Cubic Factorization

Algebra

Basic Concept
Video

Cubic Factorization

$a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$

Example 1

Factorize $x^{3} + 8$

$\begin{aligned} x^{3} + 8 \\ x^{3} + 2^{3} \\ (x + 2) (x^{2} + 2 x + 4) \end{aligned}$

Example 2

Factorize $125 x^{3} - 27 y^{3}$

$\begin{aligned} 125 x^{3} - 27 y^{3} \\ (5 x)^{3} - (3 y)^{3} \\ (5 x - 3 y) (25 x^{2} + 15 x y + 9 y^{2}) \end{aligned}$

Example 3

Expand $(x + 4) (x^{2} - 4 x + 16)$

$\begin{aligned} (x + 4) (x^{2} - 4 x + 16) \\ x^{3} + 4^{3} \\ x^{3} + 64 \end{aligned}$

Example 4

Expand $(m - 3) (m^{2} + 3 m + 9)$

$\begin{aligned} (m - 3) (m^{2} + 3 m + 9) & = (m - 3) (m^{2} + 3 m + 3^{2}) \\ (m - 3) (m^{2} + 3 m + 9) & = m^{3} - 3^{3} \\ (m - 3) (m^{2} + 3 m + 9) & = m^{3} - 27 \end{aligned}$

Exercise

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