Question

Constant term in expansion of \([1+(2 / \mathrm{x})-(2 / \mathrm{x})]^{4}\) is ........ (a) 5 (b) \(-5\) (c) 4 (d) \(-4\)

Short answer

The constant term in the expansion of the expression \((1+\frac{2}{x}-\frac{2}{x})^4\) is 1.

Step-by-step solution

Simplify the Expression

First, we can simplify the expression: \( (1+\frac{2}{x}-\frac{2}{x})^4 = (1)^4 = 1 \)

Identify Constant Term

Since we have simplified the expression to 1, it is clear that the constant term in the expansion of the expression will also be 1.

Choose the Correct Answer

Now that we have found the constant term in the expansion, we can choose the correct answer from the given options: (a) 5 (b) -5 (c) 4 (d) -4 Since the constant term is 1, none of the options (a), (b), (c), or (d) are correct. The problem seems to have incorrect answer choices. Nevertheless, the constant term in the expansion of the expression \((1+\frac{2}{x}-\frac{2}{x})^4\) is 1.

Key concepts

Constant Term

Finding the constant term in a polynomial expansion involves identifying terms in the polynomial that do not depend on any variables. This means they do not contain any variables like \( x \) or \( y \), and remain unchanged as the value of the variables changes. In the context of the binomial expansion, the constant term will be the term where the variable

• has an exponent of zero or,
• is completely canceled out.

In the original exercise, the constant term in the expansion of the simplified expression \[ [1+(2 / x)-(2 / x)]^{4} = 1^4 = 1 \]was clearly identified as 1. This happened because adding and subtracting \( \frac{2}{x} \) cancels itself out, leaving only a constant (1) in the expression.

Simplification

Simplification involves manipulating an expression to a more manageable form, usually one that is easier to work with or understand. Issues such as combining like terms, eliminating complex fractions, or cancelling out terms are common tasks.
In the given exercise, simplification is a key step. The expression \[ [1+(2 / x)-(2 / x)]^4 \]was simplified first by realizing that \( \frac{2}{x} - \frac{2}{x} = 0 \),leaving the simplified form as \( 1^4 = 1 \).Successfully simplifying an expression often makes subsequent calculations or identifications, such as finding the constant term, much easier. It aids in minimizing errors and creates clarity in solving mathematical problems.

Polynomial Expansion

The polynomial expansion refers to expressing a power of a binomial as a sum of terms generated by the binomial theorem.The binomial theorem provides a reliable way for expanding expressions that are raised to a power, like \( (a + b)^n \).In the provided exercise, the expansion seemed complicated at first due to the presence of fractions, \( [1+(2 / x)-(2 / x)]^4 \),but through simplification, it was reduced to \( 1^4 \),which means there was no need for further polynomial expansion as the expression was entirely simplified to a constant.In general, expansions can become intricate, but identifying patterns or simplifying parts of an expression can considerably reduce complexity, guiding towards efficient solutions to problems.