详解代數分式及因式分解 | DSE 甲部 30 題操練

\(\displaystyle \frac{6x-12}{3}\)

解：

\(6x-12 = 6(x-2)\)
\(\displaystyle \frac{6(x-2)}{3} = 2(x-2)\)
\(= 2x-4\)

答案： \(2x-4\)

\(\displaystyle \frac{x^2-9}{x^2+6x+9}\)

解：

\(x^2-9 = (x-3)(x+3)\)
\(x^2+6x+9 = (x+3)^2\)
\(\displaystyle \frac{(x-3)(x+3)}{(x+3)^2} = \frac{x-3}{x+3}\)

答案： \(\displaystyle \frac{x-3}{x+3}\)

\(\displaystyle \frac{2x^2-8}{x^2-4x+4}\)

解：

\(2x^2-8 = 2(x^2-4) = 2(x-2)(x+2)\)
\(x^2-4x+4 = (x-2)^2\)
\(\displaystyle \frac{2(x-2)(x+2)}{(x-2)^2} = \frac{2(x+2)}{x-2}\)

答案： \(\displaystyle \frac{2(x+2)}{x-2}\)

\(\displaystyle \frac{3x^2-12x+12}{x-2}\)

解：

\(3x^2-12x+12 = 3(x^2-4x+4) = 3(x-2)^2\)
\(\displaystyle \frac{3(x-2)^2}{x-2} = 3(x-2)\)

答案： \(3(x-2)\)

\(\displaystyle \frac{x^2+5x+6}{x^2-4}\)

解：

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

答案： \(\displaystyle \frac{x+3}{x-2}\)

\(\displaystyle \frac{4x^2-9}{2x^2+3x-2}\)

解：

\(4x^2-9 = (2x-3)(2x+3)\)
\(2x^2+3x-2 = (2x-1)(x+2)\)
沒有共同因式，已是最簡： \(\displaystyle \frac{(2x-3)(2x+3)}{(2x-1)(x+2)}\)

答案： \(\displaystyle \frac{(2x-3)(2x+3)}{(2x-1)(x+2)}\)

\(\displaystyle \frac{x^3-8}{x^2-4}\)

解：

\(x^3-8 = (x-2)(x^2+2x+4)\)
\(x^2-4 = (x-2)(x+2)\)
\(\displaystyle \frac{(x-2)(x^2+2x+4)}{(x-2)(x+2)} = \frac{x^2+2x+4}{x+2}\)

答案： \(\displaystyle \frac{x^2+2x+4}{x+2}\)

\(\displaystyle \frac{2x^2-18}{4x^2-12x}\)

解：

\(2x^2-18 = 2(x^2-9) = 2(x-3)(x+3)\)
\(4x^2-12x = 4x(x-3)\)
\(\displaystyle \frac{2(x-3)(x+3)}{4x(x-3)} = \frac{2(x+3)}{4x} = \frac{x+3}{2x}\)

答案： \(\displaystyle \frac{x+3}{2x}\)

\(\displaystyle \frac{x^2-2x-15}{x^2-7x+12}\)

解：

\(x^2-2x-15 = (x-5)(x+3)\)
\(x^2-7x+12 = (x-3)(x-4)\)
沒有共同因式。 \(\displaystyle \frac{(x-5)(x+3)}{(x-3)(x-4)}\)

答案： \(\displaystyle \frac{(x-5)(x+3)}{(x-3)(x-4)}\)

\(\displaystyle \frac{6x^2-24}{3x^2-6x}\)

解：

\(6x^2-24 = 6(x^2-4) = 6(x-2)(x+2)\)
\(3x^2-6x = 3x(x-2)\)
\(\displaystyle \frac{6(x-2)(x+2)}{3x(x-2)} = \frac{2(x+2)}{x}\)

答案： \(\displaystyle \frac{2(x+2)}{x}\)

\(\displaystyle \frac{2x}{5} \times \frac{15}{4x^2}\)

解：

\(\displaystyle \frac{2x \cdot 15}{5 \cdot 4x^2} = \frac{30x}{20x^2} = \frac{3}{2x}\)

答案： \(\displaystyle \frac{3}{2x}\)

\(\displaystyle \frac{x+1}{x-2} \times \frac{x^2-4}{x^2-1}\)

解：

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

答案： \(\displaystyle \frac{x+2}{x-1}\)

\(\displaystyle \frac{3x+6}{2x-4} \times \frac{x^2-4}{x^2+4x+4}\)

解：

\(3x+6 = 3(x+2)\)
\(2x-4 = 2(x-2)\)
\(x^2-4 = (x-2)(x+2)\)
\(x^2+4x+4 = (x+2)^2\)
\(\displaystyle \frac{3(x+2)}{2(x-2)} \times \frac{(x-2)(x+2)}{(x+2)^2} = \frac{3}{2}\)

答案： \(\displaystyle \frac{3}{2}\)

\(\displaystyle \frac{x^2-4x+4}{x^2-1} \times \frac{x+1}{x-2}\)

解：

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

答案： \(\displaystyle \frac{x-2}{x-1}\)

