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a) \(\left(3-xy^2\right)^2-\left(2+xy^2\right)^2\)
\(=\left(3-xy^2-2-xy^2\right)\left(3-xy^2+2+xy^2\right)\)
\(=\left(1-2xy^2\right).5=5-10xy^2\)
b) \(9x^2-\left(3x-4\right)^2\)
\(=\left(3x-3x+4\right)\left(3x+3x-4\right)\)
\(=4.\left(6x-4\right)=24x-16\)
c) \(\left(a-b^2\right)\left(a+b^2\right)\)
\(=a^2-b^{^4}\)
d) \(\left(a^2+2a+3\right)\left(a^2+2a-3\right)\)
\(=\left[\left(a^2+2a\right)^2\right]-3^2\)
\(=a^4+4a^3+4a^2-9\)
\(a,\left(2a-3\right)\left(a+1\right)+\left(a^2+6a+9\right):\left(a+3\right)\\ =2a^2-a-3+\left(a+3\right)^2:\left(a+3\right)\\ =2a^2-a-3+a+3\\ =2a^2\\ b,\left(3x-5y\right)\left(-xy\right)^2-3x^2y^2+4x^2y^3\\ =3x^3y^2-5x^2y^3-3x^2y^2+4x^2y^3\\ =3x^3y^2-3x^2y^2-x^2y^3\\ c,x\left(x-2\right)^2-\left(x+2\right)\left(x^2-2x+4\right)+4x^2\\ =x^3-4x^2+4x-x^3-8+4x^2\\ =4x-8\)
a) Ta có: \(\left(3-xy^2\right)^2-\left(2+xy^2\right)^2\)
\(=\left[\left(3-xy^2\right)-\left(2+xy^2\right)\right]\cdot\left[\left(3-xy^2\right)+\left(2+xy^2\right)\right]\)
\(=\left(3-xy^2-2-xy^2\right)\cdot\left(3-xy^2+2+xy^2\right)\)
\(=5\cdot\left(1-2xy^2\right)\)
\(=5-10xy^2\)
b) Ta có: \(9x^2-\left(3x-4\right)^2\)
\(=\left[3x-\left(3x-4\right)\right]\left[3x+\left(3x-4\right)\right]\)
\(=\left(3x-3x+4\right)\cdot\left(3x+3x-4\right)\)
\(=4\cdot\left(6x-4\right)\)
\(=24x-16\)
c) Ta có: \(\left(a-b^2\right)\left(a+b^2\right)\)
\(=a^2-b^4\)
d) Ta có: \(\left(a^2+2a+3\right)\left(a^2+2a-3\right)\)
\(=\left(a^2+2a\right)^2-9\)
\(=a^4+4a^3+4a^2-9\)
e) Ta có: \(\left(x-y+6\right)\left(x+y-6\right)\)
\(=x^2+xy-6x-yx-y^2+6y+6x+6y-36\)
\(=x^2-y^2+12y-36\)
f) Ta có: \(\left(y+2z-3\right)\left(y-2z-3\right)\)
\(=\left(y-3\right)^2-\left(2z\right)^2\)
\(=y^2-6y+9-4z^2\)
g) Ta có: \(\left(2y-5\right)\left(4y^2+10y+25\right)\)
\(=\left(2y\right)^3-5^3\)
\(=8y^3-125\)
h) Ta có: \(\left(3y+4\right)\left(9y^2-12y+16\right)\)
\(=\left(3y\right)^3+4^3\)
\(=27y^3+64\)
i) Ta có: \(\left(x-3\right)^3+\left(2-x\right)^3\)
\(=\left(x-3\right)^3-\left(x-2\right)^3\)
\(=x^3-9x^2+27x-27-\left(x^3-6x^2+12x-8\right)\)
\(=x^3-9x^2+27x-27-x^3+6x^2-12x+8\)
\(=-3x^2+15x-19\)
j) Ta có: \(\left(x+y\right)^3-\left(x-y\right)^3\)
\(=\left[\left(x+y\right)-\left(x-y\right)\right]\cdot\left[\left(x+y\right)^2+\left(x+y\right)\left(x-y\right)+\left(x-y\right)^2\right]\)
\(=\left(x+y-x+y\right)\left(x^2+2xy+y^2+x^2-y^2+x^2-2xy+y^2\right)\)
\(=2y\cdot\left(3x^2+y^2\right)\)
\(=6x^2y+2y^3\)
a) \(\left(3+xy^2\right)^2=9+6xy^2+x^2y^4\)
b) \(\left(10-2m^2n\right)^2=100-40m^2n+4m^4n^2\)
c) \(\left(a-b^2\right)\left(a+b^2\right)=a^2-\left(b^2\right)^2=a^2-b^4\)
e) \(\left(xy-1\right)^3=x^3y^3-3x^2y^2+3xy-1\)
h) Sửa lại đề bài chút xíu:
$(xy+ab)^2+(ay-bx)^2=x^2y^2+a^2b^2+2abxy+a^2y^2-2aybx+b^2x^2$
$=x^2y^2+a^2b^2+a^2y^2+b^2x^2$
$=(x^2y^2+b^2x^2)+(a^2b^2+a^2y^2)$
$=x^2(y^2+b^2)+a^2(b^2+y^2)=(a^2+x^2)(b^2+y^2)$
j)
$ab(x^2+y^2)+xy(a^2+b^2)=abx^2+aby^2+xya^2+xyb^2$
$=(abx^2+xya^2)+(aby^2+xyb^2)$
$=ax(bx+ay)+by(ay+bx)=(ax+by)(ay+bx)$
k)
$(xy-ab)^2+(bx+ay)^2=x^2y^2-2xyab+a^2b^2+b^2x^2+2bxay+a^2y^2$
$=x^2y^2+a^2b^2+b^2x^2+a^2y^2$
$=(x^2y^2+b^2x^2)+(a^2b^2+a^2y^2)=x^2(y^2+b^2)+a^2(b^2+y^2)$
$=(a^2+x^2)(b^2+y^2)$
e)
$x^2-(2a+b)xy+2aby^2=x^2-2axy-bxy+2aby^2$
$=x(x-2ay)-by(x-2ay)=(x-by)(x-2ay)$
g)
$y^2-(3a+2b)xy+6abx^2=(y^2-2bxy)-(3axy-6abx^2)$
$=y(y-2bx)-3ax(y-2bx)=(y-3ax)(y-2bx)$
f)
$3xy(a^2+b^2)-ab(x^2+9y^2)=3xya^2+3xyb^2-abx^2-9aby^2$
$=(3xya^2-abx^2)-(9aby^2-3xyb^2)$
$=ax(3ay-bx)-3by(3ay-bx)=(3ay-bx)(ax-3by)$