Упр.123 Вариант 2 Дидактические материалы ГДЗ Мерзляк Полонский 7 класс (Алгебра)

1. Разложите на множители:
   1) $$x^2-25$$;
   2) $$36-16y^2$$;
   3) $$4x^2-81y^2$$;
   4) $$0{,}09t^2-121p^2$$;
   5) $$a^2b^2-\frac{16}{9}$$;
   6) $$a^8-x^{10}$$;
   7) $$0{,}04b^4-a^{12}$$;
   8) $$1{,}69y^{14}-900z^8$$;
   9) $$-1+36a^6b^4$$;
   10) $$1\frac{24}{25}m^6n^4-1\frac{9}{16}a^2b^8$$.

Используем формулу разности квадратов:

$$u^2-v^2=(u-v)(u+v).$$

1. $$x^2-25=x^2-5^2=(x-5)(x+5).$$

2. $$36-16y^2=6^2-(4y)^2=(6-4y)(6+4y).$$

3. $$4x^2-81y^2=(2x)^2-(9y)^2=(2x-9y)(2x+9y).$$

4. $$0{,}09t^2-121p^2=(0{,}3t)^2-(11p)^2=(0{,}3t-11p)(0{,}3t+11p).$$

5. $$a^2b^2-\frac{16}{9}=(ab)^2-\left(\frac{4}{3}\right)^2=\left(ab-\frac{4}{3}\right)\left(ab+\frac{4}{3}\right).$$

6. $$a^8-x^{10}=(a^4)^2-(x^5)^2=(a^4-x^5)(a^4+x^5).$$

7. $$0{,}04b^4-a^{12}=(0{,}2b^2)^2-(a^6)^2=(0{,}2b^2-a^6)(0{,}2b^2+a^6).$$

8. $$1{,}69y^{14}-900z^8=(1{,}3y^7)^2-(30z^4)^2=(1{,}3y^7-30z^4)(1{,}3y^7+30z^4).$$

9. $$-1+36a^6b^4=36a^6b^4-1=(6a^3b^2)^2-1^2=(6a^3b^2-1)(6a^3b^2+1).$$

10. $$1\frac{24}{25}m^6n^4-1\frac{9}{16}a^2b^8=\frac{49}{25}m^6n^4-\frac{25}{16}a^2b^8$$

    $$=\left(\frac{7}{5}m^3n^2\right)^2-\left(\frac{5}{4}ab^4\right)^2=\left(\frac{7}{5}m^3n^2-\frac{5}{4}ab^4\right)\left(\frac{7}{5}m^3n^2+\frac{5}{4}ab^4\right).$$