MOG 2015 1

1

Part (a)

\begin{align} (a-b)(a^2+ab+b^2)&=a^3+a^2b+ab^2-a^2b-ab^2-b^3\\ &=a^3-b^3 \end{align}

Part (b)
Using the similar factorisation for a^3+b^3 :

\begin{align} \frac{2016^3+2015^3}{2016^2-2015^2}&=\frac{(2016+2015)(2016^2-2016\times 2015+2015^2)}{(2016+2015)(2016-2015)}\\ &=\frac{2016^2-2016\times 2015+2015^2}{2016-2015}\\ &=2016^2-2016\times 2015+2015^2\\ &=2016^2-2\times 2016\times 2015+2015^2+2016\times 2015\\ &=(2016-2015)^2+2016\times 2015\\ &=1+4062240\\ &=4062241 \end{align}
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