# Factoring Formulas

• Real numbers: $$a$$, $$b$$, $$c$$, $$x$$
Natural numbers: $$n$$
Roots of quadratic equation: $${x_1}$$, $${x_2}$$
1. Difference of squares $${a^2} – {b^2} = \left( {a + b} \right)\left( {a – b} \right)$$
2. Difference of cubes $${a^3} – {b^3} =$$ $$\left( {a – b} \right)\left( {{a^2} + ab + {b^2}} \right)$$
3. Sum of cubes $${a^3} + {b^3} =$$ $$\left( {a + b} \right)\left( {{a^2} – ab + {b^2}} \right)$$
4. $${a^4} – {b^4} =$$ $$\left( {{a^2} – {b^2}} \right)\left( {{a^2} + {b^2}} \right) =$$ $$\left( {a – b} \right)\left( {a + b} \right)\left( {{a^2} + {b^2}} \right)$$
5. $${a^5} – {b^5} =$$ $$\big( {a – b} \big)\big( {{a^4} + {a^3}b \,+}$$ $${ {a^2}{b^2} + a{b^3} + {b^4}} \big)$$
6. $${a^5} + {b^5} = \big( {a + b} \big)\big( {{a^4} – {a^3}b \,+}$$ $${ {a^2}{b^2} – a{b^3} + {b^4}} \big)$$
7. If the power $$n$$ is odd, then
$${a^n} + {b^n} =$$ $$\big( {a + b} \big)$$ $$\big( {{a^{n – 1}} – {a^{n – 2}}b \,+}$$ $${ {a^{n – 3}}{b^2} – \ldots }$$ $${-\, a{b^{n – 2}} + {b^{n – 1}}} \big)$$
8. If the power $$n$$ is even, then
$${a^n} + {b^n} =$$ $$\big( {a + b} \big)$$ $$\big( {{a^{n – 1}} – {a^{n – 2}}b \,+}$$ $${ {a^{n – 3}}{b^2} – \ldots }$$ $${+\, a{b^{n – 2}} – {b^{n – 1}}} \big)$$
9. $${a^n} – {b^n} =$$ $$\big( {a – b} \big)$$ $$\big( {{a^{n – 1}} + {a^{n – 2}}b \,+}$$ $${ {a^{n – 3}}{b^2} + \ldots }$$ $${+\, a{b^{n – 2}} + {b^{n – 1}}} \big)$$
$$a{x^2} + bx + c =$$ $$a\left( {x – {x_1}} \right)\left( {x – {x_2}} \right),$$
where $${x_1}$$, $${x_2}$$ are the roots of the quadratic equation $$a{x^2} + bx + c = 0.$$