Phân tích đa thức ra nhân tử
\(\left(x^2+8x+7\right)\left(x+3\right)\left(x+5\right)+15\)
\(=\left(x^2+8x+7\right)\left(x^2+8x+15\right)+15\)
\(=\left(x^2+8x+7\right)\left(x^2+8x+7+8\right)+15\)
\(=\left(x^2+8x+7\right)^2+8\left(x^2+8x+7\right)+15\)
\(=\left(x^2+8x+7\right)^2+3\left(x^2+8x+7\right)+5\left(x^2+8x+7\right)+15\)
\(=\left(x^2+8x+7\right)\left(x^2+8x+10\right)+5\left(x^2+8x+10\right)\)
\(=\left(x^2+8x+10\right)\left(x^2+8x+12\right)\)
\(=\left(x^2+8x+10\right)\left(x+2\right)\left(x+6\right)\)
\(2x^2-5xy-3y^2=2x^2-6xy+xy-3y^2\)
\(=2x\left(x-3y\right)+y\left(x-3y\right)=\left(2x+y\right)\left(x-3y\right)\)
\(12x^3-4x^2-5x+2=\left(12x^3-12x^2+3x\right)+\left(8x^2-8x+2\right)\)
\(=3x\left(4x^2-4x+1\right)+2\left(4x^2-4x+1\right)\)
\(=\left(3x+2\right)\left(4x^2-4x+1\right)\)
\(=\left(3x+2\right)\left(2x-1\right)^2\)
\(x^2y+xy^2+x^2z+xz^2+y^2z+yz^2+3xyz\)
\(=\left(x^2y+xy^2+xyz\right)+\left(x^2z+xz^2+xyz\right)+\left(y^2z+yz^2+xyz\right)\)
\(=xy\left(x+y+z\right)+xz\left(x+y+z\right)+yz\left(x+y+z\right)\)
\(=\left(x+y+z\right)\left(xy+yz+zx\right)\)
1) \(2x^2-5xy-3y^2\)
\(=2x^2-6xy+xy-3y^2\)
\(=2x\left(x-3y\right)+y\left(x-3y\right)\)
\(=\left(x-3y\right)\left(2x+y\right)\)
2) \(12x^3-4x^2-5x+2\)
\(=12x^3-6x^2+2x^2-x-4x+2\)
\(=6x^2\left(2x-1\right)+x\left(2x-1\right)-2\left(2x-1\right)\)
\(=\left(2x-1\right)\left(6x^2+x-2\right)\)
\(=\left(2x-1\right)\left(6x^2+4x-3x-2\right)\)
\(=\left(2x-1\right)\left[2x\left(3x+2\right)-\left(3x+2\right)\right]\)
\(=\left(2x-1\right)^2\cdot\left(3x+2\right)\)
3) Ta có: \(\left(x^2+8x+7\right)\left(x+3\right)\left(x+5\right)+15\)
\(=\left(x^2+8x+7\right)\left(x^2+8x+15\right)+15\)
\(=\left(x^2+8x\right)^2+22\left(x^2+8x\right)+105+15\)
\(=\left(x^2+8x\right)^2+12\left(x^2+8x\right)+10\left(x^2+8x\right)+120\)
\(=\left(x^2+8x\right)\left(x^2+8x+12\right)+10\left(x^2+8x+12\right)\)
\(=\left(x+2\right)\left(x+6\right)\left(x^2+8x+10\right)\)