\(\displaystyle \frac{2x^2+5x+2}{x^2-4} \times \frac{x-2}{2x+1}\)

解：

\(2x^2+5x+2 = (2x+1)(x+2)\)
\(x^2-4 = (x-2)(x+2)\)
\(\displaystyle \frac{(2x+1)(x+2)}{(x-2)(x+2)} \times \frac{x-2}{2x+1} = 1\)

答案： \(1\)

\(\displaystyle \frac{7x-14}{3x} \div \frac{4x-8}{7}\)

解：

\(\displaystyle \frac{7x-14}{3x} \times \frac{7}{4x-8}\)
\(7x-14 = 7(x-2)\)
\(4x-8 = 4(x-2)\)
\(\displaystyle \frac{7(x-2)}{3x} \times \frac{7}{4(x-2)} = \frac{49}{12x}\)

答案： \(\displaystyle \frac{49}{12x}\)

\(\displaystyle \frac{x^2-9}{x^2-4} \div \frac{x+3}{x+2}\)

解：

\(\displaystyle \frac{x^2-9}{x^2-4} \times \frac{x+2}{x+3}\)
\(x^2-9 = (x-3)(x+3)\)
\(x^2-4 = (x-2)(x+2)\)
\(\displaystyle \frac{(x-3)(x+3)}{(x-2)(x+2)} \times \frac{x+2}{x+3} = \frac{x-3}{x-2}\)

答案： \(\displaystyle \frac{x-3}{x-2}\)

\(\displaystyle \frac{x}{x+1} \div \frac{x^2}{x^2-1}\)

解：

\(\displaystyle \frac{x}{x+1} \times \frac{x^2-1}{x^2}\)
\(x^2-1 = (x-1)(x+1)\)
\(\displaystyle \frac{x}{x+1} \times \frac{(x-1)(x+1)}{x^2} = \frac{x-1}{x}\)

答案： \(\displaystyle \frac{x-1}{x}\)

\(\displaystyle \frac{2x^2-8}{x^2-4x+4} \div \frac{x+2}{x-2}\)

解：

\(\displaystyle \frac{2x^2-8}{x^2-4x+4} \times \frac{x-2}{x+2}\)
\(2x^2-8 = 2(x-2)(x+2)\)
\(x^2-4x+4 = (x-2)^2\)
\(\displaystyle \frac{2(x-2)(x+2)}{(x-2)^2} \times \frac{x-2}{x+2} = 2\)

答案： \(2\)

\(\displaystyle \frac{x^2-5x+6}{x^2-3x+2} \div \frac{x^2-4x+3}{x^2-1}\)

解：

\(\displaystyle \frac{x^2-5x+6}{x^2-3x+2} \times \frac{x^2-1}{x^2-4x+3}\)
因式分解每個二次式：
\(x^2-5x+6 = (x-2)(x-3)\)
\(x^2-3x+2 = (x-1)(x-2)\)
\(x^2-1 = (x-1)(x+1)\)
\(x^2-4x+3 = (x-1)(x-3)\)
\(\displaystyle \frac{(x-2)(x-3)}{(x-1)(x-2)} \times \frac{(x-1)(x+1)}{(x-1)(x-3)}\)
約去公因式：
\(\displaystyle \frac{x-3}{x-1} \times \frac{x+1}{x-3} = \frac{x+1}{x-1}\)

答案： \(\displaystyle \frac{x+1}{x-1}\)

\(\displaystyle \frac{3}{x} + \frac{2}{x}\)

解：

\(\displaystyle \frac{3+2}{x} = \frac{5}{x}\)

答案： \(\displaystyle \frac{5}{x}\)

\(\displaystyle \frac{2}{x-1} + \frac{3}{x+2}\)

解：

LCD：\((x-1)(x+2)\)
\(\displaystyle \frac{2(x+2)}{(x-1)(x+2)} + \frac{3(x-1)}{(x-1)(x+2)}\)
\(\displaystyle = \frac{2x+4+3x-3}{(x-1)(x+2)} = \frac{5x+1}{(x-1)(x+2)}\)

答案： \(\displaystyle \frac{5x+1}{(x-1)(x+2)}\)

\(\displaystyle \frac{5}{x} - \frac{2}{x+3}\)

解：

LCD：\(x(x+3)\)
\(\displaystyle \frac{5(x+3)}{x(x+3)} - \frac{2x}{x(x+3)}\)
\(\displaystyle = \frac{5x+15-2x}{x(x+3)} = \frac{3x+15}{x(x+3)}\)

答案： \(\displaystyle \frac{3x+15}{x(x+3)}\)

\(\displaystyle \frac{4}{x} - \frac{2}{x-1}\)

解：

LCD：\(x(x-1)\)
\(\displaystyle \frac{4(x-1)}{x(x-1)} - \frac{2x}{x(x-1)}\)
\(\displaystyle = \frac{4x-4-2x}{x(x-1)} = \frac{2x-4}{x(x-1)}\)

答案： \(\displaystyle \frac{2x-4}{x(x-1)}\)

\(\displaystyle \frac{1}{x^2-1} + \frac{1}{x^2+2x+1}\)

