Question

Разложить на множители 9x^2-16y^2 x^3-8y^3 8x^3-125y^3 x^2-b^2-ax-ab a^2x^2-y^4 c^2-4c+4-9x^2 4c^2+20c+25-9a^2 3x^2+2x-xy-2y^2+y^3-3xy^2 a^2x+a+ax^2+x+2ax+2

Answer 1

9x^2-16y^2 = (3x)^2 - (4y)^2 = (3x-4y)(3x+4y) разность квадратов

x^3-8y^3 = x^3 - (2y)^3 = (x-2y)(x^2+2xy+y^2) разность кубов

8x^3-125y^3 = (2x)^3 - (5y)^3 = (2x -5y)(4x^2+10xy + 25y^2)

x^2-b^2-ax-ab = (x-b)(x+b) -a(x-b) = (x-b)( (x+b) -a ) = (x-b)(x+b-a)

a^2x^2-y^4 = (ax)^2 -(y^2)^2 = (ax-y^2)(ax+y^2)

c^2-4c+4-9x^2 = c^2 -2*2c+2^2 -(3x)^2 = (c-2)^2 -(3x)^2 = ( (c-2) -3x  )( (c-2) +3x ) = (c-2-3x)
(c-2+3x)

4c^2+20c+25-9a^2 = (2c+5)^2 -(3a)^2 = (2c+5-3a)(2c+5+3a)

$3x^2+2x-xy-2y^2+y^3-3xy^2= \\\\=y(y^2-x)+ 3x^2+2x-2y^2-3xy^2= \\\\= y(y^2-x)+2x-2y^2+3x(x-y^2)= \\\\= y(y^2-x)+2(x-y^2)+3x(x-y^2)= \\\\= -y(x-y^2)+2(x-y^2)+3x(x-y^2)= \\\\=(-y+2+3x)(x-y^2)$

$a^2x+a+ax^2+x+2ax+2 = ax(a+x)+(a+x)+2(ax+1) = \\\\=(a+x)(ax+1)+2(ax+1) =((a+x)+2) (ax+1)= \\\\=(a+x+2) (ax+1)$