By using binomial theorem, expand \((1 + x + x^2)^3\).

Solution :

Let y = x + \(x^2\). Then,

\((1 + x + x^2)^3\) = \((1 + y)^3\)

= \(^3C_0\) + \(^3C_1 y\) + \(^3C_2 y^2\) + \(^3C_3 y^3\)

= \(1 + 3y + 3y^2 + y^3\) = 1 + 3\((x + x^2)\) + 3\((x + x^2)^2\) + \((x + x^2)^3\)

= \(x^6 + 3x^5 + 6x^4 + 7x^3 + 6x^2 + 3x + 1\)


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