Question

If middle term is \(\mathrm{kx}^{\mathrm{m}}\) in expansion of \([\mathrm{x}+(1 / \mathrm{x})]^{12}\) then \(\mathrm{m}=\) (a) \(-2\) (b) \(-1\) (c) 0 (d) 1

Short answer

The middle term of the expansion \([x + (1/x)]^{12}\) is computed using the binomial theorem as the 7th term, which is \(T_7 = \frac{12!}{(6!)(6!)} \cdot x^0\). Comparing this with the given form of the middle term (\(kx^m\)), we find that \(m = 0\). Therefore, the correct answer is (c) 0.

Step-by-step solution

Recall the binomial theorem

The binomial theorem tells us that for any positive integer n, \((a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r\), where \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\). Here, we have \(n = 12\), \(a = x\), and \(b = \frac{1}{x}\).

Find the middle term

Since the expansion has \(12+1=13\) terms, the middle term will be the 7th term (because 13 is an odd number and the middle term is simply the term in the middle). So, let's compute the 7th term of the expansion: \(T_7 = \binom{12}{6}(x)^{(12-6)}\left(\frac{1}{x}\right)^6 = \frac{12!}{6!(12-6)!} x^6 \times x^{-6}\). Now, simplify the term: \(T_7 = \frac{12!}{(6!)(6!)} \cdot x^{6-6} = \frac{12!}{(6!)(6!)}\cdot x^0\).

Compare the term with the given form of the middle term and find the value of \(m\)

Now, let's compare our middle term with the given form (\(kx^m\)): \(T_7 = \frac{12!}{(6!)(6!)} \cdot x^0 = kx^m\). Since x has a power of 0 in the middle term, we can infer that: \(m = 0\). Therefore, the correct answer is: \(m =\) (c) 0.

Key concepts

Expansion of Binomials

The expansion of binomials is a fundamental concept in algebra, used to expand expressions raised to a power. The binomial theorem provides a formula to expand powers of binomials, which are expressions of the form \((a + b)^n\), where \(a\) and \(b\) are any numbers, terms, or variables, and \(n\) is a positive integer.
This expansion is expressed as:

• \((a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r\)

Here, \(\binom{n}{r}\) represents the binomial coefficient, calculated using \(\frac{n!}{r!(n-r)!}\), where \(!\) denotes factorial, which is the product of all positive integers up to that number.
For example, expanding \((x + \frac{1}{x})^{12}\) involves determining each term using this formula, where each term is formed by varying \(r\) from 0 to 12.
This technique is particularly useful in finding specific terms in the expansion without expanding the entire expression, making complex algebraic manipulations manageable.

Middle Term in Binomial Expansion

Finding the middle term in a binomial expansion can be important, especially when you are asked for specific features of a polynomial's terms. In a binomial expansion like \((x + \frac{1}{x})^{12}\), the total number of terms is given by \(n + 1\), where \(n\) is the exponent.
So for the expansion \((x + \frac{1}{x})^{12}\), there are 13 terms. Since 13 is an odd number, the middle term is the 7th term. This is calculated using the formula from the binomial theorem:

• The middle term is determined as \(T_7 = \binom{12}{6} x^{12-6} \left(\frac{1}{x}\right)^6\).

After simplifying this expression, the result is \(T_7 = \frac{12!}{6!\cdot 6!} \cdot x^{0}\), highlighting the role of careful calculation in resolving potential algebraic complexities.
This approach illustrates how the binomial theorem helps isolate particular terms without evaluating the entire expression, which can be computationally expensive.

Exponent Calculation

In the context of binomial expansions, exponent calculation focuses on determining the power, or exponent, of a term within a polynomial expansion. To find the exponent of a middle term in an expression like \((x + \frac{1}{x})^{12}\), first locate the desired term through the binomial expansion formula.
For this expansion, the middle term (7th term) is calculated using:

• \(T_7 = \binom{12}{6} x^{12-6} \left(\frac{1}{x}\right)^6\), which simplifies to \(x^{6-6} = x^0\).

This indicates that the exponent of \(x\) in the middle term is 0, determined by balancing the exponents in both portions of the term. Thus, \(T_7\) results in \(x^0\), and the exponent \(m\) in \(kx^m\) is 0.
Understanding exponent calculations in binomial expansions ensures precise identification of term powers, crucial for mathematical problem-solving across algebra and calculus. It empowers you to swiftly locate and verify terms in polynomial expressions.