Упр.123 Вариант 1 Дидактические материалы ГДЗ Мерзляк Полонский 7 класс (Алгебра)

1. Разложите на множители:
   1) $$x^2-4$$;
   2) $$25-9a^2$$;
   3) $$36m^2-100n^2$$;
   4) $$0{,}04p^2-1{,}69q^2$$;
   5) $$x^2y^2-\frac{4}{9}$$;
   6) $$a^4-b^6$$;
   7) $$0{,}01c^2-d^8$$;
   8) $$0{,}81y^{10}-400z^{12}$$;
   9) $$-1+49a^4b^8$$;
   10) $$1\frac{7}{9}m^2n^2-1\frac{11}{25}a^6b^2$$.

Используем формулу разности квадратов: $$A^2-B^2=(A-B)(A+B).$$

1. $$x^2-4=x^2-2^2=(x-2)(x+2);$$
2. $$25-9a^2=5^2-(3a)^2=(5-3a)(5+3a);$$
3. $$36m^2-100n^2=(6m)^2-(10n)^2=(6m-10n)(6m+10n);$$
4. $$0{,}04p^2-1{,}69q^2=(0{,}2p)^2-(1{,}3q)^2=(0{,}2p-1{,}3q)(0{,}2p+1{,}3q);$$
5. $$x^2y^2-\frac{4}{9}=(xy)^2-\left(\frac{2}{3}\right)^2=\left(xy-\frac{2}{3}\right)\left(xy+\frac{2}{3}\right);$$
6. $$a^4-b^6=(a^2)^2-(b^3)^2=(a^2-b^3)(a^2+b^3);$$
7. $$0{,}01c^2-d^8=(0{,}1c)^2-(d^4)^2=(0{,}1c-d^4)(0{,}1c+d^4);$$
8. $$0{,}81y^{10}-400z^{12}=(0{,}9y^5)^2-(20z^6)^2=(0{,}9y^5-20z^6)(0{,}9y^5+20z^6);$$
9. $$-1+49a^4b^8=49a^4b^8-1=(7a^2b^4)^2-1^2=(7a^2b^4-1)(7a^2b^4+1);$$
10. $$1\frac{7}{9}m^2n^2-1\frac{11}{25}a^6b^2=\frac{16}{9}m^2n^2-\frac{36}{25}a^6b^2=\left(\frac{4}{3}mn\right)^2-\left(\frac{6}{5}a^3b\right)^2$$
    $$=\left(\frac{4}{3}mn-\frac{6}{5}a^3b\right)\left(\frac{4}{3}mn+\frac{6}{5}a^3b\right).$$