A factorization

By Tolaso in Uncategorized on July 30, 2022.

Let $ABC$ be a right triangle at $A$ . Factor $a^3 + b^3 + c^3$ .

Solution

Since $a^2 = b^2 + c^2$ we have successively:

$\begin{align*} a^3 + b^3 + c^3 &= a \cdot a^2 + b^3 + c^3 \\ &= a \left ( b^2 + c^2 \right ) + b^3 + c^3 \\ &= ab^2 + ac^2 + b^3 + c^3 \\ &= b^2 \left ( a + b \right ) + c^2 \left ( a + c \right ) \\ &= \left ( a^2 - c^2 \right ) \left ( a + b \right ) + \left ( a^2 - b^2 \right ) \left ( a + c \right ) \\ &= \left ( a + c \right ) \left ( a + b \right ) \left ( 2a - b - c \right ) \end{align*}$