Factorization

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

Solution

The following equations hold:

(1) $\begin{equation*} a^2 = b^2 + c^2 \end{equation*}$

(2) $\begin{equation*} b^2 = a^2 -c^2 \end{equation*}$

(3) $\begin{equation*} c^2 = a^2 - b^2 \end{equation*}$

We have successively:

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

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