Algebra

Factoring Formulas:

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

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

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

If n is odd then (a + b) is a factor of $a^n + b^n$.

If n is even then (a - b) is a factor of $a^n - b^n$.

Product Formulas:

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

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

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

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

\[(a+b+c)^2 = a^2 + b^2 +c^2+2ab+2bc+2ca\]

Logarithms: 

\[ y = \log_a x \Leftrightarrow x = a^y , \; a>0, a \neq  1 \]

\[ \log_a 1 = 0\]

\[ \log_a a =1\]

\[ \log_a (xy) = \log_a x + \log_a y\]

\[ \log_a \frac{x}{y} = \log_a x - \log_a y\]

\[ \log_a x^n = n \log_a x\]

\[ \log_a x = \frac{ \log_c x}{ \log_c a} = \log_c x.\log_a c \]

\[ a^{ \log_a x} = x\]

Compound Interest Formulas:

A : Future value

P: Initial investment

r : Annual interest rate

t : Number of years invested

n : Number of times compounded per year

\[ A = P \left ( 1 + \frac{r}{n} \right )^{nt}\]

Continuous compound interest then,

\[A = P e^{rt}\]

Simple interest then,

\[A = P(1+rt) \]