103 Вправа 103Алгебра 8 клас Тарасенкова 2025 · § 2 · стор. 32

103. Скоротіть дріб.

Використовуємо формули квадрата двочлена й різниці квадратів.

1) ${\displaystyle \frac{x^3-2x^2+x}{x^3-x^2}=}$${\displaystyle \frac{x(x^2-2x+1)}{x^2(x-1)}=}$${\displaystyle \frac{x{(x-1)}^2}{x^2(x-1)}=}$${\displaystyle \frac{\cancel{x}(x-1)\cancel{\left(x-1\right)}}{\cancel{x}\cdot x\cancel{\left(x-1\right)}}=}$${\displaystyle \frac{x-1}{x}}$;

2) ${\displaystyle \frac{y^4-4y^2}{y^3+4y^2+4y}=}$${\displaystyle \frac{y^2(y^2-4)}{y(y^2+4y+4)}=}$${\displaystyle \frac{y^2(y-2)(y+2)}{y{(y+2)}^2}=}$${\displaystyle \frac{\cancel{y}\cdot y(y-2)\cancel{\left(y+2\right)}}{\cancel{y}\cancel{\left(y+2\right)}(y+2)}=}$${\displaystyle \frac{y\left(y-2\right)}{y+2}}$;

3) ${\displaystyle \frac{a^3-16a{b}^2}{a^{2}b+8a{b}^2+16b^3}=}$${\displaystyle \frac{a(a^2-16b^2)}{b(a^2+8ab+16b^2)}=}$${\displaystyle \frac{a(a-4b)(a+4b)}{b{(a+4b)}^2}=}$${\displaystyle \frac{a(a-4b)\cancel{\left(a+4b\right)}}{b\cancel{\left(a+4b\right)}(a+4b)}=}$${\displaystyle \frac{a\left(a-4b\right)}{b\left(a+4b\right)}}$;

4) ${\displaystyle \frac{c^4-6c^3+9c^2}{9c-c^3}=}$${\displaystyle \frac{c^2(c^2-6c+9)}{c(9-c^2)}=}$${\displaystyle \frac{c^2{(c-3)}^2}{c(3-c)(3+c)}=}$${\displaystyle \frac{c^2{(3-c)}^2}{c(3-c)(3+c)}=}$${\displaystyle \frac{\cancel{c}\cdot c\cancel{\left(3-c\right)}(3-c)}{\cancel{c}\cancel{\left(3-c\right)}(3+c)}=}$${\displaystyle \frac{c\left(3-c\right)}{3+c}}$.

(Тут ${\displaystyle {(c-3)}^2=}$${\displaystyle {(3-c)}^2}$.)