IDENTIDADES NOTABLES (FORMULARIOS)

Identidades Notables

$$\left\{\begin{array}{l} \mathbf{\hbox{Cuadrado de una suma}} \\ (a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2 \\  \mathbf{\hbox{Cuadrado de una diferencia}} \\ (a - b)^2 = (a - b)(a - b) = a^2 - 2ab + b^2 \\  \mathbf{\hbox{Suma por diferencia / Diferencia de cuadrados}} \\ (a + b)(a-b) = a^2 - b^2 \\  \mathbf{\hbox{Cubo de una suma}} \\ (a + b)^3 = (a + b)(a + b)(a + b) = a^3 + 3a^2b + 3ab^2 + b^3 \\  \mathbf{\hbox{Cubo de una diferencia}} \\ (a - b)^3 = (a - b)(a - b)(a - b) = a^3 - 3a^2b + 3ab^2 - b^3 \\ \mathbf{\hbox{Diferencia de cubos}} \\ (a - b)(a^2 + ab + b^2) = a^3 - b^3 \\  \end{array}\right.$$

Binomio de Newton (Potencia n-ésima de un binomio)

$$(a + b)^n = \displaystyle{\binom{n}{0}} a^{n}b^{0} + \displaystyle{\binom{n}{1}} a^{n-1}b^{1} + \displaystyle{\binom{n}{2}} a^{n-2}b^{2} + \cdots + \displaystyle{\binom{n}{n-2}} a^{2}b^{n-2} + \displaystyle{\binom{n}{n-1}} a^{1}b^{n-1} + \displaystyle{\binom{n}{n}} a^{0}b^{n}$$ $$(a - b)^n = \displaystyle{\binom{n}{0}} a^{n}b^{0} - \displaystyle{\binom{n}{1}} a^{n-1}b^{1} + \cdots + (-1)^{n-1}\displaystyle{\binom{n}{n-1}} a^{1}b^{n-1} + (-1)^n\displaystyle{\binom{n}{n}} a^{0}b^{n}$$

Números Combinatorios

$$\left\{\begin{array}{l} \mathbf{\hbox{(Factorial de n) }} n! = n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1 \hbox{ , } \forall n \in \mathbb{N}\\ \\ \mathbf{\hbox{(Propiedades)}} \left\{\begin{array}{l} n! = n \cdot (n-1)! \\ \\ 0! = 1 \\ \end{array}\right.\\ \\ \mathbf{\hbox{(Número combinatorio) }} \hbox{Dados n, m} \in \mathbb{N},n \geq m, \displaystyle{\binom{n}{m}} = \displaystyle{\frac{n!}{m!(n-m)!}} \\ \\ \mathbf{\hbox{(Propiedades)}} \left\{\begin{array}{l} \displaystyle{\binom{n}{0}} = 1, \displaystyle{\binom{n}{1}} = n, \displaystyle{\binom{n}{n}} = 1 \\ \\ \displaystyle{\binom{n}{m}} = \displaystyle{\binom{n}{n-m}} \\ \\ \displaystyle{\binom{n}{m}} + \displaystyle{\binom{n}{m+1}} = \displaystyle{\binom{n+1}{m+1}} \\ \end{array}\right.\\ \end{array}\right.$$

Triángulo de Tartaglia (o de Pascal)

$$\left.\begin{array}{l} \left.\begin{array}{l} & & & & 1 & & & & \\ & & & 1 & & 1 & & & \\ & & 1 & & 2 & & 1 & & \\ & 1 & & 3 & & 3 & & 1 & \\ 1 & & 4 & & 6 & & 4 & & 1\\ & & & & \vdots & & & & \\ \end{array}\right. & \Leftrightarrow & \left.\begin{array}{l} & & & & \displaystyle{\binom{0}{0}} & & & & \\ & & & \displaystyle{\binom{1}{0}} & & \displaystyle{\binom{1}{1}} & & & \\ & & \displaystyle{\binom{2}{0}} & & \displaystyle{\binom{2}{1}} & & \displaystyle{\binom{2}{2}} & & \\ & \displaystyle{\binom{3}{0}} & & \displaystyle{\binom{3}{1}} & & \displaystyle{\binom{3}{2}} & & \displaystyle{\binom{3}{3}} & \\ \displaystyle{\binom{4}{0}} & & \displaystyle{\binom{4}{1}} & & \displaystyle{\binom{4}{2}} & & \displaystyle{\binom{4}{3}} & & \displaystyle{\binom{4}{4}}\\ & & & & \vdots & & & & \\ \end{array}\right.\\ \end{array}\right.$$

Otras identidades (y sus aplicaciones)

Indeterminaciones en límites con raíces cúbicas

$$\left\{\begin{array}{l} (a - b)=(\sqrt[3]{a} - \sqrt[3]{b}) \cdot (\sqrt[3]{a^2} + \sqrt[3]{ab} + \sqrt[3]{b^2}) \\ (a + b)=(\sqrt[3]{a} + \sqrt[3]{b}) \cdot (\sqrt[3]{a^2} - \sqrt[3]{ab} + \sqrt[3]{b^2}) \\ \end{array}\right.$$

Indeterminaciones en límites con raíces cuartas

$$\left\{\begin{array}{l} (a - b)=(\sqrt[4]{a} - \sqrt[4]{b}) \cdot (\sqrt[4]{a^3} + \sqrt[4]{a^2b} + \sqrt[4]{ab^2} + \sqrt[4]{b^3}) \\ (a + b)=(\sqrt[4]{a} + \sqrt[4]{b}) \cdot (\sqrt[4]{a^3} - \sqrt[4]{a^2b} + \sqrt[4]{ab^2} - \sqrt[4]{b^3}) \\ \end{array}\right.$$