Problem 23 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\left(1+x+x^{2}\right)^{n}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{2 n} x^{2 n}\), then \(a_{0}+a_{3}+a_{6}+\ldots=\) (A) \(3^{n}\) (B) \(3^{n-1}\) (C) \(3^{n-2}\) (D) None of these

Step-by-Step Solution

Answer

The answer is (B) \(3^{n-1}\).

Step 1: Understand the problem statement

We need to find the sum of coefficients of terms where the power of \(x\) is a multiple of 3 in the expansion of \( (1 + x + x^2)^n \). The coefficients \(a_0, a_3, a_6, \ldots\) correspond to terms \(x^0, x^3, x^6, \ldots\) in the expansion.

Step 2: Use roots of unity

Using the roots of unity, let's evaluate the given polynomial at \(x = 1\), \(x = \omega\), and \(x = \omega^2\), where \(\omega = e^{2\pi i / 3}\) is a primitive third root of unity.

Step 3: Evaluate the expression at x = 1

Plugging \(x = 1\) into \(P(x) = (1 + x + x^2)^n\), we get \((1 + 1 + 1)^n = 3^n\).

Step 4: Evaluate the expression at x = \(\omega\)

Plugging \(x = \omega\) into \(P(x)\), we obtain \((1 + \omega + \omega^2)^n\). Since \(1 + \omega + \omega^2 = 0\), it evaluates to \(0^n = 0\).

Step 5: Evaluate the expression at x = \(\omega^2\)

Similarly, for \(x = \omega^2\), \((1 + \omega^2 + (\omega^2)^2)^n = (1 + \omega^2 + \omega)^n\). It also evaluates to \(0^n = 0\), because \(1 + \omega + \omega^2 = 0\).

Step 6: Calculate the sum using symmetries

The sum of the coefficients we seek is \(\frac{P(1) + P(\omega) + P(\omega^2)}{3} = \frac{3^n + 0 + 0}{3} = 3^{n-1}\).

Step 7: Select the correct option

Therefore, the sum \(a_0 + a_3 + a_6 + \ldots\) is \( 3^{n-1} \), which corresponds to option (B).

Key Concepts

Polynomial Expansion

Polynomial expansion is a powerful algebraic tool used to express expressions raised to a power in an expanded form. When we look at the polynomial \( (1 + x + x^2)^n \), we can understand it as a series of terms generated when this expression is multiplied with itself \( n \) times. This technique is crucial in simplifying expressions and solving various mathematical problems.

The expanded form of a polynomial like this produces a sequence of terms where each term is some combination of the elements \( 1, x, \) and \( x^2 \). In this scenario, the task involves finding specific coefficients in this expansion. Each of these terms emerges by choosing factors from each binomial, following rules of combination and multiplication.

A beautiful symmetry in polynomial expansions is exemplified when the concept of roots of unity is applied to solve problems like finding the sum of coefficients of powers of powers of three. This makes polynomial expansion a versatile and essential tool in the field of mathematics.

Complex Numbers

Complex numbers introduce an imaginary unit, denoted as \( i \), where \( i^2 = -1 \. \) They help in solving equations that do not have real solutions and provide a meaningful context in polynomial equations and expansions. One special subset of complex numbers, known as the roots of unity, is particularly useful in this context.

Roots of unity are complex numbers that correspond to points on the unit circle in the complex plane. In our problem, we use the third roots of unity, especially \( \omega = e^{2\pi i/3} \), to simplify our polynomial. The third roots satisfy the equation \(1 + \omega + \omega^2 = 0 \). This identity plays a crucial role in simplifying the evaluation of the polynomial \( (1 + x + x^2)^n \) at different values of \( x \.\)

By evaluating the polynomial at \( x = 1, x = \omega, \) and \( x = \omega^2 \,\) complex numbers enable a simpler calculation process that leverages their properties, ultimately helping to solve complex algebraic problems with ease.

Coefficients of Powers of x

In polynomial expressions, each term has a coefficient which is the multiplier of a specific power of \( x \). The coefficients are the key numbers that dictate the terms of a polynomial. To find these, one would typically employ methods like polynomial expansion and later analyze or simplify for specific powers.

For the given problem, the primary interest is the sum of coefficients where the exponent of \( x \) is a multiple of three, such as \( x^0, x^3, x^6, \,\) etc. To extract these coefficients efficiently, summation strategies involving roots of unity can be applied: by setting \( x = 1, \omega, \) and \( \omega^2 \,\) the coefficients cycle through expressions that sum symmetrically over their roots, resulting in neat and computable forms.

The roots of unity shift the polynomial's expression to sum zero except when handled specifically, precisely selecting those coefficients that impact terms \( x^0, x^3, x^6, \,\) so the answer applies efficiently to the situation and yields the identified result \( 3^{n-1} \). Such techniques highlight how focusing on particular coefficients can simplify calculations in polynomial expressions.