vế phải hình như bạn viết bị sai, mik sửa là vế phải nha.
\(\left(a+b\right)\left(a^2-ab+b^2\right)=a^3+b^3\)
Biến đổi vế trái, ta có:
\(VT=\left(a+b\right)\left(a^2-ab+b^2\right)\)
\(=a^3-a^2b+ab^2+a^2b-ab^2+b^3\)
\(=\left(a^3+b^3\right)+\left(-a^2b+a^2b\right)+\left(ab^2-ab^2\right)\)
\(=a^3+b^3=VP\left(đpcm\right)\)
2) Để sau đi (em chưa nghĩ ra)
3) \(A=\left(x+y\right)\left(x^2-y^2\right)+\left(y+z\right)\left(y^2-z^2\right)+\left(z+x\right)\left(z^2-x^2\right)\)
\(=\left(x+y\right)^2\left(x-y\right)+\left(y+z\right)^2\left(y-z\right)+\left(z+x\right)^2\left(z-x\right)\)
Đặt x - y = a; y - z = b => z - x = -(a+b)
\(A=\left(x+y\right)^2a+\left(y+z\right)^2b-\left(z+x\right)^2a-\left(z+x\right)^2b\)
\(=a\left[\left(x+y\right)^2-\left(z+x\right)^2\right]+b\left[\left(y+z\right)^2-\left(z+x\right)^2\right]\)
\(=\left(x-y\right)\left(x+y-z-x\right)\left(x+y+z+x\right)+\left(y-z\right)\left(y+z-z-x\right)\left(y+z+z+x\right)\)
\(=\left(x-y\right)\left(y-z\right)\left(2x+y+z\right)-\left(y-z\right)\left(x-y\right)\left(2z+x+y\right)\)
\(=\left(x-y\right)\left(y-z\right)\left(x-z\right)\)
Em tính sai sót chỗ nào thì thông cảm cho em ạ :>
1) a4+b4+c4−2a2b2−2a2c2−2b2c2
=2(a4+b4+c4-4a2b2-4a2c2-4b2c2)
=2a4+2b4+2c4-4a2b2-4a2c2-4b2c2
=(a4-2a2b2+b4)+(a4-2a2c2+c4)+(b4-2b2c2+c4
a ) Ta có : \(\left(ab+1\right)^2\ge4ab\)
\(\Leftrightarrow a^2b^2+2ab+1-4ab\ge0\)
\(\Leftrightarrow\left(ab-1\right)^2\ge0\)
=> BĐT luôn đúng
Dấu " = " xảy ra \(\Leftrightarrow ab=1\)
b ) Áp dụng BĐT Bunhiacopxki , ta có :
\(\left(ab+1.2\right)^2\le\left(a^2+1^2\right)\left(b^2+2^2\right)=\left(a^2+1\right)\left(b^2+4\right)\)
Dấu " = " xảy ra \(\Leftrightarrow2a=b\)
c ) Áp dụng BĐT Cô - si cho 2 số không âm , ta có :
\(4a^2+b^2\ge2\sqrt{4a^2.b^2}=4ab\)
\(\Rightarrow2\left(4a^2+b^2\right)\ge4a^2+4ab+b^2=\left(2a+b\right)^2\)
Dấu " = " xảy ra \(\Leftrightarrow2a=b\)
d ) \(x^5+y^5\ge xy\left(x^3+y^3\right)\)
\(\Leftrightarrow x^5-x^4y-y^4x+y^5\ge0\)
\(\Leftrightarrow\left(x^4-y^4\right)\left(x-y\right)\ge0\)
\(\Leftrightarrow\left(x-y\right)^2\left(x+y\right)\left(x^2+y^2\right)\ge0\)
Vì x ; y > 0 => BĐT luôn đúng
Dấu " = " xảy ra \(\Leftrightarrow x=y\)
\(\left(ab+bc+ca\right)^2=a^2b^2+2ab^2c+2a^2bc+b^2c^2+2c^2ba+c^2a^2=\left(a^2b^2+b^2c^2+c^2a^2\right)+2abc\left(a+b+c\right)=a^2b^2+b^2c^2+c^2a^2\)