Wzory skróconego mnożenia

Podstawowe wzory:

(1)
\begin{equation} (a+b)^2=a^2+2ab+b^2 \end{equation}
(2)
\begin{equation} (a-b)^2=a^2-2ab+b^2 \end{equation}
(3)
\begin{equation} (a+b)^3=a^3+3a^2b+3ab^2+b^3 \end{equation}
(4)
\begin{equation} (a-b)^3=a^3-3a^2b+3ab^2-b^3 \end{equation}
(5)
\begin{equation} (a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4 \end{equation}
(6)
\begin{equation} (a-b)^4=a^4-4a^3b+6a^2b^2-4ab^3+b^4 \end{equation}

(7)
\begin{equation} a^2-b^2=(a-b)(a+b) \end{equation}
(8)
\begin{equation} a^3-b^3=(a-b)(a^2+ab+b^2) \end{equation}
(9)
\begin{equation} a^3+b^3=(a+b)(a^2-ab+b^2) \end{equation}
(10)
\begin{equation} a^4-b^4=(a-b)(a^3+a^2b+ab^2+b^3) \end{equation}

(11)
\begin{equation} (a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc \end{equation}
(12)
\begin{equation} (a+b+c)^3=a^3+b^3+c^3+3a^2b+3ab^2+3a^2c+3ac^2+3b^2c+3bc^2+6abc \end{equation}
(13)
\begin{equation} (a+b+c+d)^2=a^2+b^2+c^2+d^2+2ab+2ac+2ad+2bc+2bd+2cd \end{equation}

Uogólnione wzory:

(14)
\begin{align} \forall_{a,b\in\mathbb R}:\forall_{n\in\mathbb N_+}:(a+b)^n=\displaystyle\sum_{i=0}^n {i \choose n}a^{n-i}b^{i} \end{align}
(15)
\begin{align} \forall_{a,b\in\mathbb R}:\forall_{n\in\mathbb N_+}:(a-b)^n=\displaystyle\sum_{i=0}^n {i \choose n}(-1)^ia^{n-i}b^{i} \end{align}

(16)
\begin{align} \forall_{a,b\in\mathbb R}:\forall_{n\in\mathbb N_+}:a^n-b^n=(a-b)\displaystyle\sum_{i=1}^n a^{n-i}b^{i-1} \end{align}
(17)
\begin{align} \forall_{a,b\in\mathbb R}:\forall_{k\in\mathbb N}:\forall_{n=2k+1}:a^n+b^n=(a+b)\displaystyle\sum_{i=1}^n (-1)^{i-1}a^{n-i}b^{i-1} \end{align}

(18)
\begin{align} \forall_{n\in\mathbb N_+}:\forall_{a_1, a_2,..., a_n\in\mathbb R}:\left(\displaystyle\sum_{i=1}^n a_i\right)^2=\displaystyle\sum_{i=1}^n a_i^2+2\displaystyle\sum_{i=1}^{n-1}\displaystyle\sum_{j=i+1}^n a_ia_j \end{align}
O ile nie zaznaczono inaczej, treść tej strony objęta jest licencją Creative Commons Attribution-ShareAlike 3.0 License