【高校数学Ⅱ】1-1-4 分数式｜問題集
1.次の計算をしなさい。
(1)\(\displaystyle \frac{15ab^4}{6a^3b^2}\)
\(\displaystyle =\frac{5b^2}{2a^2}\)
(2)\(\displaystyle \frac{8xy^3}{6x^2y}\)
\(\displaystyle =\frac{4y^2}{3x}\)
(3)\(\displaystyle \frac{9xy^2}{12y^3}\)
\(\displaystyle =\frac{3x}{4y}\)
(4)\(\displaystyle \frac{x^2-2x-3}{2x^2-7x+3}\)
\(\displaystyle =\frac{(x+1)(x-3)}{(2x-1)(x-3)}\)
\(\displaystyle =\frac{x+1}{2x-1}\)
\(\displaystyle =\frac{x+1}{2x-1}\)
(5)\(\displaystyle \frac{x^2-1}{x^2+x-2}\)
\(\displaystyle =\frac{(x+1)(x-1)}{(x+2)(x-1)}\)
\(\displaystyle =\frac{x+1}{x+2}\)
\(\displaystyle =\frac{x+1}{x+2}\)
(6)\(\displaystyle \frac{x^2+x}{x^2-1}\)
\(\displaystyle =\frac{x(x+1)}{(x+1)(x-1)}\)
\(\displaystyle =\frac{x}{x-1}\)
\(\displaystyle =\frac{x}{x-1}\)
(7)\(\displaystyle \frac{2x}{2x+1}×\frac{2x^2-3x-2}{x-2}\)
\(\displaystyle =\frac{2x}{2x+1}×\frac{(2x+1)(x-2)}{x-2}\)
\(\displaystyle =2x\)
\(\displaystyle =2x\)
(8)\(\displaystyle \frac{3x+6}{x^2+x+1}×\frac{x^3-1}{2x^2+3x-2}\)
\(\displaystyle =\frac{3(x+2)}{x^2+x+1}×\frac{(x-1)(x^2+x+1)}{(2x-1)(x+2)}\)
\(\displaystyle =\frac{3(x-1)}{2x-1}\)
\(\displaystyle =\frac{3(x-1)}{2x-1}\)
(9)\(\displaystyle \frac{x^2-9}{2y}×\frac{y^2}{2x^2-9x+9}\)
\(\displaystyle =\frac{(x+3)(x-3)}{2y}×\frac{y^2}{(2x-3)(x-3)}\)
\(\displaystyle =\frac{y(x+3)}{2(2x-3)}\)
\(\displaystyle =\frac{y(x+3)}{2(2x-3)}\)
(10)\(\displaystyle \frac{x-2}{x^2+3x}÷\frac{x^2-3x}{x^2-9}\)
\(\displaystyle =\frac{x-2}{x(x+3)}×\frac{(x+3)(x-3)}{x(x-3)}\)
\(\displaystyle =\frac{x-2}{x^2}\)
\(\displaystyle =\frac{x-2}{x^2}\)
(11)\(\displaystyle \frac{2x^2+3x-9}{x^2+x}÷\frac{2x^2+x-6}{x^3-x}\)
\(\displaystyle =\frac{(2x-3)(x+3)}{x(x+1)}×\frac{x(x+1)(x-1)}{(2x-3)(x+2)}\)
\(\displaystyle =\frac{(x+3)(x-1)}{x+2}\)
\(\displaystyle =\frac{(x+3)(x-1)}{x+2}\)
(12)\(\displaystyle \frac{x^2+5x+6}{x+1}÷\frac{2x^2-6x-20}{3x+3}\)
\(\displaystyle =\frac{(x+2)(x+3)}{x+1}×\frac{3(x+1)}{2(x+2)(x-5)}\)
\(\displaystyle =\frac{3(x+3)}{2(x-5)}\)
\(\displaystyle =\frac{3(x+3)}{2(x-5)}\)
(13)\(\displaystyle \frac{x^2-6}{x+3}+\frac{x}{x+3}\)
\(\displaystyle =\frac{x^2-6+x}{x+3}\)
\(\displaystyle =\frac{(x+3)(x-2)}{x+3}\)
\(\displaystyle =x-2\)
\(\displaystyle =\frac{(x+3)(x-2)}{x+3}\)
\(\displaystyle =x-2\)
(14)\(\displaystyle \frac{1}{2x-1}-\frac{2}{4x^2-1}\)
\(\displaystyle =\frac{1}{2x-1}-\frac{2}{(2x+1)(2x-1)}\)
\(\displaystyle =\frac{2x-1}{(2x+1)(2x-1)}\)
\(\displaystyle =\frac{1}{2x+1}\)
\(\displaystyle =\frac{2x-1}{(2x+1)(2x-1)}\)
\(\displaystyle =\frac{1}{2x+1}\)
(15)\(\displaystyle \frac{x^2-7}{x+7}+\frac{6x}{x+7}\)
