Bài 1:
a: \(\left(2x+3y\right)^2\)
\(=\left(2x\right)^2+2\cdot2x\cdot3y+\left(3y\right)^2\)
\(=4x^2+12xy+9y^2\)
b: \(\left(5x-y\right)^2=\left(5x\right)^2-2\cdot5x\cdot y+y^2=25x^2-10xy+y^2\)
c: \(\left(2x+y^2\right)^3\)
\(=\left(2x\right)^3+3\cdot\left(2x\right)^2\cdot y^2+3\cdot2x\cdot\left(y^2\right)^2+\left(y^2\right)^3\)
\(=8x^3+12x^2y^2+6xy^4+y^6\)
d: \(\left(3x^2-2y\right)^3\)
\(=\left(3x^2\right)^3-3\cdot\left(3x^2\right)^2\cdot2y+3\cdot3x^2\cdot\left(2y\right)^2-\left(2y\right)^3\)
\(=27x^6-54x^4y+36x^2y^2-8y^3\)
e: \(\left(x-3y\right)\left(x^2+3xy+9y^2\right)\)
\(=\left(x-3y\right)\left(x^2+x\cdot3y+\left(3y\right)^2\right)\)
\(=x^3-\left(3y\right)^3=x^3-27y^3\)
f: \(\left(x^2-3\right)\left(x^4+3x^2+9\right)\)
\(=\left(x^2-3\right)\left[\left(x^2\right)^2+x^2\cdot3+3^2\right]\)
\(=\left(x^2\right)^3-3^3=x^6-27\)
g: \(\left(x^2+\dfrac{2}{5}y\right)\left(x^2-\dfrac{2}{5}y\right)=\left(x^2\right)^2-\left(\dfrac{2}{5}y\right)^2=x^4-\dfrac{4}{25}y^2\)
h: \(\left(x+\dfrac{1}{4}\right)^2=x^2+2\cdot x\cdot\dfrac{1}{4}+\left(\dfrac{1}{4}\right)^2=x^2+\dfrac{1}{2}x+\dfrac{1}{16}\)
i: \(\left(\dfrac{2}{3}x^2-\dfrac{1}{2}y\right)^3\)
\(=\left(\dfrac{2}{3}x^2\right)^3-3\cdot\left(\dfrac{2}{3}x^2\right)^2\cdot\dfrac{1}{2}y+3\cdot\dfrac{2}{3}x^2\cdot\left(\dfrac{1}{2}y\right)^2-\left(\dfrac{1}{2}y\right)^3\)
\(=\dfrac{8}{27}x^6-\dfrac{2}{3}x^4y+\dfrac{1}{2}x^2y^2-\dfrac{1}{8}y^3\)
k: \(\left(x+2y+z\right)\left(x+2y-z\right)\)
\(=\left(x+2y\right)^2-z^2\)
\(=x^2+4xy+4y^2-z^2\)
l: \(\left(2x-1\right)\left(4x^2+2x+1\right)\)
\(=\left(2x-1\right)\left[\left(2x\right)^2+2x\cdot1+1^2\right]\)
\(=\left(2x\right)^3-1^3=8x^3-1\)
Bài 2:
a: \(x^2-6x+9=x^2-2\cdot x\cdot3+3^2=\left(x-3\right)^2\)
b: \(25+10x+x^2=5^2+2\cdot5\cdot x+x^2=\left(5+x\right)^2\)
c: \(16x^2+24xy+9y^2\)
\(=\left(4x\right)^2+2\cdot4x\cdot3y+\left(3y\right)^2\)
\(=\left(4x+3y\right)^2\)
d: \(25x^2+90x+81\)
\(=\left(5x\right)^2+2\cdot5x\cdot9+9^2\)
\(=\left(5x+9\right)^2\)
e: \(9x^2-42xy+49y^2\)
\(=\left(3x\right)^2-2\cdot3x\cdot7y+\left(7y\right)^2\)
\(=\left(3x-7y\right)^2\)
f: \(64x^2-48x+9\)
\(=\left(8x\right)^2-2\cdot8x\cdot3+3^2\)
\(=\left(8x-3\right)^2\)
g: \(x^3+15x^2+75x+125\)
\(=x^3+3\cdot x^2\cdot5+3\cdot x\cdot5^2+5^3\)
\(=\left(x+5\right)^3\)
h: \(x^3-9x^2+27x-27\)
\(=x^3-3\cdot x^2\cdot3+3\cdot x\cdot3^2-3^3\)
\(=\left(x-3\right)^3\)
i: \(8x^3+12x^2y^2+6xy^4+y^6\)
\(=\left(2x\right)^3+3\cdot\left(2x\right)^2\cdot y^2+3\cdot2x\cdot\left(y^2\right)^2+\left(y^2\right)^3\)
\(=\left(2x+y^2\right)^3\)
k: \(27x^6-54x^4y+36x^2y^2-8y^3\)
\(=\left(3x^2\right)^3-3\cdot\left(3x^2\right)^2\cdot2y+3\cdot3x^2\cdot\left(2y\right)^2-\left(2y\right)^3\)
\(=\left(3x^2-2y\right)^3\)
l: \(\dfrac{x^3}{27}+\dfrac{x^2y}{6}+\dfrac{xy^2}{4}+\dfrac{y^3}{8}\)
\(=\left(\dfrac{1}{3}x\right)^3+3\cdot\left(\dfrac{1}{3}x\right)^2\cdot\dfrac{y}{2}+3\cdot\dfrac{1}{3}x\cdot\left(\dfrac{1}{2}y\right)^2+\left(\dfrac{1}{2}y\right)^3\)
\(=\left(\dfrac{1}{3}x+\dfrac{1}{2}y\right)^3\)
