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\(A=3\left(x^2+y^2\right)-2\left(x^3+y^3\right)\)
\(=3x^2+3y^2-2\left(x+y\right)\left(x^2-xy+y^2\right)\)
\(=3x^2+3y^2-2.1\left(x^2-xy+y^2\right)\)
\(=3x^2+3y^2-2x^2+2xy-2y^2\)
\(=x^2+2xy+y^2=\left(x+y\right)^2=1^2=1\)
\(B=x^3+y^3+3xy\left(x^2+y^2\right)+6x^2y^2\left(x+y\right)\)
\(=x^3+y^3+3xy\left[\left(x+y\right)^2-2xy\right]+6x^2y^2.1\)
\(=x^3+y^3+3xy\left(x+y\right)^2-6x^2y^2+6x^2y^2\)
\(=\left(x+y\right)\left(x^2-xy+y^2\right)+3xy\)
\(=x^2-xy+y^2+3xy\)
\(=x^2+2xy+y^2=\left(x+y\right)^2=1^2=1\)
\(x+y=1\)
\(\Leftrightarrow\)\(\left(x+y\right)^2=1\)
\(\Leftrightarrow\)\(x^2+y^2=1-2xy\)
\(x+y=1\)
\(\Leftrightarrow\)\(\left(x+y\right)^3=1\)
\(\Leftrightarrow\)\(x^3+y^3=1-3xy\)
\(H=1-3xy+3xy\left(1-2xy\right)+6x^2y^2\left(xy+y\right)\)
\(=1-6x^2y^2+6x^2y^2\left(xy+y\right)\)
\(=1-6x^2y^2\left(1-xy-y\right)\)
\(=1-6x^2y^2\left(x+y-xy-y\right)\)
\(=1-6x^2y^2\left(x-xy\right)\)
\(=1-6x^3y^2\left(1-y\right)\)
\(=1-6x^3y^2\left(x+y-y\right)\)
\(=1-6x^4y^2\)
mới ra đc đến đây
\(N=x^3+y^3+6x^2y^2\left(x+y\right)+3xy\left(x^2+y^2\right)\)
\(N=x^3+y^3+6x^2y^2+3xy\left[\left(x+y\right)^2-2xy\right]\)
\(N=\left(x+y\right)\left(x^2-xy+y^2\right)+6x^2y^2+3xy-6x^2y^2\)
\(N=x^2-xy+y^2+3xy\)
\(N=\left(x+y\right)^2\)
\(N=1\)
\(x^3+y^3+6x^2y^2\left(x+y\right)+3xy\left(x^2+y^2\right)\)
\(=\left(x+y\right)\left(x^2-xy+y^2\right)+6x^2y^2\left(x+y\right)+3xy\left[\left(x+y\right)^2-2xy\right]\)
\(=x^2-xy+y^2+6x^2y^2+3xy-6x^2y^2\)( Do \(x+y=1\))
\(=\left(x+y\right)^2-2xy-xy+3xy+6x^2y^2-6x^2y^3\)
\(=\left(x+y\right)^2=1^2=1\)
\(C=\left(x^3+y^3\right)+3xy\left(x^2+y^2+2xy\left(x+y\right)\right)\)
\(C=\left(x^3+y^3+3x^2y+3xy^2-3x^2y-3xy^2\right)+3xy\left(x^2+y^2+2xy\right)\) (vì x+y=1)
\(C=\left(x+y\right)^3-3x^2y-3xy^2+3xy\left(x+y\right)^2\)
\(C=1^3-3xy\left(x+y\right)+3xy.1^2\) (vì x+y=1)
\(C=1-3xy+3xy\)(vì x+y=1)
\(C=1\)
\(D=2\left(\left(x+y\right)^3-3xy\left(x+y\right)\right)-3\left(\left(x+y\right)^2-2xy\right)\)
\(D=2\left(1^3-3xy\right)-3\left(1^2-2xy\right)\)(vì x+y=1)
\(D=2-6xy-3+6xy\)
\(D=-1\)
\(3,x=\dfrac{1}{2},y=-1\)
\(\Rightarrow C=\dfrac{1}{2}\left[\left(\dfrac{1}{2}\right)^2+1\right]-\left(\dfrac{1}{2}\right)^2\left(\dfrac{1}{2}-1\right)-1\left[\left(\dfrac{1}{2}\right)^2-\dfrac{1}{2}\right]\)
\(\Rightarrow C=\dfrac{1}{2}\left(\dfrac{1}{4}+1\right)-\dfrac{1}{4}\left(-\dfrac{1}{2}\right)-\left(\dfrac{1}{4}-\dfrac{1}{2}\right)\)
\(\Rightarrow C=\dfrac{1}{2}.\dfrac{5}{4}+\dfrac{1}{8}-\left(-\dfrac{1}{4}\right)\)
\(\Rightarrow C=\dfrac{5}{8}+\dfrac{1}{8}+\dfrac{1}{4}\)
\(\Rightarrow C=1\)
\(4,x=\dfrac{1}{2},y=-100\)
\(\Rightarrow D=\dfrac{1}{2}\left[\left(\dfrac{1}{2}\right)^2+100\right]-\left(\dfrac{1}{2}\right)^2\left(\dfrac{1}{2}-100\right)-100\left[\left(\dfrac{1}{2}\right)^2-\dfrac{1}{2}\right]\)
\(\Rightarrow D=\dfrac{1}{2}\left(\dfrac{1}{4}+100\right)-\dfrac{1}{4}\left(-\dfrac{199}{2}\right)-100\left(\dfrac{1}{4}-\dfrac{1}{2}\right)\)
\(\Rightarrow D=\dfrac{1}{2}.\dfrac{401}{4}+\dfrac{199}{8}-100.\left(-\dfrac{1}{4}\right)\)
\(\Rightarrow D=\dfrac{401}{8}+\dfrac{199}{8}+25\)
\(\Rightarrow D=100\)
3: C=x^3-xy-x^3-x^2y+x^2y-xy
=-2xy=-2*1/2*(-1)=1
4: D=x^3-xy-x^3-x^2y+x^2y-xy
=-2xy
=-2*1/2*(-100)=100