\(\displaystyle =\frac{x^2-7+6x}{x+7}\)
\(\displaystyle =\frac{(x+7)(x-1)}{x+7}\)
\(\displaystyle =x-1\)
\(\displaystyle =\frac{(x+7)(x-1)}{x+7}\)
\(\displaystyle =x-1\)
(16)\(\displaystyle \frac{1}{x-1}-\frac{3}{x^3-1}\)
\(\displaystyle =\frac{1}{x-1}-\frac{3}{(x-1)(x^2+x+1)}\)
\(\displaystyle =\frac{x^2+x+1-3}{(x-1)(x^2+x+1)}\)
\(\displaystyle =\frac{(x-1)(x+2)}{(x-1)(x^2+x+1)}\)
\(\displaystyle =\frac{x+2}{x^2+x+1}\)
\(\displaystyle =\frac{x^2+x+1-3}{(x-1)(x^2+x+1)}\)
\(\displaystyle =\frac{(x-1)(x+2)}{(x-1)(x^2+x+1)}\)
\(\displaystyle =\frac{x+2}{x^2+x+1}\)
(17)\(\displaystyle \frac{2}{x+1}+\frac{3}{x-2}\)
\(\displaystyle =\frac{2(x-2)+3(x+1)}{(x+1)(x-2)}\)
\(\displaystyle =\frac{2x-4+3x+3}{(x+1)(x-2)}\)
\(\displaystyle =\frac{5x-1}{(x+1)(x-2)}\)
\(\displaystyle =\frac{2x-4+3x+3}{(x+1)(x-2)}\)
\(\displaystyle =\frac{5x-1}{(x+1)(x-2)}\)
(18)\(\displaystyle \frac{x}{x+1}+\frac{3x-1}{x^2-2x-3}\)
\(\displaystyle =\frac{x}{x+1}+\frac{3x-1}{(x+1)(x-3)}\)
\(\displaystyle =\frac{x(x-3)+3x-1}{(x+1)(x-3)}\)
\(\displaystyle =\frac{x^2-1}{(x+1)(x-3)}\)
\(\displaystyle =\frac{(x+1)(x-1)}{(x+1)(x-3)}\)
\(\displaystyle =\frac{x-1}{x-3}\)
\(\displaystyle =\frac{x(x-3)+3x-1}{(x+1)(x-3)}\)
\(\displaystyle =\frac{x^2-1}{(x+1)(x-3)}\)
\(\displaystyle =\frac{(x+1)(x-1)}{(x+1)(x-3)}\)
\(\displaystyle =\frac{x-1}{x-3}\)
(19)\(\displaystyle \frac{3x+5}{x^2-1}-\frac{1}{x^2+x}\)
\(\displaystyle =\frac{3x+5}{(x+1)(x-1)}-\frac{1}{x(x+1)}\)
\(\displaystyle =\frac{x(3x+5)-(x-1)}{x(x+1)(x-1)}\)
\(\displaystyle =\frac{3x^2+4x+1}{x(x+1)(x-1)}\)
\(\displaystyle =\frac{(3x+1)(x+1)}{x(x+1)(x-1)}\)
\(\displaystyle =\frac{3x+1}{x(x-1)}\)
\(\displaystyle =\frac{x(3x+5)-(x-1)}{x(x+1)(x-1)}\)
\(\displaystyle =\frac{3x^2+4x+1}{x(x+1)(x-1)}\)
\(\displaystyle =\frac{(3x+1)(x+1)}{x(x+1)(x-1)}\)
\(\displaystyle =\frac{3x+1}{x(x-1)}\)
(20)\(\displaystyle \frac{\frac{x}{2}-1}{1-\frac{2}{x}}\)
\(\displaystyle =\frac{x^2-2x}{2x-4}\)
\(\displaystyle =\frac{x(x-2)}{2(x-2)}\)
\(\displaystyle =\frac{x}{2}\)
\(\displaystyle =\frac{x(x-2)}{2(x-2)}\)
\(\displaystyle =\frac{x}{2}\)
(21)\(\displaystyle \frac{3x+6}{1+\frac{2}{x}}\)
\(\displaystyle =\frac{x(3x+6)}{x+2}\)
\(\displaystyle =\frac{3x(x+2)}{x+2}\)
\(\displaystyle =3x\)
\(\displaystyle =\frac{3x(x+2)}{x+2}\)
\(\displaystyle =3x\)
(22)\(\displaystyle \frac{1}{1-\frac{1}{1-\frac{1}{x}}}\)
\(\displaystyle =\frac{1}{1-\frac{x}{x-1}}\)
\(\displaystyle =\frac{x-1}{x-1-x}\)
\(\displaystyle =1-x\)
\(\displaystyle =\frac{x-1}{x-1-x}\)
\(\displaystyle =1-x\)
次の学習に進もう！