Elementary Algebra Review Sheet

Basic Rules of Algebra

  • Commutative Property of Addition
    $a+b=b+a$
  • Commutative Property of Multiplication
    $ab = ba$
  • Associative property of Addition
    $(a+b)+c=a+(b+c)$
  • Associative Property of Multiplication
    $(ab)c = a(bc)$
  • Left Distributive Property
    $a(b+c) = ab + ac$
  • Right Distributive Property
    $(a+b)c = ac+bc$
  • Additive Identity Property
    $a + 0 = a$
  • Multiplicative Identity Property
    $a\cdot 1 = 1\cdot a = a$
  • Additive Inverse Property
    $a+(-a) = 0$
  • Multiplicative Inverse Property
    $a \cdot \frac{1}{a} = 1, a\neq 0$

Properties of Equality

  • Addition Property of Equality
    If $a=b$, then $a+c=b+c$
  • Multiplication Property of Equality
    If $a=b$, then $ac=bc$
  • Cancellation Property of Addition
    If $a+c = b+c$, then $a=b$
  • Cancellation Property of Multiplication
    If $ab=bc$ and $c\neq 0$, then $a=b$

Zero-Factor Property

  • If $ab=0$, then $a=0$ or $b=0$

Properties of Negation

  • Multiplication by $-1$
    $(-1)(a) = -a$ and $(-1)(-a) = a$
  • Placement of Minus Signs
    $(-a)(b) = -(ab) = (a)(-b)$
  • Product of Two Opposites
    $(-a)(-b)=ab$

Operations with Fractions

  • $\displaystyle \frac{a}{b}\cdot\frac{c}{d} = \frac{a\cdot c}{b\cdot d}$
  • $\displaystyle \frac{a}{b}+\frac{c}{d} = \frac{ad+ bc}{b d}$
  • $\displaystyle \frac{a/b}{c/d} = \frac{a}{b}\cdot \frac{d}{c}=\frac{a}{b}\div\frac{c}{d}$
  • $\displaystyle \frac{a}{b}-\frac{c}{d} = \frac{ad- bc}{b d}$

Properties of Exponents

  • $a^0 = 1$
  • $(ab)^m = a^m \cdot b^m$
  • $\dfrac{a^m}{a^n} = a^{m-n}$, $a\neq 0$
  • $a^{-n} = \dfrac{1}{a^n}$, $a\neq 0$
  • $a^m \cdot a^n = a^{m+n}$
  • $\left( a^m \right)^n = a^{m\cdot n}$
  • $\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}$, $b\neq 0$

Special Products

  • Square of a Binomial
    $(u+v)^2 = u^2+2uv+v^2$
    $(u-v)^2 = u^2-2uv+v^2$
  • Difference of Two Squares
    $u^2-v^2 = (u+v)(u-v)$
  • Difference of Two Cubes
    $u^3-v^3 = (u-v)(u^2+uv+v^2)$
  • Sum of Two Cubes
    $u^3+v^3 = (u+v)(u^2-uv+v^2)$

The Quadratic Formula

  • Solutions of $ax^2+bx+c=0$
    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Common Formulas

Temperature

  • $F = \frac{9}{5}C +32$
    $F = $ degrees Farenheit
    $C = $ degrees Celsius

Simple Interest

  • $I = P\cdot r\cdot t$
    $I = $ interest
    $P = $ principal
    $r = $ annual interest rate
    $t = $ time in years

Distance

  • $d = r\cdot t$
    $d = $ distance traveled
    $r = $ rate of travel
    $t = $ time traveled

Compound Interest

  • $A = P\left( 1+\frac{r}{n} \right)^{nt}$
    $A = $ distance traveled
    $P = $ rate of travel
    $r = $ time traveled
    $n = $ rate of travel
    $rt= $ time traveled