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a: Ta có: \(10x^4-27x^3y-110x^2y^2-27xy^3+10y^4\)
\(=10x^4+20x^2y^2+10y^4-27xy\left(x^2+y^2\right)-130x^2y^2\)
\(=10\left(x^2+y^2\right)^2-27xy\left(x^2+y^2\right)-130x^2y^2\)
\(=10\left(x^2+y^2\right)^2-52xy\left(x^2+y^2\right)+25xy\left(x^2+y^2\right)-130x^2y^2\)
\(=2\left(x^2+y^2\right)\left(5x^2+5y^2-26xy\right)+5xy\left(5x^2+5y^2-26xy\right)\)
\(=\left(5x^2-26xy+5y^2\right)\left(2x^2+5xy+2y^2\right)\)
\(=\left(5x^2-25xy-xy+5y^2\right)\left(2x^2+4xy+xy+2y^2\right)\)
\(=\left\lbrack5x\left(x-5y\right)-y\left(x-5y\right)\right\rbrack\left\lbrack2x\left(x+2y\right)+y\left(x+2y\right)\right\rbrack\)
=(5x-y)(x-5y)(2x+y)(x+2y)
b: \(x^5-4x^4+3x^3+3x^2-4x+1\)
\(=x^5+x^4-5x^4-5x^3+8x^3+8x^2-5x^2-5x+x+1\)
\(=\left(x+1\right)\left(x^4-5x^3+8x^2-5x+1\right)\)
\(=\left(x+1\right)\left(x^4-x^3-4x^3+4x^2+4x^2-4x-x+1\right)\)
\(=\left(x+1\right)\left(x-1\right)\left(x^3-4x^2+4x-1\right)\)
\(=\left(x+1\right)\left(x-1\right)\left\lbrack\left(x^3-x^2\right)-3x^2+3x+x-1\right\rbrack\)
\(=\left(x+1\right)\left(x-1\right)\cdot\left(x-1\right)\left(x^2-3x+1\right)=\left(x+1\right)\left(x-1\right)^2\cdot\left(x^2-3x+1\right)\)
\(3z^2+6zy+3y^2-27x^2\)
\(=3\left(z^2+2zy+y^2-9x^2\right)\)
\(=3\left(\left(z+y\right)^2-\left(3x\right)^2\right)\)
\(=3\left(z+y-3x\right)\left(z+y+3x\right)\)
a) \(x^2-2x-15\)
\(\Leftrightarrow x^2-2x+1-16\)
\(\Leftrightarrow\left(x-1\right)^2-4^2\)
\(\Leftrightarrow\left(x-5\right)\left(x-3\right)\)
\(a,x^2-2x-15=\left(x^2-2x+1\right)-16.\)
\(=\left(x-1\right)^2-4^2\)
\(=\left(x-5\right)\left(x+3\right)\)
b) 2x^2 + 7x - 15
2x^2 + 10x - 3x -15
2x(x+5) - 3(x+5)
(x+5)(2x-3)
a, 5x^3y - 10x^2y^2 + 5xy^3 = 5xy. ( x^2 - 2xy + y^2) = 5xy.( x-y)^2
b, 2x^2 + 7x -15 = 2x^2 + 10X - 3x -15
= 2x( x+5) - 3( x+5)
= ( 2x-3) (x+5)
=X4-3X3 +6X3-18X2+11X2-33X+6X-18
=(X-3)(X3+6X2+11X+6)
=(X-3)(X+3)(X+1)(X+2)
\(x^4+3x^3-7x^2-27x-18.\)
\(=\left(x^4-9x^2\right)+\left(3x^3-27x\right)+\left(2x^2-18\right)\)
\(=x^2\left(x-3\right)\left(x+3\right)+3x\left(x-3\right)\left(x+3\right)+2\left(x-3\right)\left(x+3\right).\)
\(=\left(x-3\right)\left(x+3\right)\left(x^2+3x+2\right)\)
\(=\left(x-3\right)\left(x+3\right)\left(x^2+x+2x+2\right)\)
\(=\left(x-3\right)\left(x+3\right)\left(x+1\right)\left(x+2\right).\)
1. = (3x)3 - (ab)3
= (3x - ab)[(3x)2 + 3x . ab + ab2)
= (3x - ab)(9x2 + 3xab + ab2)