Question

DAU COROANA!Descompuneti in factori:L a) $x^{2} y^{2} -36 x^{2} + y^{4}-37 y^{2} +36$ b)$25 x^{2} - x^{4}-25 y^{2} + x^{2} y^{2}$ c)$x^{3} y^{2} -100x- x^{2} y^{3}+100y+ x^{2} y^{2}z-100z$ d) $4x^{2} y^{2} -( x^{2} + y^{2} - z^{2})^{2}$ e) $x^{4} +y^{4}-( x^{2} - y^{2} )+ x^{2} +y^{2}-(x-y)^{2}$ f)$( x^{2} -3) ( x^{2} -7)+4$ g)$( x^{2} -3)( x^{2} -6)+2$ h)$x^{4} +2x^{3}+2 x^{2} +2x+1$ i) $x^{4} -3 x^{3} -3 x^{2} -3x-4$ j) $x^{2} y^{4}z^{2}- x^{2} y^{2} z^{4}- x^{4} y^{2} z^{2} +x^{4}z^{4}$ k)$4(x-2) ^{2} +9+12(x-2)$ l)$9 x^{2} -4+4y^{4}+12xy$. Sa fie pe lung ca sa stiu cum ati rezolvat!

Answer 1

a)$x^{2} y^{2}+y^{4}-36x^{2}-36 y^{2}-y^{2}+36=$$y^{2}(x^{2}+ y^{2})-36( x^{2}+y^{2})-(y^{2}-36)=}(x^{2}+ y^{2})(y^{2}-36)-(y^{2}-36)$$(y^{2}-36)(x^{2}+ y^{2}-1)=(y+6)(y-6)(x^{2}+ y^{2}-1)$
b)$x^{2}(25- x^{2})-y^{2}(25- x^{2}) =( x^{2}-y^{2})(25- x^{2})=$$(x+y)(x-y)(5+x)(5-x)$
c)$x^{3}y^{2}-x^{2}y^{3}-100x+100y+x^{2}y^{2}z-100z=$$x^{2}y^{2}(x-y)-100(x-y)+z(x^{2}y^{2}-100)=$$(x-y)( x^{2}y^{2}-100)+z( x^{2}y^{2}-100)=(x^{2}y^{2}-100)(x-y+z)=$$(xy+10)(xy-10)(x-y+z)$
d)$(2xy- x^{2}-y^{2}+z^{2})(2xy+x^{2}+y^{2}-z^{2})$
f)($x^{4}-7x^{2}-3x^{2}+21+4= x^{4}-10 x^{2}+25=( x^{2}-5)^{2}$
g)$x^{4}-6x^{2}-3x^{2}+18+2=x^{4}-9x^{2}+20= x^{4}-4x^{2}-5(x^{2}-4)=$$x^{2}( x^{2}-4)-5( x^{2}-4)=(x^{2}-4)( x^{2}-5)=(x-2)(x+2)( x^{2}-5)$
h)$(x^{4}+x^{3})+( x^{3}+x^{2})+(x^{2}+2x+1)=$$x^{3}(x+1)+ x^{2}(x+1)+(x+1)^{2}=(x+1)(x^{3}+x^{2}+x+1)$
i)$(x^{4}-1)-(3x^{3}+3x^{2})-(3x+3)=$$(x^{2}-1)(x^{2}+1)-3x^{2}(x+1)-3(x+1)=$$(x^{2}-1)(x^{2}+1)-(x+1)(3x^{2}-3)=$$(x^{2}+1)(x^{2}-1)-3(x+1)(x^{2}-1)=(x^{2}-1)(x^{2}+1-3x-3)=$$(x^{2}-1)(x^{2}-3x-2)$
j)$x^{2}y^{4}z^{2}-x^{4}y^{2}z^{2}-(x^{2}y^{2}z^{4}-x^{4}z^{4})=$$y^{2}z^{2}(x^{2}y^{2}-x^{4})-z^{4}(x^{2}y^{2}-x^{4})=$$(x^{2}y^{2}-x^{4})(y^{2}z^{2}-z^{4})=$$x^{2}z^{2}(y-x)(y+x)(y-z)(y+z)$
k)$4x^{2}-16x+16+9+12x-24=4x^{2}-4x+1=(2x-1)^{2}$
l)$9x^{2}+12xy+4y^{2}-4=(3x+2y)^{2}-4=(3x+2y-2)(3x+2y+2)$