Algebraic Identities & Expressions

• $(a+b)^2 = a^2 + b^2 + 2ab$
• $(a-b)^2 = a^2 + b^2 - 2ab$
• $a^2 - b^2 = (a-b)(a+b)$
• $(a+b)^3 = a^3 + b^3 + 3ab(a+b)$
• $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$
• $a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)$
• Special Case: If $a+b+c=0$, then $a^3+b^3+c^3 = 3abc$.

Convert abstract algebra into simple arithmetic by assuming values for variables.

• Rule: Never let the denominator become zero.
• Golden Values: Try $a=1, b=1, c=0$ or $a=2, b=1, c=-3$.
• Symmetry: If all variables are symmetric in the expression, try $a=b=c$.

• If $x + 1/x = k$, then $x^2 + 1/x^2 = k^2 - 2$
• If $x - 1/x = k$, then $x^2 + 1/x^2 = k^2 + 2$
• If $x + 1/x = 1$, then $x^3 = -1$
• If $x + 1/x = sqrt{3}$, then $x^6 = -1$