Problem 57 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(n\) be a positive integer such that \(\left(1+x+x^{2}\right)^{\mathrm{n}}=a_{0}+a_{1} x+a_{z} x^{2}+\ldots+a_{2 a} x^{2 \mathrm{n}}\), then \(a_{\mathrm{r}}=\) (A) \(a_{\text {n- }-r}, 0 \leq r \leq 2 n\) (B) \(a_{2 n}, 0 \leq r \leq 2 n\) (C) \(a_{2 n-r}, 0 \leq r \leq 2 n\) (D) none of these

Step-by-Step Solution

Answer

(C) \(a_{2n-r}, 0 \leq r \leq 2n\)

Step 1: Understanding the Problem

We are given an expression \( (1 + x + x^{2})^{n} \), and we need to determine the general form of the coefficients \( a_r \) in the expanded polynomial. Specifically, we are asked to determine which option among (A), (B), (C), or (D) correctly describes \( a_r \).

Step 2: Binomial Expansion by Trinomial Theorem

The expression \( (1 + x + x^{2})^n \) can be expanded using the multinomial theorem, which considers the expression as a trinomial. The expansion involves all terms of the form \( \binom{n}{k_1, k_2, k_3} (1)^{k_1} (x)^{k_2} (x^2)^{k_3} \), where \( k_1 + k_2 + k_3 = n \) and the exponent of \( x \) is given by \( k_2 + 2k_3 = r \).

Step 3: Analyzing Coefficient Symmetry

The expression \( (1 + x + x^2)^n \) has a symmetry in its coefficients, similar to how a standard binomial expression \( (1 + x)^n \) has symmetry. Specifically, \( a_r = a_{2n - r} \), due to the symmetry of the trinomial expansion around the midpoint \( x^n \).

Step 4: Matching the Symmetry to Options

The symmetry identified in Step 3 matches option (C), which states \( a_r = a_{2n - r} \) for \( 0 \leq r \leq 2n \). Therefore, the correct answer is (C).

Key Concepts

Trinomial Theorem

The Trinomial Theorem is an extension of the well-known Binomial Theorem. While the Binomial Theorem deals with powers of the sum of two terms, the Trinomial Theorem addresses the expansion of expressions like \[ (1 + x + x^2)^n \]This expression has three terms, hence the name 'trinomial'. The expansion process involves using the multinomial theorem, which allows for generalizing the binomial coefficients to multinomial coefficients.Multinomial coefficients are denoted as:\[ \binom{n}{k_1, k_2, k_3} \]Here, \( k_1, k_2, \) and \( k_3 \) represent the different terms' exponents, and their sum must be equal to \( n \). Each term in the expansion \[ (1)^{k_1} (x)^{k_2} (x^2)^{k_3} \]contributes to finding the expanded form. The exponent of \( x \) can be calculated as \( r = k_2 + 2k_3 \). This process helps us understand the number of ways each power of \( x \) can appear in the trinomial expansion.

Polynomial Expansion

Polynomial expansion involves expressing a polynomial raised to a power as a sum of terms. Each term is a product of coefficients and variables raised to some power. This concept is central to understanding how expressions like\[ (1 + x + x^2)^n \]get expanded into a series of terms, where each term involves powers of \( x \) and constants. The goal is to find each coefficient corresponding to powers of \( x \) in the polynomial.When dealing with a trinomial polynomial, the expansion results in terms with the structure:

• Coefficients from multinomial terms are calculated as \( \binom{n}{k_1, k_2, k_3} \).
• The structure of the exponents of \( x \) and the sum of contributions leads to varying powers of \( x \) depending on \( k_2 \) and \( k_3 \).

This systematic expansion helps us find terms for specific powers without directly computing each term, which is useful for large \( n \).

Symmetry of Coefficients

The symmetry of coefficients in polynomial expansions is a fascinating topic. It suggests that coefficients across the expanded polynomial can mirror each other in certain conditions. For trinomial expansions, this symmetry is particularly prominent.In our trinomial polynomial \[ (1 + x + x^2)^n \]we observe symmetry such that \[ a_r = a_{2n - r} \]This means, for a polynomial of degree \( 2n \), the coefficient of \( x^r \) is equal to the coefficient of \( x^{2n - r} \). This reflection property can simplify computations significantly, as it reduces the number of coefficients that need to be directly computed. Instead, by determining half of them, the rest can be inferred by symmetry.