Problem 41 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The coefficient of \(x^{n}\) in expansion of \((1+x)(1-x)^{n}\) is (a) \((-1)^{n-1} n\) (b) \((-1)^{n}(1-n)\) (c) \((-1)^{n-1}(n-1)^{2}\) (d) \((n-1)\)

Step-by-Step Solution

Answer

The coefficient is \((-1)^n (1-n)\).

Step 1: Expand individual terms

We begin by expanding each term in the expression \((1+x)(1-x)^{n}\). The first term, \((1+x)\), expands as is: \(1 + x\.\)

Step 2: Apply the Binomial Theorem to \((1-x)^{n}\)

Next, we expand \((1-x)^{n}\) using the Binomial Theorem. The theorem states that \((1-x)^{n} = \sum_{k=0}^{n} \binom{n}{k} (-x)^{k}\). Substituting in \(-x\), we get: \[(1-x)^{n} = \sum_{k=0}^{n} \binom{n}{k} (-1)^{k}x^{k}.\]

Step 3: Combine expansions

We now multiply the expansions from Step 1 and Step 2:\[(1+x)\left(\sum_{k=0}^{n} \binom{n}{k} (-1)^{k} x^{k} \right) = \sum_{k=0}^{n} \binom{n}{k} (-1)^{k} x^{k} + \sum_{k=0}^{n} \binom{n}{k} (-1)^{k} x^{k+1}.\]In combining these, we are looking for the coefficient of \(x^n\).

Step 4: Identify the coefficients for \(x^n\) terms

To find the coefficient of \(x^n\), examine each sum separately:- In \(\sum_{k=0}^{n} \binom{n}{k} (-1)^{k} x^{k}\), the coefficient of \(x^n\) is \((-1)^n \binom{n}{n} = (-1)^n\).- In \(\sum_{k=0}^{n} \binom{n}{k} (-1)^{k} x^{k+1}\), the term that contributes to \(x^n\) is \((-1)^{n-1} \binom{n}{n-1} x^n\). Here, the coefficient is \((-1)^{n-1} n\).Thus, combining the coefficients for \(x^n\), we have:\[(-1)^n + (-1)^{n-1} n.\]

Step 5: Simplify the expression

Simplify the expression for the coefficient by factoring out \((-1)^n\):\[(-1)^n (1 - n).\]This matches choice (b) in the problem: \((-1)^n (1 - n)\.\)

Key Concepts

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial in its full form, usually as a result of multiplication or other operations. In our exercise, we examine the expression

• \((1+x)(1-x)^{n}\).

The first part,

• \((1+x)\),

requires no further expansion, already being in its simplest form. However, the term

• \((1-x)^{n}\)

is expanded using the Binomial Theorem.
The theorem assists in breaking down this term into a sum of simpler polynomials.By expanding and analyzing these two parts individually, then combining them,we can better understand how different components of the polynomial interact.
This step is crucial as it lays the groundwork for finding specific coefficients within the polynomial.

Coefficient Determination

Determining coefficients within a polynomial expansion allows for the extraction of specific terms, such as constants or terms with a certain power of a variable.The core of our exercise asks us to find the coefficient of

• \(x^n\)

in the expanded expression.Once expanded, multiple terms might contribute to a specific power of \(x\).
In this case, the two relevant parts come from:

• \((1-x)^{n}\) contributing \((-1)^n\)
• and \((1-x)^{n} \cdot x\) contributing \((-1)^{n-1} n\).

Combining both, we obtain the total coefficient for \(x^n\):

• \((-1)^n + (-1)^{n-1} n\)

Simplifying provides the final result, which highlights the importance of accurate coefficient determination in understanding polynomial expressions.

Combinatorics

Combinatorics plays a vital role in expanding polynomials, particularly through the use of binomial coefficients, which are the

• numbers found in Pascal's Triangle.

These coefficients, \(\binom{n}{k}\),represent the different ways of choosing \(k\) items from \(n\) items without regard to order.
In the Binomial Theorem,this forms the basis for identifying the coefficients in

• \((1-x)^n\).

Using combinatorics helps simplify the calculation of these coefficients by reducing complex iterations into a single, manageable number.
By understanding how these coefficients influence the polynomial, students can move elegantly from expression to expanded form, and solve exercises like determining the coefficient of \(x^n\) in a large polynomial easily and effectively.