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Algebraic Formulas

\begin{equation} a + b = b + a \end{equation} \begin{equation} a \times b = b \times a \end{equation}

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\begin{equation} a \times (b + c) = a \times b + a \times c \end{equation}

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\begin{equation} \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \end{equation}

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\begin{equation} \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \end{equation}

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\begin{equation} \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} \end{equation}

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\begin{equation} \sqrt[n]{a} = \frac{a^{\frac{1}{n}}}{1} \end{equation}

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\begin{equation} (a + b)^2 = a^2 + 2ab + b^2 \end{equation} \begin{equation} (a - b)^2 = a^2 - 2ab + b^2 \end{equation}

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\begin{align} (a + b + c)^2 &= a^2 + b^2 + c^2 + 2ab + 2ac + 2bc \\&= a^2 + b^2 + c^2 + 2(ab + ac + bc) \end{align} \begin{align} (a - b - c)^2 &= a^2 - 2ab - 2ac + b^2 + 2bc + c^2 \end{align}

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\begin{align} (a+b)^2 + (a-b)^2 & = 2a^2 + 2b^2 \\& = 2(a^2 + b^2) \end{align} \begin{equation} (a+b)^2 - (a-b)^2 = 4ab \end{equation}

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\begin{align} (a+b)^3 & = a^3 + 3a^2b + 3ab^2 + b^3 \\& = a^3 + b^3 + 3a^2b + 3ab^2 \\& = a^3 + b^3 + 3(a^2b + ab^2) \\& = a^3 + b^3 + 3ab(a + b) \end{align} \begin{align} (a-b)^3 & = a^3 - 3a^2b + 3ab^2 - b^3 \\& = a^3 - b^3 - 3a^2b + 3ab^2 \\& = a^3 - b^3 - 3(a^2b - ab^2) \\& = a^3 - b^3 - 3ab(a - b) \end{align}

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\begin{align} a^2 - b^2 &= (a + b)(a - b) \\ (a^2 + b^2) &= (a + b)^2 - 2ab \end{align}

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\begin{align} (a ^3 + b ^3) &= (a + b) (a^2 - ab + b^2) \\ &= (a + b) (a^2 + b^2 - ab ) \end{align} \begin{align} (a^3 - b^3) &= (a - b)(a^2 + ab + b^2) \\ &= (a - b)(a^2 + b^2 + ab ) \end{align} \begin{equation} \frac{a^3 + b^3}{a + b} = a^2 - ab + b^2 \end{equation} \begin{equation} \frac{a^3 - b^3}{a - b} = a^2 + ab + b^2 \end{equation}

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\begin{equation} (a + b + c)^3 = a^3 + b^3 + c^3 + 3(a^2b + ab^2 + a^2c + bc^2 + ac^2 + b^2c) \end{equation}

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\begin{align} a^3 + b^3 + c^3 &= (a + b + c)^3 - 3(a^2b + ab^2 + a^2c + bc^2 + ac^2 + b^2c) \\ a^3 + b^3 + c^3 - 3abc &= (a + b + c)(a^2 + b^2 + c^2 - ab -bc - ca) \end{align} \begin{equation} if (a +b + c) = 0 \: then \: \: a ^3 + b ^3 + c^3 = 3abc \end{equation}

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