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\(P=x^3+y^3+2021xy\)
\(=\left(x+y\right)^3-3xy\left(x+y\right)+2021xy\)
\(=\left(\dfrac{2021}{3}\right)^3-2021xy+2021xy\)
\(=\dfrac{8254655261}{27}\)
\(P=x^3+y^3+2021xy\)
\(=\left(x+y\right)^3-3xy\left(x+y\right)+2021xy\)
\(=\left(\dfrac{2021}{3}\right)^3=\dfrac{8254655261}{27}\)
\(A=2x^2+y^2-2x+2xy+2y+3=y^2+2y\left(x+1\right)+\left(x+1\right)^2+\left(x^2-4x+4\right)-2=\left(y+x+1\right)^2+\left(x-2\right)^2-2\ge-2\)
\(minA=-2\Leftrightarrow\)\(\left\{{}\begin{matrix}x=2\\y=-3\end{matrix}\right.\)
\(P=x^3+2021xy+y^3\)
\(=\left(x+y\right)^3-3xy\left(x+y\right)+2021xy\)
\(=\left(\dfrac{2021}{3}\right)^3\)
\(=\dfrac{8254655261}{27}\)
P(x,y) = x^3 - 3x^2 + 3x^2y + 3xy^2 + y^3 - 3y^2 - 6xy + 3x + 3y
= ( x^3 + 3x^2y + 3xy^2 + y^3 ) - ( 3x^2 + 3y^2 + 6xy ) + ( 3x + 3y)
= ( x+ y)^3 - 3 ( x^2 + 2xy + y^2) + 3 ( x+ y)
= ( x+ y)^3 - 3 ( x+ y)^2 + 3(x +y)
Thay x+ y = 101 ta có :
= 101^3 - 3.101^2 + 3.101
= 101 . ( 101^2 - 3.101 + 3 )
= 101 .9901
= 1000001
1/ \(3x^2+6x+3-3y^2=3x^2+3x+3x+3-3y^2\)
\(=3\left(x^2+2x+1-y^2\right)\)
\(=3\left[\left(x^2+2x+1\right)-y^2\right]\)
\(=3\left[\left(x+1\right)^2-y^2\right]\)
\(=3\left(x+1-y\right)\left(x+1+y\right)\)
2/ \(25-x^2-y^2+2xy=5^2-\left(x^2+y^2-2xy\right)\)
\(=5^2-\left(x-y\right)^2\)
\(=\left[5-\left(x-y\right)\right]\left(5+x+y\right)\)
\(=\left(5-x+y\right)\left(5+x+y\right)\)
3/ \(3x-3y-x^2+2xy-y^2=3\left(x-y\right)-\left(x^2-2xy+y^2\right)\)
\(=3\left(x-y\right)-\left(x-y\right)^2\)
\(=\left(x-y\right)\left[3-\left(x-y\right)\right]\)
\(=\left(x-y\right)\left(3-x+y\right)\)
a/ \(=3y^2-6y-2x+1\)
b/ \(=-\left(x^3-3x^2+3x-1\right)=-\left(x-1\right)^3\)
c/ \(=\left(2-x\right)^3\)
d/ \(=xy^2+x^2y+3xy+x^2y+x^3+3x^2-3xy-3x^2-9x\)
\(=xy\left(y+x+3\right)+x^2\left(y+x+3\right)-3x\left(y+x+3\right)\)
\(=\left(xy+x^2-3x\right)\left(y+x+3\right)=x\left(y+x-3\right)\left(y+x+3\right)\)
e/ \(=xy-x^2+2x-y^2+xy-2y\)
\(=x\left(y-x+2\right)-y\left(y-x+2\right)=\left(x-y\right)\left(y-x+2\right)\)
a) =(2x+3y-1)2
b)=-(x-1)3
c)=-(x3-6x2+12x-8)=-(x-2)3
d)x3 + 2x2y + xy2 – 9x
= x(x2 + 2xy + y2 -9)
= x[(x2 + 2xy + y2) - 32]
= x[(x + y)2 - 32]
= x (x + y – 3)(x + y + 3)
e) 2x-2y-x2+2xy-y2=2(x-y)-(x-y)2=(x-y)(2-x+y)
a: \(A=x^3+3x^2+3x+1-1\)
\(=\left(x+1\right)^3-1\)
\(=100^3-1=999999\)
b: \(B=3\left[\left(x+y\right)^2-2xy\right]-2\left[\left(x+y\right)^3-3xy\left(x+y\right)\right]\)
\(=3\left(1-2xy\right)-2\left(1-3xy\right)\)
\(=3-6xy-2+6xy=1\)
c: \(C=\left(x^3+3x^2y+3xy^2+y^3\right)-3\left(x^2+2xy+y^2\right)+3\left(x+y\right)+2017\)
\(=101^3-3\cdot101^2+3\cdot101+2017\)
\(=101^3-3\cdot101^2+3\cdot101-1+2018\)
\(=100^3+2018=1002018\)
\(3x+3y=2021\)
\(\Leftrightarrow x+y=\dfrac{2021}{3}\)
\(P=x^3+y^3+2021xy\)
\(=\left(x+y\right)^3-3xy\cdot\left(x+y\right)+2021xy\)
\(=\left(\dfrac{2021}{3}\right)^3-3xy\cdot\dfrac{2021}{3}+2021xy\)
\(=\dfrac{8254655261}{27}\)