phân tích đa thức thành nhân tử
a, \(x^3y+x-y-1\)
b, \(x^2\left(x-2\right)+4\left(2-x\right)\)
c, \(x^3-x^2-20x\)
d, \(\left(x^2+1\right)^2-\left(x+1\right)^2\)
e, \(6x^2-7x+2\)
f, \(x^4+8x^2+12\)
g, \(\left(x^3+x+1\right)\left(x^3+x\right)-2\)
h, \(\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)+1\)
i, \(-\left(x^2+2\right)^2+4x\left(x^2+2\right)-3x^2\)
j, \(81x^4+4y^4\)
a: \(x^3y+x-y-1\)
\(=\left(x^3y-y\right)+\left(x-1\right)\)
\(=y\left(x^3-1\right)+\left(x-1\right)\)
\(=y\left(x-1\right)\left(x^2+x+1\right)+\left(x-1\right)\)
\(=\left(x-1\right)\left[y\left(x^2+x+1\right)+1\right]\)
b: \(x^2\left(x-2\right)+4\left(2-x\right)\)
\(=x^2\left(x-2\right)-4\left(x-2\right)\)
\(=\left(x-2\right)\left(x^2-4\right)=\left(x-2\right)\cdot\left(x-2\right)\left(x+2\right)\)
\(=\left(x+2\right)\left(x-2\right)^2\)
c: \(x^3-x^2-20x\)
\(=x\left(x^2-x-20\right)\)
\(=x\left(x^2-5x+4x-20\right)\)
\(=x\left[x\left(x-5\right)+4\left(x-5\right)\right]\)
\(=x\left(x-5\right)\left(x+4\right)\)
d: \(\left(x^2+1\right)^2-\left(x+1\right)^2\)
\(=\left(x^2+1+x+1\right)\left(x^2+1-x-1\right)\)
\(=\left(x^2+x+2\right)\left(x^2-x\right)\)
\(=x\left(x-1\right)\left(x^2+x+2\right)\)
e: \(6x^2-7x+2\)
\(=6x^2-3x-4x+2\)
\(=3x\left(2x-1\right)-2\left(2x-1\right)\)
\(=\left(2x-1\right)\left(3x-2\right)\)
f: \(x^4+8x^2+12\)
\(=x^4+2x^2+6x^2+12\)
\(=x^2\left(x^2+2\right)+6\left(x^2+2\right)\)
\(=\left(x^2+2\right)\left(x^2+6\right)\)
g: \(\left(x^3+x+1\right)\left(x^3+x\right)-2\)
\(=\left(x^3+x\right)^2+\left(x^3+x\right)-2\)
\(=\left(x^3+x+2\right)\left(x^3+x-1\right)\)
\(=\left(x^3+x^2-x^2-x+2x+2\right)\left(x^3+x-1\right)\)
\(=\left[x^2\left(x+1\right)-x\left(x+1\right)+2\left(x+1\right)\right]\left(x^3+x-1\right)\)
\(=\left(x+1\right)\cdot\left(x^2-x+2\right)\left(x^3+x-1\right)\)
h: \(\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)+1\)
\(=\left(x+1\right)\left(x+4\right)\left(x+2\right)\left(x+3\right)+1\)
\(=\left(x^2+5x+4\right)\left(x^2+5x+6\right)+1\)
\(=\left(x^2+5x+5-1\right)\left(x^2+5x+5+1\right)+1\)
\(=\left(x^2+5x+5\right)^2-1+1\)
\(=\left(x^2+5x+5\right)^2\)
i: \(-\left(x^2+2\right)^2+4x\left(x^2+2\right)-3x^2\)
\(=-\left[\left(x^2+2\right)^2-4x\left(x^2+2\right)+3x^2\right]\)
\(=-\left[\left(x^2+2\right)^2-3x\left(x^2+2\right)-x\left(x^2+2\right)+3x^2\right]\)
\(=-\left[\left(x^2+2\right)\left(x^2+2-3x\right)-x\left(x^2+2-3x\right)\right]\)
\(=-\left(x^2-x+2\right)\left(x^2-3x+2\right)\)
\(=-\left(x^2-x+2\right)\left(x-1\right)\left(x-2\right)\)
j: \(81x^4+4y^4\)
\(=81x^4+36x^2y^2+4y^4-36x^2y^2\)
\(=\left(9x^2+2y^2\right)^2-\left(6xy\right)^2\)
\(=\left(9x^2-6xy+2y^2\right)\left(9x^2+6xy+2y^2\right)\)
