Question

Verify $({\mathrm{x}}^3-{\mathrm{y}}^3)=(\mathrm{x}-\mathrm{y})({\mathrm{x}}^2+\mathrm{xy}+{\mathrm{y}}^2)$using some non-zero positive integers and check by actual multiplication.

Answer 1

Step 1: Verify $({\mathrm{x}}^3-{\mathrm{y}}^3)=(\mathrm{x}-\mathrm{y})({\mathrm{x}}^2+\mathrm{xy}+{\mathrm{y}}^2)$by actual multiplication.

Let us expand R.H.S.

$\begin{array}{rcl}(\mathrm{x}-\mathrm{y})({\mathrm{x}}^2+\mathrm{xy}+{\mathrm{y}}^2)& =& {\mathrm{x}}^3-{\mathrm{x}}^2\mathrm{y}+{\mathrm{x}}^2\mathrm{y}–{\mathrm{xy}}^2+{\mathrm{xy}}^2–{\mathrm{y}}^3\\ & =& {\mathrm{x}}^3–{\mathrm{y}}^3\\ & =& \mathrm{LHS}\end{array}$

Hence Proved.

Step 2: Substituting some non-zero positive integers and check by actual multiplication.

Put $\mathrm{x}=3,\mathrm{y}=2$

$\begin{array}{rcl}\mathrm{L}.\mathrm{H}.\mathrm{S}.& =& {\mathrm{x}}^3–{\mathrm{y}}^3\\ & =& 3^3–2^3\\ & =& 27–8\\ & =& 19\end{array}$

$\begin{array}{rcl}\mathrm{R}.\mathrm{H}.\mathrm{S}.& =& (\mathrm{x}-\mathrm{y})({\mathrm{x}}^2+\mathrm{xy}+{\mathrm{y}}^2)\\ & =& (3-2)(3^2+6+2^2)\\ & =& 9+6+4\\ & =& 19\end{array}$

$\mathrm{R}.\mathrm{H}.\mathrm{S}.=\mathrm{L}.\mathrm{H}.\mathrm{S}.$

Hence proved.