Real numbers: \(a\), \(b\), \(c\), \(x\)

Natural numbers: \(n\)

Natural numbers: \(n\)

Roots of quadratic equation: \({x_1}\), \({x_2}\)

- Difference of squares \({a^2} – {b^2} = \left( {a + b} \right)\left( {a – b} \right)\)
- Difference of cubes \({a^3} – {b^3} =\) \( \left( {a – b} \right)\left( {{a^2} + ab + {b^2}} \right)\)
- Sum of cubes \({a^3} + {b^3} =\) \( \left( {a + b} \right)\left( {{a^2} – ab + {b^2}} \right)\)
- \({a^4} – {b^4} =\) \( \left( {{a^2} – {b^2}} \right)\left( {{a^2} + {b^2}} \right) =\) \( \left( {a – b} \right)\left( {a + b} \right)\left( {{a^2} + {b^2}} \right)\)
- \({a^5} – {b^5} =\) \( \big( {a – b} \big)\big( {{a^4} + {a^3}b \,+}\) \({ {a^2}{b^2} + a{b^3} + {b^4}} \big)\)
- \({a^5} + {b^5} = \big( {a + b} \big)\big( {{a^4} – {a^3}b \,+}\) \({ {a^2}{b^2} – a{b^3} + {b^4}} \big)\)
- If the power \(n\) is odd, then

\({a^n} + {b^n} =\) \( \big( {a + b} \big) \) \(\big( {{a^{n – 1}} – {a^{n – 2}}b \,+}\) \({ {a^{n – 3}}{b^2} – \ldots }\) \({-\, a{b^{n – 2}} + {b^{n – 1}}} \big)\) - If the power \(n\) is
*even*, then

\({a^n} + {b^n} =\) \( \big( {a + b} \big)\) \( \big( {{a^{n – 1}} – {a^{n – 2}}b \,+}\) \({ {a^{n – 3}}{b^2} – \ldots }\) \({+\, a{b^{n – 2}} – {b^{n – 1}}} \big)\) - \({a^n} – {b^n} =\) \( \big( {a – b} \big)\) \( \big( {{a^{n – 1}} + {a^{n – 2}}b \,+}\) \({ {a^{n – 3}}{b^2} + \ldots }\) \({+\, a{b^{n – 2}} + {b^{n – 1}}} \big)\)
- Factoring a quadratic trinomial

\(a{x^2} + bx + c =\) \( a\left( {x – {x_1}} \right)\left( {x – {x_2}} \right),\)

where \({x_1}\), \({x_2}\) are the roots of the quadratic equation \(a{x^2} + bx + c = 0.\)