Đáp án:
$\displaystyle \begin{array}{{>{\displaystyle}l}} Bài\ 1:\\ a.\ A=x^{2}\left( x^{2} +10x+1\right)\\ b.\ B=( x+y)( x+y-1)( x+y+1)\\ c.\ C=5( x-y-2z)( x-y+2z)\\ d.\ D=( -5x+1)( x-3)\\ e.\ E=( x+2)( x+3)( x+4)\\ Bài\ 2:\\ a.x=-1\ hoặc\ x=3\ \\ b.x=3\ hoặc\ x=\frac{1}{2} \ \\ Bài\ 3:\\ a.\ 3x^{2} y^{2}\left( y^{2} -2xy+3x^{2}\right)\\ b.\ ( x+y+z-3y+2z)( x+y+x+3y-2z)\\ c.\ ( x-2)( x+2)( x-3)( x+3)\\ d.\ \left( x^{2} +2x\right)( x-7)\\ e.\ ( x-1)\left( x^{2} -3x+5\right)\\ f.\ ( x-1)( x+1)( x-4)( x-2)\\ \\ \end{array}$
Giải thích các bước giải:
$\displaystyle \begin{array}{{>{\displaystyle}l}} Bài\ 1:\\ a.A=x^{4} +2x^{3} +x^{2} +8x^{3}\\ =x^{2}\left( x^{2} +10x+1\right)\\ b.\ B=\ x^{3} -x+3x^{2} y+3xy^{2} +y^{3} -y\\ =( x+y)^{3} -( x+y)\\ =( x+y)\left(( x+y)^{2} -1\right)\\ =( x+y)( x+y-1)( x+y+1)\\ c.\ C=\ 5x^{2} -10xy+5y^{2} -20z^{2}\\ =5\left( x^{2} -2xy+y^{2} -4z^{2}\right)\\ =5\left(( x-y)^{2} -( 2z)^{2}\right)\\ =5( x-y-2z)( x-y+2z)\\ d.\ D=16x-5x^{2} -3\\ =-5x^{2} +15x+x-3\\ =-5x( x-3) +( x-3)\\ =( -5x+1)( x-3)\\ e.\ E=x^{3} +9x^{2} +26x+24\\ =\ x^{3} +4x^{2} +5x^{2} +20x+6x+24\\ =x^{2}( x+4) +5x( x+4) +6( x+4)\\ =\left( x^{2} +5x+6\right)( x+4)\\ =\left( x^{2} +2x+3x+6\right)( x+4)\\ =( x+2)( x+3)( x+4)\\ Bài\ 2:\\ a.\ x^{2} -2x-3=0\\ \Leftrightarrow x^{2} -3x+x-3=0\\ \Leftrightarrow x( x-3) +( x-3) =0\\ \Leftrightarrow ( x+1)( x-3) =0\\ \Leftrightarrow x=-1\ hoặc\ x=3\\ b.\ 2x^{2} +5x-3=0\\ \Leftrightarrow 2x^{2} -6x+x-3=0\\ \Leftrightarrow ( x-3)( 2x-1) =0\\ \Leftrightarrow x=3\ hoặc\ x=\frac{1}{2}\\ Bài\ 3:\\ a.\ 3x^{2} y^{4} -6x^{3} y^{3} +9x^{4} y^{2}\\ =3x^{2} y^{2}\left( y^{2} -2xy+3x^{2}\right)\\ b.\ x^{2} +2xy-8y^{2} +2xz+14yz-3z^{2}\\ =\left( x^{2} +y^{2} +z^{2} +2xy+2xz+2yz\right) +\left( -9y^{2} +12yz-4z^{2}\right)\\ =( \ x+y+x)^{2} -( 3y-2z)^{2}\\ =( x+y+z-3y+2z)( x+y+x+3y-2z)\\ c.\ x^{4} -13x^{2} +36\\ =x^{4} -3x^{3} +3x^{3} -9x^{2} -4x^{2} +12x-12x+36\\ =x^{3}( x-3) +3x^{2}( x-3) -4x( x-3) -12( x-3)\\ =\left( x^{3} +3x^{2} -4x-12\right)( x-3)\\ =\left( x^{2}( x+3) -4( x+3)\right)( x-3)\\ =\left( x^{2} -4\right)( x+3)( x-3)\\ =( x-2)( x+2)( x-3)( x+3)\\ d.\ x^{3} -5x^{2} -14x\\ =x^{3} -7x^{2} +2x^{2} -14x\\ =x^{2}( x-7) +2x( x-7)\\ =\left( x^{2} +2x\right)( x-7)\\ e.\ x^{3} -4x^{2} +4x-1\\ =x^{3} -1-4\left( x^{2} -1\right)\\ =( x-1)\left( x^{2} +x+1\right) -4( x-1)( x+1)\\ =( x-1)\left( x^{2} +x+1-4x+4\right)\\ =( x-1)\left( x^{2} -3x+5\right)\\ f.\ x^{4} -6x^{3} +7x^{2} +6x-8\\ =\ x^{4} -2x^{3} -4x^{3} +8x^{2} -x^{2} +2x+4x-8\\ =x^{3}( x-2) -4x^{2}( x-2) -x( x-2) +4( x-2)\\ =\left( x^{3} -4x^{2} -x+4\right)( x-2)\\ =\left( x^{2}( x-4) -( x-4)\right)( x-2)\\ =\left( x^{2} -1\right)( x-4)( x-2)\\ =( x-1)( x+1)( x-4)( x-2) \end{array}$
