Упр.21.15 ГДЗ Мерзляк Поляков 7 класс (Алгебра)

1) a^2-b^2-a-b; 6) a^2-10a+25-ab+5b;
2) x-y-x^2+y^2; 7) 8mp+8np-m^2-2mn-n^2;
3) 4m^2-9n^2+2m+3n; 8) a^3+b^3-a^2 b-ab^2;
4) c^2-d^2+4c-4d; 9) m^3-8n^3-m^2+4mn-4n^2;
5) 5x^2 y-5xy^2-x^2+y^2; 10) a^3-4a^2+4a-1.

1. $$a^2-b^2-a-b=(a-b)(a+b)-(a+b)=(a+b)(a-b-1)$$
2. $$x-y-x^2+y^2=(x-y)-(x^2-y^2)=(x-y)-(x-y)(x+y)$$
   $$=(x-y)(1-x-y)$$
3. $$4m^2-9n^2+2m+3n=(2m-3n)(2m+3n)+(2m+3n)$$
   $$=(2m+3n)(2m-3n+1)$$
4. $$c^2-d^2+4c-4d=(c-d)(c+d)+4(c-d)$$
   $$=(c-d)(c+d+4)$$
5. $$5x^2y-5xy^2-x^2+y^2=5xy(x-y)-(x^2-y^2)$$
   $$=5xy(x-y)-(x-y)(x+y)$$
   $$=(x-y)(5xy-x-y)$$
6. $$a^2-10a+25-ab+5b=(a-5)^2-b(a-5)$$
   $$=(a-5)(a-5-b)$$
7. $$8mp+8np-m^2-2mn-n^2=8p(m+n)-(m^2+2mn+n^2)$$
   $$=8p(m+n)-(m+n)^2$$
   $$=(m+n)(8p-m-n)$$
8. $$a^3+b^3-a^2b-ab^2=(a^3+b^3)-ab(a+b)$$
   $$=(a+b)(a^2-ab+b^2)-ab(a+b)$$
   $$=(a+b)(a^2-2ab+b^2)$$
   $$=(a+b)(a-b)^2$$
9. $$m^3-8n^3-m^2+4mn-4n^2=(m^3-8n^3)-(m^2-4mn+4n^2)$$
   $$=(m-2n)(m^2+2mn+4n^2)-(m-2n)^2$$
   $$=(m-2n)\bigl(m^2+2mn+4n^2-m+2n\bigr)$$
10. $$a^3-4a^2+4a-1=(a^3-1)-(4a^2-4a)$$
    $$=(a-1)(a^2+a+1)-4a(a-1)$$
    $$=(a-1)(a^2-3a+1)$$

Ответ

1) $$ (a+b)(a-b-1) $$;
2) $$ (x-y)(1-x-y) $$;
3) $$ (2m+3n)(2m-3n+1) $$;
4) $$ (c-d)(c+d+4) $$;
5) $$ (x-y)(5xy-x-y) $$;
6) $$ (a-5)(a-5-b) $$;
7) $$ (m+n)(8p-m-n) $$;
8) $$ (a+b)(a-b)^2 $$;
9) $$ (m-2n)(m^2+2mn+4n^2-m+2n) $$;
10) $$ (a-1)(a^2-3a+1) $$.