Algebras formulas — teorija. Matemātika, 12. klase.

Saīsinātās reizināšanas formulas $\begin{array}{l} {(a \pm b)}^{3} = a^{3} \pm 3 a^{2} b + 3 a b^{2} \pm b^{3} \\ a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2}) \\ a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2}) \end{array}$ ________________________________ Kvadrāttrinoms $a x^{2} + bx + c = a (x - x_{1}) (x - x_{2})$	Kvadrātvienādojums $\begin{array}{l} a x^{2} + bx + c = 0 \\ a \neq 0 \\ \{\begin{cases} x_{1} + x_{2} = - \frac{b}{a} \\ x_{1} \cdot x_{2} = \frac{c}{a} \end{cases} \end{array}$	Modulis $\begin{array}{l} \|a\| = \{\begin{cases} a, ja a \geq 0 \\ - a, ja a < 0 \end{cases} \\ \|a\| \geq 0 \\ \|a + b\| \leq \|a\| + \|b\| \end{array}$
Aritmētiskā progresija $\begin{array}{l} a_{n} = a_{1} + (n - 1) \cdot d \\ S_{n} = \frac{(a_{1} + a_{n}) \cdot n}{2} \\ a_{k} = \frac{a_{k + 1} + a_{k - 1}}{2} \end{array}$	Ģeometriskā progresija $\begin{array}{l} b_{n} = b_{1} \cdot q^{n - 1} \\ S_{n} = \frac{b_{1} \cdot (q^{n} - 1)}{q - 1} \\ b_{k}^{2} = b_{k - 1} \cdot b_{k + 1} \end{array}$	Bezgalīgi dilstoša ģeometriskā progresija $\begin{array}{l} \|q\| < 1 \\ S = \frac{b_{1}}{1 - q} \end{array}$
Sakņu īpašības $\begin{array}{l} \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \\ \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \\ \sqrt[n \cdot m]{a^{k \cdot m}} = \sqrt[n]{a^{k}} \\ \sqrt[n]{\sqrt[m]{a}} = \sqrt[n \cdot m]{a} \\ \sqrt[n]{a} \cdot \sqrt[k]{b} = \sqrt[n \cdot k]{a^{k} \cdot b^{n}} \\ \sqrt{a^{2}} = \|a\| \end{array}$	Pakāpju īpašības $\begin{array}{l} a^{0} = 1 \\ a^{- n} = \frac{1}{a^{n}} \\ a^{\frac{m}{n}} = \sqrt[n]{a^{m}} \\ a^{m} \cdot a^{n} = a^{m + n} \\ a^{m} \cdot b^{m} = {(ab)}^{m} \\ a^{m} : a^{n} = a^{m - n} \\ {(a^{m})}^{n} = a^{mn} \end{array}$	Logaritmu īpašības $\begin{array}{l} a^{\log_{a} b} = b \\ \log_{a} (x \cdot y) = \log_{a} x + \log_{a} y \\ \log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y \\ \log_{a} x^{k} = k \cdot \log_{a} x \\ \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \\ \log_{a^{k}} x = \frac{1}{k} \log_{a} x \end{array}$
Kombinatorika $\begin{array}{l} P_{n} = n! \\ A_{n}^{k} = \frac{n!}{(n - k)!} \end{array}$ $\begin{array}{l} C_{n}^{k} = \frac{n!}{k! (n - k)!} \\ C_{n}^{m} = C_{n}^{n - m} \\ C_{n}^{0} + C_{n}^{1} + C_{n}^{2} + ... + C_{n}^{n - 1} + C_{n}^{n} = 2^{n} \end{array}$		Varbūtību teorija $P (A) = \frac{k}{n}$ , $k$ - labvēlīgo notikumu skaits, $n$ - visu iespējamo notikumu skaits. $P (A \cup B) = P (A) + P (B)$ , kur $A$ , $B$ - nesavienojami notikumi. $P (A \cap B) = P (A) \cdot P (B)$ , kur $A$ , $B$ - neatkarīgi notikumi.

Trigonometrija

\begin{array}{l} \sin^{2} α + \cos^{2} α = 1 \\ tg α = \frac{\sin α}{\cos α} ctg α = \frac{\cos α}{\sin α} \\ 1 + {tg}^{2} α = \frac{1}{\cos^{2} α} \\ 1 + {ctg}^{2} α = \frac{1}{\sin^{2} α} \\ tg α \cdot ctg α = 1 \end{array}

\begin{array}{l} \sin 2 α = 2 \sin α \cdot \cos α \\ \cos 2 α = \cos^{2} α - \sin^{2} α \\ tg 2 α = \frac{2 tg α}{1 - {tg}^{2} α} \\ \sin (α \pm β) = \sin α \cos β \pm \cos α \sin β \\ \cos (α + β) = \cos α \cos β - \sin α \sin β \\ \cos (α - β) = \cos α \cos β + \sin α \sin β \\ tg (α + β) = \frac{tg α + tg β}{1 - (tg α \cdot tg β)} \\ tg (α - β) = \frac{tg α - tg β}{1 + (tg α \cdot tg β)} \\ \sin α + \sin β = 2 \sin \frac{α + β}{2} \cdot \cos \frac{α - β}{2} \\ \sin α - \sin β = 2 \sin \frac{α - β}{2} \cdot \cos \frac{α + β}{2} \\ \cos α + \cos β = 2 \cos \frac{α + β}{2} \cdot \cos \frac{α - β}{2} \\ \cos α - \cos β = - 2 \sin \frac{α + β}{2} \cdot \sin \frac{α - β}{2} \end{array}

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1. Algebras formulas

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