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Algebra Level 2

$\big(1+2x+x^2\big)^n$

How many terms are there in the expansion of the above trinomial, when expanded in descending powers of $x?$

n

n+1

2n

2n+1

3 solutions

Tanishq Varshney
Apr 2, 2015

$(1+x)^{2n}$ Number of terms = $2n+1$

Akansha Dubey
Dec 18, 2017

$(1+2x+x^2) ^{n}=(1+x)^{2n}$ now, number of ways to distribute powers to 1 and $x$ in such a way that sum of their powers always comes to be equal to 2 is:- ${2n+2-1\choose 2-1} =2n+1$

Testing out case n=2 yields a degree 4 polynomial, and since the last term is a constant, namely x^0, we count from 4 down to 0 which is 5. the only solution that is equivalent to 5 when n=2 is 2n+1

Mounir Baroudi - 11 months, 2 weeks ago

Tristan Goodman
Jun 7, 2021

Since x^2 is the highest power of x in the parenthesis, the highest power of x in the expansion will be 2n. Since there is also a term that is linear in x and a term that is constant, the expansion will therefore contain every power of x smaller than 2n, all the way down to 0.

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