Chapter 1 代数方法 - 公式表

\[\frac{P(x)}{Q(x)} \text{ 其中 } Q(x) \neq 0\]

\[\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}\] \[\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\] \[\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}\]

\[\frac{A \cdot C}{B \cdot C} = \frac{A}{B} \text{ (当 } C \neq 0\text{)}\]

\[ax + ay = a(x + y)\]

\[a^2 - b^2 = (a + b)(a - b)\]

\[a^2 + 2ab + b^2 = (a + b)^2\] \[a^2 - 2ab + b^2 = (a - b)^2\]

\[x^2 + (p+q)x + pq = (x+p)(x+q)\] \[ax^2 + bx + c = a(x - x_1)(x - x_2)\]

\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\] \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]

\[f(p) = 0 \Leftrightarrow (x - p) \text{ 是 } f(x) \text{ 的因式}\]

\[f\left(\frac{b}{a}\right) = 0 \Leftrightarrow (ax - b) \text{ 是 } f(x) \text{ 的因式}\]

\[f(x) \div (x - a) \text{ 的余数 } = f(a)\]

\[f(x) \div (ax - b) \text{ 的余数 } = f\left(\frac{b}{a}\right)\]

\[P \Rightarrow Q\]

\[\neg Q \Rightarrow \neg P\]

\[P(1) \text{ 成立，且 } P(k) \Rightarrow P(k+1)\]

\[P = P_1 \land P_2 \land \cdots \land P_n\]