Factors and Products Formulas

1. ${(a + b)^2} = {a^2} + 2ab + {b^2}$
2. ${(a - b)^2} = {a^2} - 2ab + {b^2}$
3. ${(a + b)^2} = {(a - b)^2} + 4ab$
4. ${(a - b)^2} = {(a + b)^2} - 4ab$
5. ${(a + b)^2} + {(a - b)^2} = 2{a^2} + 2{b^2}$
6. ${(a + b + c)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ac$
7. ${(a + b + c + \cdots )^2} = {a^2} + {b^2} + {c^2} + \cdots + 2(ab + ac + bc + \cdots )$
8. ${(a + b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3} = {a^3} + {b^3} + 3ab(a + b)$
9. ${(a - b)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3} = {a^3} - {b^3} - 3ab(a - b)$
10. $(a + b)(a - b) = {a^2} - {b^2}$
11. ${a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})$
12. ${a^3} + {b^3} = (a + b)({a^2} - ab + {b^2})$
13. $(a + b)(a + c) = {a^2} + (b + c)a + bc$
14. $(x + b)(x + c) = {x^2} + (b + c)x + bc$
15. $(a + b + c)({a^2} + {b^2} + {c^2} - ac - bc - ca) = {a^3} + {b^3} + {c^3} - 3abc$
16. ${a^n} - {b^n} = (a - b)({a^{n - 1}} + {a^{n - 2}}b + {a^{n - 3}}{b^2} + \cdots + {b^{n - 1}})$ if $n$ is odd.
17. ${a^n} - {b^n} = (a + b)({a^{n - 1}} - {a^{n - 2}}b + {a^{n - 3}}{b^2} - \cdots - {b^{n - 1}})$ if $n$ is even.
18. ${a^n} + {b^n} = (a + b)({a^{n - 1}} - {a^{n - 2}}b + {a^{n - 3}}{b^2} - \cdots - {b^{n - 1}})$ if $n$ is odd.
19. $(x + a)(x + b)(x + c) = {x^3} + (a + b + c){x^2} + (ab + bc + ac)x + abc$