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Model Paper 5 — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSModel Paper 53 Marks
Question
Expand $\left(1+x+x^2\right)^3$ using binomial expansion.
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Answer

Let $y=x+x^2$. Then,
$\begin{array}{l}\left(1+x+x^2\right)^3=(1+y)^3={ }^3 C_0+{ }^3 C_1 y+{ }^3 C_2 y^2+{ }^3 C_3 y^3=1+3 y+3 y^2+y^3 \\ =1+3\left(x+x^2\right)+3\left(x+x^2\right)^2+\left(x+x^2\right)^3 \\ =1+3\left(x+x^2\right)+3\left(x^2+2 x^3+x^4\right)+\left\{{ }^3 C_0 x^3\left(x^2\right)^0+{ }^3 C_1 x^{3-1}\left(x^2\right)^1+{ }^3 C_2 x^{3-2}\left(x^2\right)^2+{ }^3 C_3 x^0\left(x^2\right)^3\right\} \\ =1+3\left(x+x^2\right)+3\left(x^2+2 x^3+x^4\right)+\left(x^3+3 x^4+3 x^5+x^6\right) \\ =x^6+3 x^5+6 x^4+7 x^3+6 x^2+3 x+1\end{array}$

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