解：

\(x^2-1 = (x-1)(x+1)\)
\(x^2+2x+1 = (x+1)^2\)
LCD：\((x-1)(x+1)^2\)
\(\displaystyle \frac{1}{(x-1)(x+1)} + \frac{1}{(x+1)^2}\)
\(\displaystyle = \frac{(x+1)}{(x-1)(x+1)^2} + \frac{(x-1)}{(x-1)(x+1)^2}\)
\(\displaystyle = \frac{x+1+x-1}{(x-1)(x+1)^2} = \frac{2x}{(x-1)(x+1)^2}\)

答案： \(\displaystyle \frac{2x}{(x-1)(x+1)^2}\)

\(\displaystyle \frac{2}{x^2-4x+3} - \frac{3}{x^2-3x+2}\)

解：

\(x^2-4x+3 = (x-1)(x-3)\)
\(x^2-3x+2 = (x-1)(x-2)\)
LCD：\((x-1)(x-2)(x-3)\)
\(\displaystyle \frac{2}{(x-1)(x-3)} - \frac{3}{(x-1)(x-2)}\)
\(\displaystyle = \frac{2(x-2)}{(x-1)(x-2)(x-3)} - \frac{3(x-3)}{(x-1)(x-2)(x-3)}\)
\(\displaystyle = \frac{2x-4-3x+9}{(x-1)(x-2)(x-3)} = \frac{-x+5}{(x-1)(x-2)(x-3)}\)

答案： \(\displaystyle \frac{-x+5}{(x-1)(x-2)(x-3)}\)

\(\displaystyle \frac{x}{x^2-1} + \frac{1}{x^2-2x+1}\)

解：

\(x^2-1 = (x-1)(x+1)\)
\(x^2-2x+1 = (x-1)^2\)
LCD：\((x-1)^2(x+1)\)
\(\displaystyle \frac{x}{(x-1)(x+1)} + \frac{1}{(x-1)^2}\)
\(\displaystyle = \frac{x(x-1)}{(x-1)^2(x+1)} + \frac{(x+1)}{(x-1)^2(x+1)}\)
\(\displaystyle = \frac{x^2-x + x+1}{(x-1)^2(x+1)} = \frac{x^2+1}{(x-1)^2(x+1)}\)

答案： \(\displaystyle \frac{x^2+1}{(x-1)^2(x+1)}\)

\(\displaystyle \frac{3}{x^2+3x+2} + \frac{2}{x^2+2x+1}\)

解：

\(x^2+3x+2 = (x+1)(x+2)\)
\(x^2+2x+1 = (x+1)^2\)
LCD：\((x+1)^2(x+2)\)
\(\displaystyle \frac{3}{(x+1)(x+2)} + \frac{2}{(x+1)^2}\)
\(\displaystyle = \frac{3(x+1)}{(x+1)^2(x+2)} + \frac{2(x+2)}{(x+1)^2(x+2)}\)
\(\displaystyle = \frac{3x+3+2x+4}{(x+1)^2(x+2)} = \frac{5x+7}{(x+1)^2(x+2)}\)

答案： \(\displaystyle \frac{5x+7}{(x+1)^2(x+2)}\)

\(\displaystyle \frac{x+2}{x^2-2x} - \frac{x}{x^2-4}\)

解：

\(x^2-2x = x(x-2)\)
\(x^2-4 = (x-2)(x+2)\)
LCD：\(x(x-2)(x+2)\)
\(\displaystyle \frac{x+2}{x(x-2)} - \frac{x}{(x-2)(x+2)}\)
\(\displaystyle = \frac{(x+2)^2}{x(x-2)(x+2)} - \frac{x^2}{x(x-2)(x+2)}\)
\(\displaystyle = \frac{x^2+4x+4 - x^2}{x(x-2)(x+2)} = \frac{4x+4}{x(x-2)(x+2)} = \frac{4(x+1)}{x(x-2)(x+2)}\)

答案： \(\displaystyle \frac{4(x+1)}{x(x-2)(x+2)}\)

\(\displaystyle \frac{4}{x^2-5x+6} - \frac{3}{x^2-4x+3} + \frac{2}{x^2-3x+2}\)

解：

因式分解每個分母：
\(x^2-5x+6 = (x-2)(x-3)\)
\(x^2-4x+3 = (x-1)(x-3)\)
\(x^2-3x+2 = (x-1)(x-2)\)
LCD：\((x-1)(x-2)(x-3)\)
\(\displaystyle \frac{4}{(x-2)(x-3)} - \frac{3}{(x-1)(x-3)} + \frac{2}{(x-1)(x-2)}\)
\(\displaystyle = \frac{4(x-1)}{(x-1)(x-2)(x-3)} - \frac{3(x-2)}{(x-1)(x-2)(x-3)} + \frac{2(x-3)}{(x-1)(x-2)(x-3)}\)
\(\displaystyle = \frac{4x-4 - 3x+6 + 2x-6}{(x-1)(x-2)(x-3)}\)
\(\displaystyle = \frac{3x-4}{(x-1)(x-2)(x-3)}\)

答案： \(\displaystyle \frac{3x-4}{(x-1)(x-2)(x-3)}\)