# Product Formulas

Real numbers: $$a$$, $$b$$, $$c$$, $$u$$, $$v$$
Natural numbers: $$n$$, $$m$$
Number of $$m$$-combinations of $$n$$ elements: $$C_n^m$$
1. Square of a difference $${\left( {a – b} \right)^2} = {a^2} – 2ab + {b^2}$$
2. Square of a sum $${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$$
3. Cube of a difference $${\left( {a – b} \right)^3} =$$ $${a^3} – 3{a^2}b + 3a{b^2} – {b^3}$$
4. Cube of a sum $${\left( {a + b} \right)^3} =$$ $${a^3} + 3{a^2}b + 3a{b^2} + {b^3}$$
5. Difference of numbers to the 4th power $${\left( {a – b} \right)^4} =$$ $${a^4} – 4{a^3}b + 6{a^2}{b^2} -$$ $$4a{b^3} + {b^4}$$
6. Sum of numbers to the 4th power $${\left( {a + b} \right)^4} =$$ $${a^4} + 4{a^3}b + 6{a^2}{b^2} +$$ $$4a{b^3} + {b^4}$$
7. Difference of numbers to the 5th power $${\left( {a – b} \right)^5} =$$ $${a^5} – 5{a^4}b + 10{a^3}{b^2} -$$ $$10{a^2}{b^3} + 5a{b^4} – {b^5}$$
8. Sum of numbers to the 5th power $${\left( {a + b} \right)^5} =$$ $${a^5} + 5{a^4}b + 10{a^3}{b^2} +$$ $$10{a^2}{b^3} + 5a{b^4} + {b^5}$$
9. Binomial theorem
$${\left( {a + b} \right)^n} =$$ $$C_n^0{a^n} + C_n^1{a^{n – 1}}b +$$ $$C_n^2{a^{n – 2}}{b^2} + \ldots$$ $$+\, C_n^{n – 1}a{b^{n – 1}} + C_n^n{b^n},$$ where $$C_n^m = \large\frac{{n!}}{{m!\left( {n – m} \right)!}}$$ are the binomial coefficients.
10. Square of a trinomial $${\left( {a + b + c} \right)^2} =$$ $${a^2} + {b^2} + {c^2} +$$ $$2ab + 2ac + 2bc$$
11. Square of a linear form
$${\left( {a + b + c + \ldots + u + v} \right)^2} =$$ $${a^2} + {b^2} + {c^2} + \ldots$$ $$+\, {u^2} + {v^2} +$$ $$2\big( {ab + ac + \ldots }$$ $${+\, au + av + bc + \ldots }$$ $${+\, bu + bv + \ldots + uv} \big)$$
12. Cube of a trinomial
$${\left( {a + b + c} \right)^3} =$$ $${a^3} + {b^3} + {c^3} +$$ $$3{a^2}b + 3a{b^2} +$$ $$3{a^2}c + 3a{c^2} +$$ $$3{b^2}c + 3b{c^2} + 6abc$$