Problem 97 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The coefficient of \(x^{\mathrm{n}}\) in polynomial \(\left(x+{ }^{2 \mathrm{n}+1} C_{0}\right)\left(x+{ }^{2 \mathrm{n}+1} C_{1}\right)\left(x+{ }^{2 \mathrm{n}+1} C_{2}\right) \ldots .\left(x+{ }^{2 \mathrm{n}+1} C_{\mathrm{n}}\right)\) is (A) \(2^{2 n+1}\) (B) \(2^{2 n}\) (C) \(2^{2 n-1}\) (D) none of these

Step-by-Step Solution

Answer

The correct option is (A) \(2^{2n+1}\).

Step 1: Understand the Polynomial Structure

We have a polynomial formed by the product of binomials: \((x + {}^{2n+1}C_0)(x + {}^{2n+1}C_1) \ldots (x + {}^{2n+1}C_n)\). Each binomial has a constant term \({}^{2n+1}C_k\) and a linear term \(x\).

Step 2: Consider the Coefficient of x^n

The term \(x^n\) is formed by selecting the linear term \(x\) from exactly \(n\) of the \(n+1\) binomials and the constant term \({}^{2n+1}C_k\) from the remaining \(n+1-n = 1\) binomial. Thus, the number of ways to select \(n\) binomials from \(n+1\) is \({}^{n+1}C_n = n+1\).

Step 3: Determine the Contribution of Each Choice

Each choice of \(n\) linear terms \(x\) contributes a product of the constant terms from the \(1\) remaining binomial. Thus, each such product is \({}^{2n+1}C_k\), where \(k\) is the binomial from which the constant term is selected.

Step 4: Calculate the Sum of All Coefficients

For each choice of \(n\) \(x\)'s, the remaining constant term's coefficients sum up to the symmetric sum of binomial coefficients: \(\sum_{k=0}^{n} {}^{2n+1}C_k = 2^{2n+1}\), due to the binomial theorem (sum of all binomial coefficients for a given expansion is \(2^p\) for polynomial degree \(p\)).

Step 5: Find the Coefficient of x^n

Therefore, the coefficient of \(x^n\) in the expansion is given by the number of ways to choose \(n\) factors: \({}^{n+1}C_n \times \text{sum of constant terms} = 2^{2n+1}\).

Step 6: Conclusion: Select the Correct Option

The coefficient of \(x^n\) in the given polynomial expansion is \(2^{2n+1}\). Thus, the correct option is (A) \(2^{2n+1}\).

Key Concepts

Polynomial Expansion

Polynomial expansion is a key concept in mathematics that involves expressing a polynomial raised to a certain power as a sum of terms. For example, when dealing with a polynomial like \( (x+a)^n \), expanding it means writing it out in full, such as \( x^n + nx^{n-1}a + \dots + a^n \).To perform a polynomial expansion, it's essential to understand each term's structure. Each term is typically a product of powers of the polynomial's variables and constants. You will often use the Binomial Theorem for expansion, especially where binomial expressions are involved \((a+b)^n\). This theorem provides a shortcut for expanding binomials without having to multiply them repeatedly.

• Identify the individual components: Powers of terms, like \(x^k\) or \(a^m\).
• Apply relevant equations: Use binomial coefficients, represented as \( \binom{n}{k} \), denoting combinations.
• Sum the terms: Add up all the derived terms to get the expanded polynomial.

Understanding polynomial expansion helps simplify complex algebraic expressions. It is a foundational skill, crucial for tackling more intricate mathematical challenges.

Coefficient Calculation

Calculating coefficients is a fundamental task when dealing with polynomial expressions. Coefficients are the numerical factors that multiply the variable terms in an equation. For example, in \( 3x^2 + 5x + 2 \), the coefficients are 3, 5, and 2.When you need to find the coefficient of a specific term within an expanded polynomial, you often utilize binomial coefficients. In our context, we're interested in finding the coefficient of \( x^n \) in the expanded form.Here’s how to calculate it:

• Choose the number of linear terms \(x\) from the available binomials; the rest will be constants.
• Use combinatorial counting methods to determine how many ways this can occur. The binomial coefficient \( \binom{n+1}{n} = n+1 \) illustrates the number of ways to choose \(n\) items from \(n+1\).
• Multiply the outcome of this combinatorial process with the sum of constant term contributions.

Finding coefficients requires careful use of algebraic tools like the Binomial Theorem and combinatorics, making it a crucial skill in algebra and calculus.

Combinatorics

Combinatorics is the branch of mathematics focused on counting, arranging, and finding patterns. It is vital in many fields of mathematics, including polynomial studies. Combinatorial methods are essential when working with binomials and polynomial expansions.In the problem we are addressing, combinatorics is used to determine the number of ways linear terms \((x)\) can be chosen from the polynomial's factors.

• Combinatorics helps in counting the possible arrangements or selections of items. For instance, selecting \(n\) linear terms from \(n+1\) available factors.
• Utilizing combinations, denoted by \( \binom{n}{k} \), shows different ways to pick \(k\) items from a larger set \(n\).
• This application reduces the complexity of calculating polynomial coefficients, as seen in the coefficient calculation for \(x^n\).

Through combinatorics, one can efficiently determine possible configurations within complex algebraic structures, streamlining the process of identifying coefficients in polynomial expansions.