Problem 78 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a_{0}, a_{1}, a_{2}, \ldots, a_{2 \mathrm{n}}\) be the coefficients in the expansion of \(\left(1+x+x^{2}\right)^{\mathrm{n}}\) in ascending powers of \(x\), then \(a_{0}^{2}-a_{1}^{2}+a_{2}^{2}-a_{3}^{2}+\ldots-a_{2 n-1}^{2}+a_{2 n}^{2}=\) (A) \(a_{2 \mathrm{n}}\) (B) \(a_{\mathrm{n}}\) (C) \(a_{0}\) (D) none of these

Step-by-Step Solution

Answer

The alternating sum equals \(a_n\), so the answer is (B).

Step 1: Understanding the problem

We are given the coefficients of the polynomial expansion of \((1 + x + x^2)^n\) and need to evaluate the alternating sum of squares of these coefficients: \(a_0^2 - a_1^2 + a_2^2 - a_3^2 + \ldots - a_{2n-1}^2 + a_{2n}^2\).

Step 2: Polynomial interpretation in terms of roots of unity

Rewrite the polynomial \((1 + x + x^2)^n\) using the roots of unity, i.e., express 1, \( e^{2\pi i / 3} \), and \( e^{4\pi i / 3} \). These roots satisfy the equation \(1 + x + x^2 = 0\) when \(x eq 1\).

Step 3: Coefficient behavior under roots of unity substitution

Consider the substitution of values \( x = 1, e^{2\pi i / 3}, e^{4\pi i / 3} \). Evaluate \((1 + x + x^2)^n\) under these conditions to identify transformations affecting the coefficients.

Step 4: Evaluating the alternating sum using symmetry

Since these substitutions leverage the symmetry and roots of unity properties, notice that the sum alternates, allowing simplifying identification of involved coefficients. Only coefficients affected by roots of unity (and specific cases where roots result in zero) contribute to final solutions.

Step 5: Simplifying the alternating square sum expression

The result from step 4 ensures that for most terms canceled by symmetry, only \(a_n\) correctly describes the altered series due to alternating signs and polynomial degree parity (considering coefficients' symmetry).

Step 6: Conclude with selection from given options

Conclude that the alternating sum simplifies to \(a_n\) based on the symmetry and properties of roots of unity applied to the polynomial expansions. Thus, the correct option is (B) \(a_n\).

Key Concepts

Roots of Unity

The concept of roots of unity plays a crucial role in the problem of polynomial expansions. Roots of unity are solutions to the equation \(x^n = 1\). In this scenario, we are particularly interested in the cubic roots of unity, which solve the equation \(1 + x + x^2 = 0\). These roots include:

• \(1\)
• \(e^{2\pi i / 3}\)
• \(e^{4\pi i / 3}\)

These roots allow us to express any polynomial as a sum of its values at these roots, leveraging symmetry properties. By substituting them into the polynomial \((1 + x + x^2)^n\), we can simplify the transformation and interaction of the coefficients. Practically, they help reduce complex polynomial expressions by utilizing symmetry, thereby simplifying our calculations.

Alternating Sum

An alternating sum is a sequence where consecutive terms are added and subtracted successively. In our exercise, we have an alternating sum of squared coefficients from the polynomial expansion: \(a_0^2 - a_1^2 + a_2^2 - a_3^2 + \ldots - a_{2n-1}^2 + a_{2n}^2\).
This operation affects how each term in the series contributes to the final result. The alternating pattern of addition and subtraction emphasizes selective cancellation, which is a key factor for simplifying expressions.
Using the properties of symmetry, many terms effectively cancel out, leaving only a few terms that influence the sum. The careful selection and application of alternating sums are essential for revealing the underlying relationships in coefficient calculations.

Coefficient Behavior

Understanding how coefficients behave under specific transformations is vital in polynomial analysis. The coefficients of \((1 + x + x^2)^n\) represent various interactions between the roots involved.
By substituting roots of unity, we see how different instances transform the polynomial, either cancelling terms or emphasizing others. The substitution \(x = 1, e^{2\pi i / 3}, e^{4\pi i / 3}\) particularly highlights this behavior.
Subtle changes in substitution can significantly shift coefficients' contributions due to symmetry, leading to surprising yet predictable outcomes. This behavior allows us to predict which coefficients will emerge or vanish from the expanded polynomial, simplifying our complex expressions to identify terms like \(a_n\).

Polynomial Symmetry

Symmetry is a property that allows a polynomial to mirror itself across a particular axis or point. In our context, the symmetry is governed by the roots of unity and the structure of the polynomial.
For the polynomial \((1 + x + x^2)^n\), the symmetry arises due to the balanced inclusion of terms and their exponents. This symmetry can be observed when evaluating the polynomial under specific root substitutions, which leads to the cancellation of equal and opposite terms in expressions like our alternating sum.
The remarkable result is that despite seemingly complex expressions, polynomial symmetry aids in simplifying them, ultimately identifying outcomes such as the dominant term \(a_n\). Understanding such symmetry is key in leveraging mathematical properties to simplify the evaluation of polynomials.