Rút gọn
P=\(\frac{\left(x-y\right)\left(y-z\right)\left(z-x\right)}{\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3}\)
thực hiện phép tính
a,\(x^3+\left[\frac{x\left(2y^3-x^3\right)}{x^3+y^3}\right]^3-\left[\frac{y\left(2x^3-y^3\right)}{x^3+y^3}\right]^3\)
b,\(\frac{\frac{x\left(x+y\right)}{x-y}+\frac{x\left(x+z\right)}{x-z}}{1+\frac{\left(y-z\right)^2}{\left(x-y\right)\left(x-z\right)}}+\frac{\frac{y\left(y+z\right)}{y-z}+\frac{y\left(y+x\right)}{y-x}}{1+\frac{\left(z-x\right)^2}{\left(y-z\right)\left(y-x\right)}}+\frac{\frac{z\left(z+x\right)}{z-x}+\frac{z\left(z+y\right)}{z-y}}{1+\frac{\left(x-y\right)^2}{\left(z-x\right)\left(z-y\right)}}\)
c,\(\left[\frac{y+z-2x}{\frac{\left(y-z\right)^3}{y^3-z^3}+\frac{\left(x-y\right)\left(x-z\right)}{y^2+yz+z^2}}+\frac{z+x-2y}{\frac{\left(z-x\right)^3}{z^3-x^3}+\frac{\left(y-z\right)\left(y-x\right)}{z^2+xz+x^2}}+\frac{x+y-2z}{\frac{\left(x-y\right)^3}{x^3-y^3}+\frac{\left(z-x\right)\left(z-y\right)}{x^2+xy+y^2}}\right]:\frac{1}{x+y+z}\)
Rút gọn
P = \(\frac{\left(x-y\right)\left(y-z\right)\left(z-x\right)}{\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3}\)
Rút gọn: \(\frac{x^3+y^3+z^3-3xyz}{xy^2+xz\left(2y+z\right)}.\frac{x\left(y^2+z\right)+y\left(x-xy\right)}{\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2}\)
\(x\ne y\ne z\ne0\)
1rút gọn\(\frac{x^2+y^2+z^2}{\left(y-z\right)^2+\left(z-x\right)^2+\left(x-y\right)^2}\)biết rằng x+y+z=0
2 rút gọn các phân thức
a,\(\frac{x^3-y^3+z^3+3xyz}{\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2}\)
b,\(\frac{x^3+y^3+z^3-3xyz}{\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2}\)
1. Cho các số x, y, z thỏa mãn : (x + y)(y + z)(z + x) = 4. CMR: \(\left(x^2-y^2\right)^3\)+ \(\left(y^2-z^2\right)^3\)+ \(\left(z^2-x^2\right)^3\)= 12 (x - y)(y - z)(z - x)
2. Rút gọn: \(\dfrac{\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3}{\left(x^2-y^2\right)^3+\left(y^2-z^2\right)^3+\left(z^2-x^2\right)^3}\) biết (x + y)(y + z)(z + x) = 1
3. Cho a, b, c ≠ 0 thỏa mãn: a + b + c = \(a^2+b^2+c^2\) = 2. CMR: \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{1}{abc}\)
MONG MN GIẢI GIÚP EM Ạ!!! EM ĐANG CẦN GẤP ! CẢM ƠN MN NHIỀU
Hầy mình không nghĩ lớp 7 đã phải làm những bài biến đổi như thế này. Cái này phù hợp với lớp 8-9 hơn.
1.
Đặt $x^2-y^2=a; y^2-z^2=b; z^2-x^2=c$.
Khi đó: $a+b+c=0\Rightarrow a+b=-c$
$\text{VT}=a^3+b^3+c^3=(a+b)^3-3ab(a+b)+c^3$
$=(-c)^3-3ab(-c)+c^3=3abc$
$=3(x^2-y^2)(y^2-z^2)(z^2-x^2)$
$=3(x-y)(x+y)(y-z)(y+z)(z-x)(z+x)$
$=3(x-y)(y-z)(z-x)(x+y)(y+z)(x+z)$
$=3.4(x-y)(y-z)(z-x)=12(x-y)(y-z)(z-x)$
Ta có đpcm.
Bài 2:
Áp dụng kết quả của bài 1:
Mẫu:
$(x^2-y^2)^3+(y^2-z^2)^3+(z^2-x^2)^3=3(x-y)(y-z)(z-x)(x+y)(y+z)(z+x)=3(x-y)(y-z)(z-x)(1)$
Tử:
Đặt $x-y=a; y-z=b; z-x=c$ thì $a+b+c=0$
$(x-y)^3+(y-z)^3+(z-x)^3=a^3+b^3+c^3$
$=(a+b)^3-3ab(a+b)+c^3=(-c)^3-3ab(-c)+c^3=3abc$
$=3(x-y)(y-z)(z-x)(2)$
Từ $(1);(2)$ suy ra \(\frac{(x-y)^3+(y-z)^3+(z-x)^3}{(x^2-y^2)^3+(y^2-z^2)^3+(z^2-x^2)^3}=1\)
Bài 3:
\(ab+bc+ac=\frac{(a+b+c)^2-(a^2+b^2+c^2)}{2}=\frac{2^2-2}{2}=1\)
Do đó:
\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{ab+bc+ac}{abc}=\frac{1}{abc}\)
Ta có đpcm.
\(\frac{\left(X^2-y^2\right)^3+\left(y^2-z^2\right)^3+\left(z^2-x^2\right)^3}{\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3}\)
Rút gọn phân thức
Giúp mình nha
Cho x, y, z khác 0 và x+y+z=2008
Tính P=\(\frac{x^3}{\left(x-y\right)\left(x-z\right)}+\frac{y^3}{\left(y-x\right)\left(y-z\right)}+\frac{z^3}{\left(z-x\right)\left(z-y\right)}\)
\(P=\frac{x^3}{\left(x-y\right)\left(x-z\right)}+\frac{y^3}{\left(y-x\right)\left(y-z\right)}+\frac{z^3}{\left(z-x\right)\left(z-y\right)}\)
\(=\frac{-x^3\left(y-z\right)-y^3\left(z-x\right)-z^3\left(x-y\right)}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}\)
\(=\frac{-x^3y+x^3z-y^3z+y^3x-z^3x+z^3y}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}\)
\(=\frac{\left(x-y\right)\left(z-x\right)\left(y-z\right)\left(x+y+z\right)}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}\)
\(=x+y+z=2008\)
Rút gọn phân thức
1, \(\frac{x^2+y^2-1+2xy}{x^2-y^2+1+2x}\)
2, \(\frac{x^4-y^4}{x^3+y^3}\)
3, \(\frac{x^3+y^3+z^3-3xyz}{\left(x-y\right)^2+\left(x-z\right)^2+\left(y-z\right)^2}\)
4, \(\frac{\left(x^2-y^2\right)^3+\left(y^2-z^2\right)^3+\left(z^2-x^2\right)^3}{\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3}\)
5, \(\frac{x^3-7x+6}{x^2\left(x-3\right)^2+4x\left(3-x\right)^2+4\left(x-3\right)^2}\)
Rút gọn các phân thức sau:
a) \(\frac{x^3-y^3+z^3+3xyz}{\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2}\)
b)\(\frac{x^3+y^3+z^3-3xyz}{\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2}\)