Last updated: 25 July 2022

Algebra

Arithmetic Operations

$$a(b+c) = ab + ac$$

$${{a \over b} + {c \over d}} = {{ad + bc} \over bd}$$

$${{a + c} \over b} = {{a \over b} + {c \over b}}$$

$${{a \over b } \over {c \over d}} = {{a \over b} \times {d \over c}} = {ad \over bc}$$

$${{x^m x^n} = x^{m+n}}$$

$${(x^m)^n} = {x^{mn}}$$

$${(xy)^n} = {x^n y^n}$$

$${x^{1/n}} = {\sqrt[n]{x}}$$

$${\sqrt[n]{xy}} = {\sqrt[n]{x} \sqrt[n]{y}}$$

$${{x^m} \over {x^n}} = {x^{m-n}}$$

$${x^{-n}} = {1 \over {x^n}}$$

$${\left( \frac{x}{y} \right) ^n} = {{x^n} \over {y^n}}$$

$${x^{m/n}} = {\sqrt[n]{x^m}} = { {\left( \sqrt[n]{x}\right)}^m}$$

$${\sqrt[n]{x \over y}} = {{\sqrt[n]{x}}\over{\sqrt[n]{y}}}$$

Factoring Special Polynomials

$$x^2 – x^2 = (x+y)(x-y)$$

$$x^3 + y^3 = (x+y)(x^2-xy+y^2)$$

$$x^3 – y^3 = (x-y)(x^2+xy+y^2)$$

Binomial Theorem

$${(x+y)^2} = {x^2 + 2xy + y^2}$$

$${(x-y)^2} = {x^2 – 2xy + y^2}$$

$${(x+y)^3} = {x^3 +3x^2y + 3xy^2 + y^3}$$

$${(x-y)^3} = {x^3 -3x^2y + 3xy^2 – y^3}$$

$${(x + y)^n} = x^n + nx^{n-1}y + {n(n-1) \over 2}{x^{n-2}y^2}$$

If $ax^2 + bx + c = 0$, then: $$x = {{-b \pm \sqrt{b^2 – 4ac} }\over 2a}$$

Inequalities and Absolute Value

If $a < b$ and $b < c$, then $a < c$

If $a < b$ then $a+c < b +c$

If $a < b$ and $c > 0$, then $ca < cb$

If $a < b$ and $c < 0$, then $ca > cb$

If $a > 0$, then:

$|x| = a$ means $x=a$ or $x = -a$

$|x| < a$ means $-a < x < a$

$|x| > a$ means $x>a or x<-a$