Formulas and Tables

Elementary Algebra

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
    \({\left( {a + b + c} \right)^3} =\) \( {a^3} + {b^3} + {c^3} +\) \( 3{a^2}b + 3a{b^2} +\) \( 3{a^2}c + 3a{c^2} +\) \( 3{b^2}c + 3b{c^2} + 6abc\)