Задание 4 Параграф 19 ГДЗ Рабочая тетрадь 1 Мерзляк Полонский 7 класс (Алгебра)

Разложите на множители:

1. $$m^2-4n^2-m-2n$$;
2. $$8a^2-8b^2+3a-3b$$;
3. $$x^3+x^2y-4x^2+4y^2$$;
4. $$a^2-14a+49-ac+7c$$;
5. $$10xy+10xz-y^2-2yz-z^2$$;
6. $$3m^2+3n^2-n^3+3mn+m^3$$;
7. $$5p^2+p^3+5pk-k^3+5k^2$$;
8. $$a^3-216b^3-a^2+12ab-36b^2$$.

1. $$m^2-4n^2-m-2n=(m^2-4n^2)-(m+2n)$$
   $$=(m-2n)(m+2n)-(m+2n)$$
   $$=(m+2n)(m-2n-1)$$
2. $$8a^2-8b^2+3a-3b=8(a^2-b^2)+3(a-b)$$
   $$=8(a-b)(a+b)+3(a-b)$$
   $$=(a-b)(8a+8b+3)$$
3. $$x^3+x^2y-4x^2+4y^2=x^2(x+y)-4(x^2-y^2)$$
   $$=x^2(x+y)-4(x-y)(x+y)$$
   $$=(x+y)(x^2-4x+4y)$$
4. $$a^2-14a+49-ac+7c=(a^2-14a+49)-(ac-7c)$$
   $$=(a-7)^2-c(a-7)$$
   $$=(a-7)(a-7-c)$$
5. $$10xy+10xz-y^2-2yz-z^2=(10xy+10xz)-(y^2+2yz+z^2)$$
   $$=10x(y+z)-(y+z)^2$$
   $$=(y+z)(10x-y-z)$$
6. $$3m^2+3n^2-n^3+3mn+m^3=(3m^2+3mn+3n^2)+(m^3-n^3)$$
   $$=3(m^2+mn+n^2)+(m-n)(m^2+mn+n^2)$$
   $$=(m^2+mn+n^2)(m-n+3)$$
7. $$5p^2+p^3+5pk-k^3+5k^2=(5p^2+5pk+5k^2)+(p^3-k^3)$$
   $$=5(p^2+pk+k^2)+(p-k)(p^2+pk+k^2)$$
   $$=(p^2+pk+k^2)(p-k+5)$$
8. $$a^3-216b^3-a^2+12ab-36b^2=(a^3-216b^3)-(a^2-12ab+36b^2)$$
   $$=(a-6b)(a^2+6ab+36b^2)-(a-6b)^2$$
   $$=(a-6b)\bigl(a^2+6ab+36b^2-a+6b\bigr)$$

Ответ

1) $$(m+2n)(m-2n-1)$$
2) $$(a-b)(8a+8b+3)$$
3) $$(x+y)(x^2-4x+4y)$$
4) $$(a-7)(a-7-c)$$
5) $$(y+z)(10x-y-z)$$
6) $$(m^2+mn+n^2)(m-n+3)$$
7) $$(p^2+pk+k^2)(p-k+5)$$
8) $$(a-6b)(a^2+6ab+36b^2-a+6b)$$