If (1+2 x+x^2)^n=∑r=0^2 n ar x^r, then ar= (A) ( ^n Cr)^2 (B) ^n Cr · ^n Cr+1 (C) ^2 n Cr (D)

step 1 In this problem we have 1 plus 2x plus x square power n is equals to summation r equals to 0 til

If (1+2 x+x^2)^n=∑r=0^2 n ar x^r, then ar= (A) ( ^n Cr)^2...

If $\left(1+2 x+x^{2}\right)^{n}=\sum_{r=0}^{2 n} a_{r} x^{r}$, then $a_{r}=$
(A) $\left({ }^{n} C_{r}\right)^{2}$
(B) ${ }^{n} C_{r} \cdot{ }^{n} C_{r+1}$
(C) ${ }^{2 n} C_{r}$
(D) ${ }^{2 n} C_{r+1}$

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Key Concepts

Algebraic Factorization

This concept involves recognizing that an expression can be rewritten in a simpler form by factoring. In the problem, the trinomial 1 + 2x + x² is identified as a perfect square, which can be factored as (1 + x)². Recognizing such patterns allows for a simpler approach to further manipulation or expansion of the expression.

Binomial Theorem

The binomial theorem states that the power of any binomial expression can be expanded as a sum involving binomial coefficients. Specifically, for an expression of the form (1 + x)^(k), it expands to the sum from r = 0 to k of (k choose r) x^r. This theorem is central to determining the coefficients in the expansion, as it directly provides the values of each term's coefficient.

Coefficient Extraction in Polynomial Expansion

Coefficient extraction is the process of finding the specific coefficient of a particular power of the variable in a polynomial expansion. In the problem, after factoring the quadratic into (1 + x)² and raising it to the power n to get (1 + x)^(2n), the coefficient of x^(r) is directly identified as the binomial coefficient (2n choose r). This concept underpins many techniques in combinatorics and algebra.

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