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a) \(3x\left(2x-y\right)+5y\left(y-2x\right)\)
\(=3x\left(2x-y\right)-5y\left(2x-y\right)\)
\(=\left(3x-5y\right)\left(2x-y\right)\)
b) \(\left(x-5\right)^2-9\left(x+y\right)^2\)
\(=\left(x-5\right)^2-3^2\left(x+y\right)^2\)
\(=\left(x-5\right)^2-\left(3x+3y\right)^2\)
\(=\left(x-5+3x+3y\right)\left(x-5-3x-3y\right)\)
\(=\left(4x+3y-5\right)\left(-2x-3y-5\right)\)
a: \(3x\left(2x-y\right)+5y\left(y-2x\right)=\left(2x-y\right)\left(3x-5y\right)\)
e: \(x^2-10x+24=\left(x-4\right)\left(x-6\right)\)
\(A=x\left(y^2-z^2\right)+y\left(z^2-x^2\right)+z\left(x^2-y^2\right)=x\left(y^2-z^2\right)+y\left(-y^2+z^2-x^2+y^2\right)+z\left(x^2-y^2\right)=\left(y^2-z^2\right)\left(x-y\right)+\left(x^2-y^2\right)\left(z-y\right)=\left(y-z\right)\left(y+z\right)\left(x-y\right)-\left(x-y\right)\left(x+y\right)\left(y-z\right)=\left(x-y\right)\left(y-z\right)\left(y+z-x-y\right)=\left(x-y\right)\left(y-z\right)\left(z-x\right)\)
\(B=a\left(b^3-c^3\right)+b\left(c^3-a^3\right)+c\left(a^3-b^3\right)=ab^3-ac^3+bc^3-a^3b+a^3c-b^3c=ab\left(b^2-a^2\right)-c^3\left(a-b\right)+c\left(a^3-b^3\right)=-ab\left(a-b\right)\left(a+b\right)-c^3\left(a-b\right)+c\left(a-b\right)\left(a^2+ab+b^2\right)=\left(a-b\right)\left(-a^2b-ab^2-c^3+a^2c+abc+b^2c\right)\)
Đặt \(A=2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4\)
\(A=-\left(a^4+b^4+c^4-2\left(ab\right)^2-2\left(bc\right)^2-2\left(ca\right)^2\right)\)
\(A=-\left(a^4+b^4+c^4-2\left(ab\right)^2-2\left(bc\right)^2+2\left(ca\right)^2-4\left(ca\right)^2\right)\)
Áp dụng hàng đẳng thức \(\left(a^2-b^2+c^2\right)=a^4+b^4+c^4-2\left(ab\right)^2-2\left(bc\right)^2+2\left(ca\right)^2\):
\(A=-\left[\left(a^2-b^2+c^2\right)^2-4\left(ca\right)^2\right]\)
\(A=-\left(a^2-b^2+c^2-2ca\right)\left(a^2-b^2+c^2+2ca\right)\)
2222222222222a+257222222222222222222222222222222222222222222222222222222222222222222222222222222222222222a=?
Lời giải:
a. $a^4+a^3+a^2+a=(a^4+a^3)+(a^2+a)$
$=a^3(a+1)+a(a+1)=(a+1)(a^3+a)=a(a+1)(a^2+1)$
b. $3xy^2+5y-3x^2y+(-5x)=(3xy^2-3x^2y)+(5y-5x)$
$=3xy(y-x)+5(y-x)=(y-x)(3xy+5)$
c. $xy-z+y-xz=(xy+y)-(z+xz)=y(x+1)-z(x+1)=(x+1)(y-z)$
d.
$x^2-bx+ax-ab=(a^2+ax)-(bx+ab)=a(a+x)-b(a+x)=(a+x)(a-b)$
\(a^4+b^4+c^4-2a^2b^2-2b^2c^2-2a^2c^2=\left(a^4-2a^2b^2+b^4\right)+2\left(a^2-b^2\right)c^2+c^4-4a^2c^2=\left(a^2-b^2+c^2\right)^2-\left(2ac\right)^2=\left(a^2-b^2+c^2-2ac\right)\left(a^2-b^2+c^2+2ac\right)\)
\(a^4+b^4+c^4-2a^2b^2-2b^2c^2-2a^2c^2\)
\(=\left(a^4-2a^2b^2+b^4\right)+2\left(a^2-b^2\right)c^2+c^4-4a^2c^2\)
\(=\left(a^2-b^2+c^2\right)^2-\left(2ac\right)^2\)
\(=\left(a^2-2ac+c^2-b^2\right)\left(a^2+2ac+c^2-b^2\right)\)
\(=\left(a-c-b\right)\left(a-c+b\right)\left(a+c-b\right)\left(a+c+b\right)\)
x 2 y + x y 2 + x 2 z + x z 2 + y 2 z + y z 2 + 3xyz.
= ( x 2 y + x 2 z + xyz) + (x y 2 + y 2 z + xyz) + (x z 2 + y z 2 + xyz)
= x(xy + xz + yz) + y(xy + yz + xz) + z(xz + yz + xy)
= (x + y + z)(xy + xz + yz).
\(x^2y+xy^2+x^2z+xz^2+y^2z+yz^2+3xyz\)
\(=\left(x^2y+x^2z+xyz\right)+\left(xz^2+yz^2+xyz\right)+\left(xy^2+y^2z+xyz\right)\)
\(=x\left(xy+xz+yz\right)+z\left(xz+yz+xy\right)+y\left(xy+yz+xz\right)\)
\(=\left(x+y+z\right)\left(xy+yz+xz\right)\)
