第1章 1.22 | 高専の数学解説

(1)

$\frac{1}{a-1}-\frac{1}{a^2-a}=\frac{a-1}{a(a-1)}=\frac{1}{a}$

$\frac{a}{a+b}+\frac{b}{a-b}-\frac{2ab}{a^2-b^2}=\frac{a(a-b)+b(a+b)-2ab}{a^2-b^2}=\frac{(a-b)^2}{(a+b)(a-b)}=\frac{a-b}{a+b}$

$\frac{2a}{a+b}+\frac{2b}{a-b}-\frac{a^2+b^2}{a^2-b^2}=\frac{2a(a-b)+2b(a+b)-a^2-b^2}{a^2-b^2}=\frac{a^2+b^2}{a^2-b^2}$

$\frac{y}{x(y-x)}+\frac{x}{y(x-y)}=\frac{-y^2+x^2}{xy(x-y)}=\frac{(x-y)(x+y)}{xy(x-y)}=\frac{x+y}{xy}$

$\frac{1}{x-1}-\frac{1}{x+1}-\frac{2}{x^2+1}=\frac{(x+1)(x^2+1)-(x-1)(x^2+1)-2(x^2-1)}{(x^2-1)(x^2+1)}=\frac{4}{x^4-1}$

$\frac{3x-1}{3x^2+5x-2}+\frac{5x-10}{x^2-4}=\frac{3x-1}{(3x-1)(x+2)}+\frac{5(x-2)}{(x-2)(x+2)}=\frac{6}{x+2}$