第1章 1.24 | 高専の数学解説

(1)

$\frac{4x+3}{2x+1} = \frac{2(2x+1)+1}{2x+1} = 2 + \frac{1}{2x+1}$

$\frac{x^2-2x+3}{x-2} = \frac{x(x-2)+3}{x-2} = x + \frac{3}{x-2}$

\begin{array}{r} 3x-13\phantom{00}\\[-2pt] x+3\ \overline{\smash{\big)}\ 3x^2-4x\ +5\phantom{0}}\\[-2pt] \underline{3x^2+9x\phantom{000}}\\[-2pt] -13x+5\phantom{0}\\[-2pt] \underline{-13x-39}\\[-2pt] 44 \end{array}

$\frac{3x^2-4x+5}{x+3} = 3x-13+\frac{44}{x+3}$

\begin{array}{r} 2x-1\phantom{00}\\[-2pt] 3x-1\ \overline{\smash{\big)}\ 6x^2-5x\ +6\phantom{0}}\\[-2pt] \underline{6x^2-2x\phantom{000}}\\[-2pt] -3x+6\phantom{0}\\[-2pt] \underline{-3x+1}\\[-2pt] 5 \end{array}

$\frac{6x^2-5x+6}{3x-1} = 2x-1+\frac{5}{3x-1}$

\begin{array}{r} 3x-4\phantom{000000}\\[-2pt] x^2+2x-3\ \overline{\smash{\big)}\ 3x^3+2x^2-5x+3\phantom{0}}\\[-2pt] \underline{3x^3+6x^2-9x\phantom{00000}}\\[-2pt] -4x^2+4x+3\phantom{0}\\[-2pt] \underline{-4x^2-8x+12}\\[-2pt] 12x-9 \end{array}

$\frac{3x^3+2x^2-5x+3}{x^2+2x-3} = 3x-4+\frac{12x-9}{x^2+2x-3}$