Problem 125 - A Complete Resource Book in Mathematics for JEE Main

Question

The coefficient of \(x^{\mathrm{n}}\) in expansion of \((1+x)(1-x)^{\mathrm{n}}\) is (A) \(-\frac{5}{3}\) (B) \(\frac{3}{5}\)(C) \((-1)^{n-1}(n-1)^{2}\) (D) \((-1)^{n-1} n\)

Step-by-Step Solution

Answer

(D) \((-1)^{n-1} n\)

Step 1: Expand (1+x) using the Binomial Theorem

The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Since \(1+x\) is expanded to \(1 + x\) with coefficients 1 and 1, explicit expansion doesn't change the expression.

Step 2: Expand (1-x)^n using the Binomial Theorem

Use the binomial theorem to expand \((1-x)^n\): \((1-x)^n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k x^k\). This expansion yields terms with coefficients based on \((-1)^k\) and the binomial coefficient \(\binom{n}{k}\).

Step 3: Multiply the Two Expansions

Form the product \((1+x)(1-x)^n = (1+x) \times \sum_{k=0}^{n} \binom{n}{k} (-1)^k x^k\). Distributing \((1+x)\) gives:\[\sum_{k=0}^{n} \binom{n}{k} (-1)^k x^k + \sum_{k=0}^{n} \binom{n}{k} (-1)^k x^{k+1}\]

Step 4: Collect Terms for Same Power of x

For the coefficient of \(x^n\), look at terms from both sums. In the first sum, we have the \(k=n\) term: \[\binom{n}{n} (-1)^{n} x^n = (-1)^{n} x^n\]. In the second sum, we consider \(k=n-1\): \[\binom{n}{n-1} (-1)^{n-1} x^n = n(-1)^{n-1} x^n\].

Step 5: Combine and Determine Coefficient

The coefficient of \(x^n\) is the sum of coefficients from \(x^n\) terms: \((-1)^n + n(-1)^{n-1} = (-1)^n - n(-1)^n = (-n - 1)(-1)^{n-1}\). Rewriting, we get: \((-1)^{n-1}n\).

Key Concepts

Coefficient

In algebra, a coefficient is a numerical or constant value that quantifies the influence of a variable in an expression. When discussing polynomials, the coefficient is the number in front of the variable, such as the number multiplying \(x\) in terms of \(x^n\). This is important because it describes how the variables contribute to the overall value of the polynomial when evaluated.
In our specific problem, we are interested in identifying the coefficient of \(x^n\) in the expansion \((1+x)(1-x)^n\). The step-by-step solution involves calculating these coefficients by manipulating terms as per the binomial expansions. The coefficients here stem from applying the Binomial Theorem repeatedly and matching terms of equal power from the polynomial expansion. Recognizing and calculating these coefficients lets us precisely express and predict the nature of the polynomial expression.

Polynomial Expansion

Polynomial expansion is a method used in algebra to express a polynomial as a sum of terms derived from its factors. When we perform an expansion using the Binomial Theorem, the expression is reformulated through multiplication and summation processes to list all terms.
For instance, expanding \((1+x)(1-x)^n\) involves applying the binomial theorem separately to each part, and then combining their results. The expansion of \((1+x)\) is straightforward, while for \((1-x)^n\), the theorem gives:

• Each term in the sum is derived from the general binomial expansion formula: \(\binom{n}{k} (-1)^k x^k\).
• The terms from both expansions then get multiplied together.
• The resulting summation constructs the full polynomial expansion.

Understanding polynomial expansion allows us to break down complex expressions into manageable parts, which are crucial for identifying terms like coefficients.

Power of x

When discussing a polynomial, the power of \(x\) refers to the exponent associated with \(x\). The power indicates how many times \(x\) is used as a factor in terms within the polynomial.
In the given problem, identifying the power of \(x\) means looking at terms like \(x^n\). During the expansion process, particularly with expressions like \((1-x)^n\), terms with different powers arise due to raising \(x\) to different exponents in each quantity of the binomial expansions.
To determine the coefficient of a specific power of \(x\) in a polynomial, such as \(x^n\), one must balance the terms that contribute to \(x^n\). In this case, we found the power by matching terms in a sum like \((-1)^{n} x^n\) and \(n(-1)^{n-1} x^n\), calculating the sum in order to focus only on the terms with \(x^n\) power in the final expression. Knowing the power of \(x\) helps isolate and solve for its coefficients accurately.