这部分需要会什么？
- 先确定分式的定义域，再化简或解方程
- 区分约分后的空点与竖直渐近线
- 解完分式方程检验候选解是否使分母为零
核心概念与公式
分式的原分母不能为零。约分只能消去公共因式，不能单独消去加减项；约分后仍要保留原式的限制。若 \((x-r)\) 被完全约去且剩余分母在 \(r\) 处非零，原函数在 \(x=r\) 处有空点；若 \((x-r)\) 约不掉，\(x=r\) 处是竖直渐近线。
解分式方程常用通分或交叉相乘化为多项式方程，但由此得到的候选解必须代回原分母检查，任何使分母为零的解都要舍去。
做题时怎样使用
- 先写出所有分母不为零的限制，再通分、约分或交叉相乘
- 化成多项式方程后求出候选解
- 把候选解代回原分母检查，舍去使分母为零的增根
容易混淆的地方
把 \((x^2-1)/(x-1)\) 约成 \(x+1\) 后仍需 \(x\ne1\)。交叉相乘可能引入使原分母为零的增根，必须检验后舍去。
用一个例题串起方法
知识点教学例题解 \(\frac{1}{x}+\frac{1}{x-2}=\frac{2}{x^2-2x}\)。
展开解题思路 ＋
因为 \(x^2-2x=x(x-2)\)，定义域要求 \(x\ne0\) 且 \(x\ne2\)。两边乘 \(x(x-2)\) 得 \((x-2)+x=2\)，即 \(2x-2=2\)，\(x=2\)。但 \(x=2\) 使原分母为零，必须舍去，所以原方程无解。
本知识点题库
先独立作答，再查看答案与解析。
分式方程与有理函数挑战
\( \frac{25x}{x^2-25}=\frac{x}{x-5}+\frac{20}{x+5} \)
What is the positive solution to the given equation?
原库参考解析
分式方程与有理函数挑战
Which expression is equivalent to\( \frac{y+13}{x-10}+\frac{y\left(x-10\right)}{x^2y-10xy} \) ?
原库参考解析
分式方程与有理函数挑战
\( \frac{2}{c}-\frac{7}{d}=5 \)
The given equation relates the variables c and d, where c and d are greater than 1. Which equation correctly expresses d in terms of c?
原库参考解析
分式方程与有理函数进阶
The function g is defined as \( g\left(x\right)=\frac{2x-6}{\left(x+13\right)\left(x-8\right)} \) . If \( g\left(a+7\right)=0 \), where a is a constant, what is the value of a?
原库参考解析
分式方程与有理函数挑战
\( \frac{32}{p}=\frac{32}{q}-\frac{32}{r}-\frac{32}{s} \)
The given equation relates the positive variables p, q, r, and s. Which of the following is equivalent to q?
原库参考解析
分式方程与有理函数进阶
If \( \frac{72}{x}=6x \), what is the value of \( 6x^2 \) ?
原库参考解析
分式方程与有理函数进阶
\( \frac{\left(x-8\right)\left(x-16\right)}{x-7}=0 \)
What is the sum of the solutions to the given equation?
原库参考解析
分式方程与有理函数挑战
The function \( f \) is defined by\( f\left(x\right)=\frac{x^2+ax+b}{2x+c} \) , where a, b, and c are constants. The graph of the function \( f \) in the xy-plane, where \( y=f\left(x\right) \), does not intersect the line \( x=4 \) . If \( f(5)=f(6)=0 \). What is the value of \( a+b+c \) ?
原库参考解析
分式方程与有理函数挑战
\( \frac{46}{p}=\frac{46}{q}-\frac{46}{r}-\frac{46}{s} \)
The given equation relates the positive variables p, q, r, and s. Which of the following is equivalent to q?
原库参考解析
分式方程与有理函数进阶
In the xy-plane, the graph of function \( f \), where \( y=f(x) \) , has exactly 8 x-intercepts. One of these x-intercepts is (17, 0). The rational function g is defined by \( g(x)=\frac{f(x)}{x-17} \). In the xy-plane, how many x-intercepts does the graph of \( y=g(x) \) have?
原库参考解析
分式方程与有理函数进阶
\( \frac{\left(x-7\right)\left(x-9\right)}{x-4}=0 \)
What is the sum of the solutions to the given equation?
原库参考解析
分式方程与有理函数挑战
\( \frac{x+1}{5x^2}=\frac{k}{x} \)
In the given equation, k is a constant. The solution to the given equation is \( \frac{1}{224} \). What is the value of k ?
原库参考解析
