Question

Вроде как простые (8-9 класс), но не могу решить разложить на множители: x^3+x^2y-xy^2-y^3 a^2+2ab+b^2-ac-bc m^2+2mn+n^2-p^2+2pq-q^2 x^5-x^3+x^2-1 a^3-8+6a^2-12a a^4+a^3+a+1 x^3-x^2y-xy^2+y^3

Answer 1

1) x³+x²y-xy²-y³=(x³-y³)+(x²y-xy²)=(x-y)(x²+xy+y²)+xy(x-y)=
    =(x-y)(x²+xy+y²+xy)=(x-y)(x²+2xy+y²)=(x-y)(x+y)²

2) a²+2ab+b²-ac-bc=(a²+2ab+b²)-(ac+bc)=(a+b)²-c(a+b)=
    =(a+b)(a+b-c)

3) m²+2mn+n²-p²+2pq-q²=(m²+2mn+n²)-(p²-2pq+q²)=
    =(m+n)² - (p-q)² = (m+n-p+q)(m+n+p-q)

4) x⁵-x³+x²-1=(x⁵-x³)+(x²-1)=x³(x²-1)+(x²-1)=(x²-1)(x³+1)=
    =(x-1)(x+1)(x+1)(x²-x+1)=(x+1)²(x-1)(x²-x+1)

5) a³-8+6a²-12a=(a³-8)+(6a²-12a)=(a³-2³)+6a(a-2)=
    =(a-2)(a²+2a+4)+6a(a-2)=(a-2)(a²+2a+4+6a)=
    =(a-2)(a²+8a+4)

6) a⁴+a³+a+1=(a⁴+a³)+(a+1)=a³(a+1)+(a+1)=(a+1)(a³+1)=
    =(a+1)(a³+1³)=(a+1)(a+1)(a²-a+1)=(a+1)²(a²-a+1)

7) x³-x²y-xy²+y³=(x³+y³)-(x²y+xy²)=(x+y)(x²-xy+y²)-xy(x+y)=
    =(x+y)(x²-xy+y²-xy)=(x+y)(x²-2xy+y²)=(x+y)(x-y)²

Answer 2

$x^3+x^2y-xy^2-y^3=(x^3-xy^2)+(x^2y-y^3)=\\x(x^2-y^2)+y(x^2-y^2)=(x-y)(x+y)(x+y)\\\\a^2+2ab+b^2-ac-bc=(a+b)^2-c(a+b)=\\(a+b)(a+b)-c(a+b)=(a+b)(a+b-c)\\\\m^2+2mn+n^2-p^2+2pq-q^2=(m+n)^2-(p-q)^2=\\(m+n-p+q)(m+n+p-q)\\\\x^5-x^3+x^2-1=x^3(x^2-1)+(x^2-1)1=(x^2-1)(x^3+1)=\\(x-1)(x+1)(x+1)(x^2-x+1)\\$$a^3-8+6a^2-12a=(a-2)(a^2+2a+4)+6a(a-2)=\\(a-2)(a^2+2a+4+6a)=(a-2)(a^2+8a+4)\\\\a^4+a^3+a+1=a^3(a+1)+(a+1)1=(a+1)(a^3+1)=\\(a+1)(a+1)(a^2-a+1)\\\\x^3-x^2y-xy^2+y^3=(x^3-x^2y)+(-xy^2+y^3)=\\x^2(x-y)-y^2(x-y)=(x-y)(x^2-y^2)=(x-y)(x-y)(x+y)$