Упр.410 ГДЗ Муравин 7 класс (Алгебра)

1. Разложите на множители выражение:
   
   1) $$20x^3-140x^2y+245xy^2$$;
   
   2) $$100k^4-120k^3n+36k^2n^2$$;
   
   3) $$(3a+4)^2-(a-12)^2$$;
   
   4) $$(5x-7)^2-(5-3x)^2$$;
   
   5) $$(a^2+6ab+16b^2)^2-(6ab+20b^2)^2$$;
   
   6) $$(4-6y^3+4y^6)^2-(6y^3-5y^6)^2$$.

1. $$20x^3-140x^2y+245xy^2=5x(4x^2-28xy+49y^2)=5x(2x-7y)^2.$$
2. $$100k^4-120k^3n+36k^2n^2=k^2(100k^2-120kn+36n^2)=k^2(10k-6n)^2=4k^2(5k-3n)^2.$$
3. $$(3a+4)^2-(a-12)^2=((3a+4)-(a-12))((3a+4)+(a-12))$$
   $$=(2a+16)(4a-8)=2(a+8)\cdot 4(a-2)=8(a+8)(a-2).$$
4. $$(5x-7)^2-(5-3x)^2=((5x-7)-(5-3x))((5x-7)+(5-3x))$$
   $$=(8x-12)(2x-2)=4(2x-3)\cdot 2(x-1)=8(2x-3)(x-1).$$
5. $$(a^2+6ab+16b^2)^2-(6ab+20b^2)^2$$
   $$=\bigl((a^2+6ab+16b^2)-(6ab+20b^2)\bigr)\bigl((a^2+6ab+16b^2)+(6ab+20b^2)\bigr)$$
   $$=(a^2-4b^2)(a^2+12ab+36b^2) =(a-2b)(a+2b)(a+6b)^2.$$
6. $$(4-6y^3+4y^6)^2-(6y^3-5y^6)^2$$
   $$=\bigl((4-6y^3+4y^6)-(6y^3-5y^6)\bigr)\bigl((4-6y^3+4y^6)+(6y^3-5y^6)\bigr)$$
   $$=(4-12y^3+9y^6)(4-y^6)=(2-3y^3)^2(2-y^3)(2+y^3).$$

Ответ

1) $$5x(2x-7y)^2$$
2) $$4k^2(5k-3n)^2$$
3) $$8(a+8)(a-2)$$
4) $$8(2x-3)(x-1)$$
5) $$(a-2b)(a+2b)(a+6b)^2$$
6) $$(2-3y^3)^2(2-y^3)(2+y^3)$$