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进阶
The function g is defined as g(x) =\( \frac{2x-4}{(x+13)(x-5)} \) , when x=a,the function is undefined 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
\( \frac{x+1}{5x^2}=\frac{k}{x} \)
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
分式方程与有理函数挑战
\( \frac{1}{x}+\frac{1}{x-7}=\frac{2}{x^2-7x} \)
What are all 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 1/174. What is the value of k?
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 \( \frac{1}{174} \) . What is the value of k?
原库参考解析
分式方程与有理函数挑战
\( \frac{\left(x-3\right)\left(x-6\right)}{x-2}=0 \)
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进阶
\( \frac{27\left(k+6\right)}{3\left(k+6\right)} \)
The given expression is equivalent to c, where c is a constant and \( k>−6 \) . What is the value of c?
原库参考解析
分式方程与有理函数挑战
\( \frac{1}{5xy}+xyz=\frac{1}{4yz} \)
In the given equation, x, y, and z are positive numbers. Which expression is equivalent to y?
原库参考解析
分式方程与有理函数进阶
The function g is defined as\( g\left(x\right)=\frac{2x-8}{\left(x+13\right)\left(x-6\right)} \) . If \( g(a+7)=0, \) 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
\( \frac{1}{R}=\frac{1}{p}+\frac{1}{q}+\frac{1}{s}+\frac{1}{32} \)
A parallel electric circuit contains four resistors with resistances of p ohms, q ohms, s ohms, and 32 ohms. The given equation relates the resistance R, in ohms, of the circuit, to the resistances of the four individual resistors it contains. Which equation correctly expresses R in terms of p, q, and s?
原库参考解析
分式方程与有理函数进阶
\( \frac{1}{x}+\frac{1}{x-9}=\frac{10}{x^2-9x} \)
What are all the solutions to the given equation?
原库参考解析
分式方程与有理函数进阶
If \( \frac{48}{x}=6x \), what is the value of \( 6x^2 \)?
原库参考解析
