The sum of the coefficients of the polynomial (1 + x - 3x²)²¹⁶³ is .........

Q: Wait, why does setting $x=1$ give the sum of coefficients? Shouldn't I expand the whole thing using the Binomial Theorem?

Expanding $(1 + x - 3x^2)^{2163}$ would take you a lifetime! The beauty of mathematics lies in finding the 'shortcut' hidden in the structure. When you set $x=1$, every term $a_k x^k$ becomes $a_k(1)^k$, which is just $a_k$. Thus, the expression naturally collapses into the sum of all coefficients. It is a fundamental property of polynomials, not a coincidence.

Q: What if the polynomial had a constant term like $+5$ at the end? Would $x=1$ still work?

Absolutely! If $P(x) = a_0 + a_1x + \dots + a_n x^n + C$, then $P(1) = a_0 + a_1 + \dots + a_n + C$. The constant term is just the coefficient of $x^0$, so it is included in the sum. The logic holds perfectly.

Q: Is it possible for the sum of coefficients to be zero?

Yes, it is! If the polynomial evaluates to zero at $x=1$, then the sum of its coefficients is indeed zero. This happens in many JEE problems where the coefficients are designed to cancel each other out.

Q: Why is the exponent $2163$ so large? Is it a trap?

In JEE Advanced, large numbers are often used to test your confidence in properties rather than your ability to calculate. Once you realize that $(-1)^{2163}$ is just a matter of parity (odd vs even), the magnitude of the number becomes irrelevant. Don't let the size of the exponent intimidate you!

Q: What if the question asked for the sum of coefficients of even powers only?

That is a classic variation! To find the sum of coefficients of even powers, you would calculate $\frac{P(1) + P(-1)}{2}$. By evaluating at $x=1$ and $x=-1$, you can isolate specific patterns of coefficients. Always look for these symmetry-based tricks.

Question

The sum of the coefficients of the polynomial $(1 + x - 3x^2)^{2163}$ is .........

Sigma Percentile · Accepted Answer

### The Elegance of the Hidden Identity

Imagine you are staring at the expression $P(x) = (1 + x - 3x^2)^{2163}$. At first glance, it looks like a monster.

If you were to attempt a manual expansion, you would be trapped in a labyrinth of binomial coefficients and powers for days. But here is the secret that separates the novice from the master: in mathematics, we rarely need to see the entire expansion to understand its properties.

We are looking for the 'soul' of the polynomial, which is the sum of its coefficients.

### The Magic of the Unit Substitution

Let us define our polynomial as $P(x) = a_0 + a_1x + a_2x^2 + \dots + a_nx^n$. Our goal is to find the sum $S = a_0 + a_1 + a_2 + \dots + a_n$.

Look closely at the structure. If we could somehow make every $x$ in that expression vanish, we would be left with exactly what we need.

What happens if we set $x = 1$?

$$P(1) = a_0 + a_1(1) + a_2(1)^2 + \dots + a_n(1)^n$$

Because $1$ raised to any power is still $1$, the expression simplifies beautifully to $P(1) = a_0 + a_1 + a_2 + \dots + a_n$. This is the **Magic Substitution**.

It is not just a trick; it is a fundamental bridge between the functional form of a polynomial and the sum of its constituent parts. Whenever you see a question asking for the sum of coefficients, stop and ask yourself: "Can I just evaluate this at $x=1$?"

### Executing the Calculation

Now, let us apply this to our specific problem. We have $P(x) = (1 + x - 3x^2)^{2163}$.

By our logic, the sum of the coefficients is simply $P(1)$. Let us substitute $x=1$ into the base of our expression:

$$P(1) = (1 + (1) - 3(1)^2)^{2163}$$

Focus on the inner terms first. We have $1 + 1 - 3$, which simplifies to $-1$. Now, the expression becomes much more manageable:

$$P(1) = (-1)^{2163}$$

### The Final Parity Check

We are left with a simple exponentiation. The number $2163$ is clearly an odd integer.

In the realm of negative numbers, parity is everything. We know that $(-1)^{	ext{even}} = 1$ and $(-1)^{	ext{odd}} = -1$.

Since $2163$ is odd, the result is locked in:

$$P(1) = -1$$

And there it is. The sum of the coefficients is **$-1$**.

### Reflecting on the Journey

Think about what we just achieved. We bypassed thousands of potential terms, avoided a massive algebraic expansion, and arrived at the truth by understanding the symmetry of the polynomial.

This is the essence of JEE Advanced mathematics. It is not about brute force; it is about identifying the underlying structure.

The next time you see a daunting expression, don't panic. Look for the substitution that simplifies the world, and you will find that even the most complex problems have a simple, elegant heart.

Animated Solution for Mathematics - Binomial Theorem: The sum of the coefficients of the polynomial $(1 + x - 3 x^{2})^{2163}$ is .........

Visualized Solution

Defining the Polynomial $P (x)$

Understanding the Expansion

The Goal: Sum of Coefficients

The Magic Substitution

Applying the Rule to $P (x)$

Simplifying the Inner Terms

Final Addition and Subtraction

Evaluating the Exponent

The Final Answer

The Sigma Insight: Properties of Binomial Coefficients

The Elegance of the Hidden Identity

The Magic of the Unit Substitution

Executing the Calculation

The Final Parity Check

Reflecting on the Journey

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Animated Solution for Mathematics - Binomial Theorem: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the coefficients of the polynomial (1+x−3x2)2163 is .........