Algebraic Identities
Formula Sheet 3
1. Square Identities
| Identity | Expansion |
|---|---|
| $$ (a+b)^2 $$ | $$ a^2 + b^2 + 2ab $$ |
| $$ (a-b)^2 $$ | $$ a^2 + b^2 – 2ab $$ |
| $$ a^2 – b^2 $$ | $$ (a-b)(a+b) $$ |
| $$ a^2 + b^2 $$ | $$ (a+b)^2 – 2ab $$ or $$ (a-b)^2 + 2ab $$ |
| $$ (a+b+c)^2 $$ | $$ a^2+b^2+c^2 + 2(ab+bc+ca) $$ |
| $$ (-a-b)^2 $$ | $$ (a+b)^2 $$ |
2. Cubic Identities
| Identity | Formula |
|---|---|
| $$ (a+b)^3 $$ | $$ a^3 + b^3 + 3ab(a+b) $$ |
| $$ (a-b)^3 $$ | $$ a^3 – b^3 – 3ab(a-b) $$ |
| $$ a^3 + b^3 $$ | $$ (a+b)(a^2 – ab + b^2) $$ |
| $$ a^3 – b^3 $$ | $$ (a-b)(a^2 + ab + b^2) $$ |
3. Special Identities
Standard Form:
$$ a^3 + b^3 + c^3 – 3abc = (a+b+c)(a^2 + b^2 + c^2 – ab – bc – ca) $$
Conditional Case:
If $$ a + b + c = 0 $$, then:
$$ a^3 + b^3 + c^3 = 3abc $$
Sum of Squares Identity:
$$ 2a^2 + 2b^2 + 2c^2 – 2ab – 2bc – 2ca = (a-b)^2 + (b-c)^2 + (c-a)^2 $$