Question

145.Разложите на множители:
1) х^2 + 2xy + y^2 — 64;
2) m^2 + 16n^2 + 8mn — b^2;
3) х^2у^з – xy + у — у^з;
4) а^з + 27 — За – a^2,
5) x^12 – 6х^10 + 9x^8 – 36;
6) b^3+ 64а^3 + b^2 + 8ba + 16а^2;
7) x^2 — 6xy + 9y^2 — а^2 – 2а – -1;
8) 4х^2 - y^2 — 4х + 1.​

Answer 1

$1)\ \ \underbrace {x^2+2xy+y^2}-64=(x+y)^2-8^2=(x+y-8)(x+y+8)\\\\2)\ \ \underbrace {m^2+16n^2+8mn}-b^2=(m+4n)^2-b^2=(m+4n-b)(m+4n+b)\\\\3)\ \ x^2y^3-xy+y-y^3=(x^2y^3-y^3)+(y-xy)=y^3(x^2-1)-y(x-1)=\\\\=y^3(x-1)(x+1)-y(x-1)=y(x-1)(y^2(x+1)-1)=y(x-1)(xy^2+y^2-1)\\\\4)\ \ \underbrace{a^3+27}-\underbrace{3a-a^2}=(a+3)(a^2-3a+9)-a(a+3)=\\\\=(a+3)(a^2-3a+9-a)=(a+3)(a^2-4a+9)$

$5)\ \ \underbrace {x^{12}-6x^{10}+9x^8}-36=(x^6+3x^4)^2-36=\\\\=(x^6+3x^4-6)(x^6+3x^4+6)\\\\6)\ \ \underbrace {b^3+64a^3}+\underbrace {b^2+8ba+16a^2}=(b+4a)(b^2-4ab+16a^2)+(b+4a)^2=\\\\=(b+4a)(b^2-4ab+16a^2+b+4a)\\\\7)\ \ \underbrace {x^2-6xy+9y^2}-\underbrace {a^2-2a-1}=(x+3y)^2-(a+1)^2=\\\\=(x+3y-a-1)(x+3y+a+1)\\\\8)\ \ 4x^2-y^2-4x+1=(4x^2-4x+1)-y^2=(2x-1)^2-y^2=\\\\=(2x-1-y)(2x-1+y)$