Упражнение 285 к § 10

\(1 - \dfrac{3x^2}{1 - x^2} = \dfrac{1 - x^2 - 3x^2}{1 - x^2} = \dfrac{1 - 4x^2}{1 - x^2} = \dfrac{(1 - 2x)(1 + 2x)}{(1 - x)(1 + x)}\).

\(\dfrac{x}{x + 1} + 1 = \dfrac{x + x + 1}{x + 1} = \dfrac{2x + 1}{x + 1}\).

\(\dfrac{(1 - 2x)(1 + 2x)}{(1 - x)(1 + x)} : \dfrac{2x + 1}{x + 1} = \dfrac{(1 - 2x)(1 + 2x)(x + 1)}{(1 - x)(1 + x)(2x + 1)} = \dfrac{1 - 2x}{1 - x}\)

(при \(x \ne \pm 1\), \(x \ne -\dfrac{1}{2}\)).

\(\dfrac{a + b}{b} - \dfrac{a}{a + b} = \dfrac{(a + b)^2 - ab}{b(a + b)} = \dfrac{a^2 + ab + b^2}{b(a + b)}\).

\(\dfrac{a + b}{a} - \dfrac{b}{a + b} = \dfrac{(a + b)^2 - ab}{a(a + b)} = \dfrac{a^2 + ab + b^2}{a(a + b)}\).

\(\dfrac{a^2 + ab + b^2}{b(a + b)} : \dfrac{a^2 + ab + b^2}{a(a + b)} = \dfrac{(a^2 + ab + b^2) \cdot a(a + b)}{b(a + b)(a^2 + ab + b^2)} = \dfrac{a}{b}\).

\(a^3 - 1 = (a - 1)(a^2 + a + 1)\), а \(\dfrac{2}{1 - a} = -\dfrac{2}{a - 1}\). Общий знаменатель — \(a^3 - 1\):

\(\dfrac{3a^2 - a + 3 - (a - 1)^2 - 2(a^2 + a + 1)}{a^3 - 1} = \dfrac{3a^2 - a + 3 - a^2 + 2a - 1 - 2a^2 - 2a - 2}{a^3 - 1} = \dfrac{-a}{a^3 - 1} = \dfrac{a}{1 - a^3}\)

(при \(a \ne 1\)).

\(\dfrac{2b}{1 - b} - b = \dfrac{2b - b + b^2}{1 - b} = \dfrac{b(1 + b)}{1 - b}\).

\(\dfrac{b(1 + b)}{1 - b} : \dfrac{3(b + 1)}{b - 1} = \dfrac{b(b + 1)(b - 1)}{(1 - b) \cdot 3(b + 1)} = -\dfrac{b}{3}\)

(при \(b \ne \pm 1\)), так как \(\dfrac{b - 1}{1 - b} = -1\).

\(a - x + \dfrac{x^2}{a + x} = \dfrac{(a - x)(a + x) + x^2}{a + x} = \dfrac{a^2 - x^2 + x^2}{a + x} = \dfrac{a^2}{a + x}\).

\(\dfrac{a^2}{a + x} \cdot \dfrac{a - x}{a} = \dfrac{a(a - x)}{a + x}\) (при \(a \ne 0\), \(a \ne -x\)).