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a) \(\left(x+y-z\right)^2=\left[\left(x+y\right)-z\right]^2\)
\(=\left(x+y\right)^2-2\left(x+y\right)z+z^2\)
\(=x^2+2xy+y^2-2zx-2yz+z^2\)
\(=x^2+y^2+z^2+2xy-2yz-2zx\)
b) \(\left(x-y\right)\left(x^3+x^2y+xy^2+y^3\right)\)
\(=x^4+x^3y+x^2y^2+xy^3-x^3y-x^2y^2-xy^3-y^4\)
\(=x^4-y^4\)
c) \(\left(x+y\right)\left(x^4-x^3y+x^2y^2-xy^3+y^4\right)\)
\(=x^5-x^4y+x^3y^2-x^2y^3+xy^4+x^4y-x^3y^2+x^2y^3-xy^4+y^5\)
\(=x^5+y^5\)
a) Ta có: \(VP=x^2+y^2+z^2-2xy+2yz-2zx\)
\(=\left(x^2-xy-xz\right)+\left(y^2-xy+yz\right)+\left(z^2-yz-zx\right)\)
\(=x\left(x-y-z\right)+y\left(y-x+z\right)+z\left(z-y-x\right)\)
\(=x\left(x-y-z\right)-y\left(x-y-z\right)-z\left(x-y-z\right)\)
\(=\left(x-y-z\right)\left(x-y-z\right)\)
\(=\left(x-y-z\right)^2=VT\)(đpcm)
b) Ta có: \(VP=x^2+y^2+z^2+2xy-2yz-2zx\)
\(=\left(x^2+xy-zx\right)+\left(y^2+xy-2yz\right)+\left(z^2-yz-zx\right)\)
\(=x\left(x+y-z\right)+y\left(x+y-z\right)+z\left(z-y-x\right)\)
\(=\left(x+y-z\right)\left(x+y\right)-z\left(x+y-z\right)\)
\(=\left(x+y-z\right)\left(x+y-z\right)\)
\(=\left(x+y-z\right)^2=VT\)(đpcm)
c) Ta có: \(VP=x^4-y^4\)
\(=\left(x^2-y^2\right)\left(x^2+y^2\right)\)
\(=\left(x-y\right)\left(x+y\right)\left(x^2+y^2\right)\)
\(=\left(x-y\right)\left(x^3+xy^2+x^2y+y^3\right)=VT\)(đpcm)
d) Ta có: \(VT=\left(x+y\right)\left(x^4-x^3y+x^2y^2-xy^3+y^4\right)\)
\(=x^5-x^4y+x^3y^2-x^2y^3+xy^4+x^4y-x^3y^2+x^2y^3-xy^4+y^5\)
\(=x^5+y^5=VP\)(đpcm)
a) Ta có:\(\left(x+y\right)^2=5^2\)(Vì x + y = 5)
\(\Leftrightarrow x^2+2xy+y^2=25\)
\(\Leftrightarrow x^2+2.4+y^2=25\)
\(\Leftrightarrow x^2+8+y^2=25\)
\(\Leftrightarrow x^2+y^2=17\)
b) \(\left(x+y\right)^2=3^2\)(Vì x + y = 3)
\(\Leftrightarrow x^2+2xy+y^2=9\)
\(\Leftrightarrow2xy+5=9\)
\(\Leftrightarrow2xy=4\)
\(\Leftrightarrow xy=2\)
\(x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)\)
\(=3\left(5-2\right)=9\)
a) ta có:(x+y)2=x2+2xy+y2=>x2+y2=(x+y)2-2xy
thay x+y=5;xy=4 vào biểu thức ta có:
52-2×4=25-8=17
TL:
a) \(x^2+y^2=x^2+2xy+y^2-2xy\)
\(=\left(x+y\right)^2-2xy=25-12=13\)