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How do I expand this equation: $(1+t+t^2)^5$

I formed the equation into a binomial equation this way: $(1+t+t^2)^5=\sum \binom{5}{r_1}\binom{5-r_1}{r_2}t^{r_2}t^{2r_1}$

But I cannot remember how to continue from here to solve for the $r_1$ and $r_2$ terms from here and then to further expand it.

Thanks for any help!

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    Hint: $(1+t+t^2)^5=(1+[t+t^2])^5$2011-08-21

2 Answers 2

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You could use the trinomial expansion, a specific case of the multinomial formula. Your formula would then be $ (1+t+t^2)^5=\sum_{i,j,k}\binom{n}{i,j,k}1^it^j(t^2)^k $ for $i,j,k$ nonnegative, where $i+j+k=5$. It's then just a matter of finding all possible sums and plugging in.

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You can take $(1+t)$ as $a$ and $t^2$ as $b$. Then solve by using same binomial formula.

Your answer after simplifying will be:

$1+5t+15t^2+30t^3+45t^4+51t^5+45t^6+30t^7+15t^8+5t^9+t^{10}$

Hope this will help you