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a, \(P=\left(5x-1\right)+2\left(1-5x\right)\left(4+5x\right)+\left(5x+4\right)^2\)
\(=5x-1+2\left(4+5x-20x-25x^2\right)+\left(25x^2+40x+16\right)\)
\(=5x-1+8-30x-50x^2+25x^2+40x+16\)
\(=\left(-50x^2+25x^2\right)+\left(5x-30x+40x\right)+\left(-1+8+16\right)\)
\(=-25x^2+15x+23\)
b, \(Q=\left(x-y\right)^3+\left(y+x\right)^3+\left(y-x\right)^3-3xy\left(x+y\right)\)
\(=x^3-3x^2y+3xy^2-y^3+y^3+3y^2x+3yx^2+x^3+y^3-3y^2x+3yx^2-x^3-3x^2y-3xy^2\)
\(=\left(x^3+x^3-x^3\right)+\left(-y^3+y^3+y^3\right)+\left(-3x^2y+3x^2y+3x^2y-3x^2y\right)+\left(3xy^2+3xy^2-3xy^2-3xy^2\right)\)
\(=x^3+y^3\)
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![](https://rs.olm.vn/images/avt/0.png?1311)
a)P=2(1-5x)(4+5x)+(5x+4)2
=5x-1-50x2-30x+8+25x2+40x+16
=(-50x2+25x2)+(5x-30x+40x)+8+16-1
=-25x2+15x+23
b)Q=(x-y)3+(y+x)3+(y-x)3-3xy(x+y)
=(x+y)3-3xy(x+y)
=x3+y3
![](https://rs.olm.vn/images/avt/0.png?1311)
\(a,2\left(x-y\right)\left(x+y\right)+\left(x+y\right)^2+\left(x-y\right)^2\)
\(=2x^2+2y^2+x^2+2xy+y^2+x^2-2xy+y^2=3\left(x^2+y^2\right)\)\(b,\left(5x-1\right)+2\left(1-5x\right)\left(4x+5\right)+\left(5x+4\right)\)\(=\left[\left(5x-1\right)-\left(5x+4\right)\right]^2=25\)
c)\(Q=\left(x-y\right)^3+\left(x+y\right)^3+\left(x-y\right)^3-3xy\left(x+y\right)\)
\(=x^3-3x^2y+3xy^2-y^3+x^3+3x^2y+3xy^2+y^3-x^3+3x^2y-3xy^2+y^3-3xy^2-3x^2y\)
\(=x^3+y^3\)
d)\(P=12\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)
\(2P=\left(5^2-1\right)\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)
\(2P=\left(5^4-1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)
\(2P=\left(5^8-1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)
\(2P=\left(5^{16}-1\right)\left(5^{16}+1\right)\)
\(2P=5^{32}-1\Rightarrow P=\dfrac{5^{32}-1}{2}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(M=4\left(x-1\right)\left(x+1\right)-5x\left(x-2\right)+x^2\)
\(=4x^2-4-5x^2+10x+x^2\)
\(=10x-4\)
\(M=\left(y^2+2\right)\left(y-4\right)-\left(2y^2+1\right)\left(\dfrac{1}{2}y-2\right)\)
\(=\left(y^2+2\right)\left(y-4\right)-\dfrac{1}{2}\left(2y^2+1\right)\left(y-4\right)\)
\(=\left(y-4\right)\left(y^2+2-y^2-\dfrac{1}{2}\right)\)
\(=\dfrac{3}{2}y-6\)
c)
\(C=\left(3-2x\right)\left(x-2\right)-4\left(x-1\right)\left(x-3\right)-\left(x-2\right)\left(x+2\right)\)
= 3x - 6 - 2x2 + 4x - 4x2 + 12x + 4x - 12 - x2 + 4
= - 7x2 + 23x - 14
![](https://rs.olm.vn/images/avt/0.png?1311)
a) \(5x^2-2x\left(3x+\frac{3}{2}\right)=-x^2-3x=-x\left(x+3\right)=-3\left(3+3\right)=-18\)
b) \(3x\left(x-4y\right)-\frac{12}{5}y\left(y-5x\right)=3x^2-\frac{12}{5}y^2=3\left(x^2-\frac{4}{5}y^2\right)\)
\(=3\left(4^2-\frac{4}{5}.5^2\right)=3.\left(-4\right)=-12\)
c) \(\left(x-2\right)^2-\left(x+7\right)\left(x-7\right)=x^2-4x+4-x^2+49=-4x+53=-4.3+53=41\)
d) \(x^2+12x+36=\left(x+6\right)^2=\left(64+6\right)^2=70^2=4900\)
e) \(\left(x-3\right)^2-\left(x-4\right)\left(x+4\right)=x^2-6x+9-x^2+16=-6x+25=-6\left(-1\right)+25\)
= 31
f) \(\left(3x+2y\right)^2-4y\left(3x+y\right)=9x^2+12xy+4y^2-12xy-4y^2=9x^2=9\left(-\frac{1}{3}\right)^2=1\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a: \(A=2x^2-2xy-y^2+2xy=2x^2-y^2\)
\(=2\cdot\dfrac{4}{9}-\dfrac{1}{9}=\dfrac{7}{9}\)
b: \(B=5x^2-20xy-4y^2+20xy=5x^2-4y^2\)
\(=5\cdot\dfrac{1}{25}-4\cdot\dfrac{1}{4}\)
=1/5-1=-4/5
c \(C=x^3+6x^2+12x+8=\left(x+2\right)^3=\left(-9\right)^3=-729\)
d: \(D=20x^3-10x^2+5x-20x^2+10x+4\)
\(=20x^3-30x^2+15x+4\)
\(=20\cdot5^3-30\cdot5^2+15\cdot2+4=1784\)
Q=\(\left(x-y\right)^3+x^3+3x^2y+3xy^2-\left(x-y\right)^3-3x^2y-3xy^2\)
Q=\(x^3+y^3\)
P=\(\left(5x-1-5x-4\right)^2\)
P=25