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a, x4 +x3+x2+x-4/x-1
b, xyz+xy+yz+xz+x+y+z+1/(x+1)(y+1)
c, az+by+bx+ay/a+b
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17 tháng 7 2023
\(\left(x+y+z\right)^2=x^2+y^2+z^2+2xy+2yz+2xz=x^2+y^2+z^2+2\left(xy+yz+xz\right)\)
\(\Rightarrow2\left(xy+yz+xz\right)=\left(x+y+z\right)^2+\left(x^2+y^2+z^2\right)\)
\(\Rightarrow2\left(xy+yz+xz\right)=a^2+b\)
\(\Rightarrow xy+yz+xz=\dfrac{a^2+b}{2}\)
\(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{1}{c}\Rightarrow\dfrac{xy+yz+xz}{xyz}=\dfrac{1}{c}\)
\(\Rightarrow xyz=c\left(xy+yz+xz\right)\)
\(\Rightarrow xyz=\dfrac{\left(a^2+b\right)c}{2}\)
\(x^3+y^3+z^3-3xyz=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)\)
\(\Rightarrow x^3+y^3+z^3=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)+3xyz\)
\(\Rightarrow x^3+y^3+z^3=\left(x+y+z\right)\left(x^2+y^2+z^2-\left(xy+yz+xz\right)\right)+3xyz\)
\(\Rightarrow x^3+y^3+z^3=a\left(b-\dfrac{a^2+b}{2}\right)+3\dfrac{\left(a^2+b\right)c}{2}\)
\(\Rightarrow x^3+y^3+z^3=a\dfrac{\left(b-a^2\right)}{2}+3\dfrac{\left(a^2+b\right)c}{2}\)
a: \(\frac{x^4+x^3+x^2+x-4}{x-1}\)
\(=\frac{x^4-x^3+2x^3-2x^2+3x^2-3x+4x-4}{x-1}\)
\(=\frac{\left(x-1\right)\left(x^3+2x^2+3x+4\right)}{x-1}=x^3+2x^2+3x+4\)
c: ax+by+bx+ay
=a(x+y)+b(x+y)
=(x+y)(a+b)
\(\frac{ax+by+bx+ay}{a+b}\)
\(=\frac{\left(a+b\right)\left(x+y\right)}{a+b}\)
=x+y