Question

Middle term in expansion of \([(2 / x)-3 x y]^{12}\) is (a) \(14370048 \mathrm{y}^{6}\) (b) \(14370024 \mathrm{y}^{6}\) (c) \(43110144 \mathrm{y}^{6}\) (d) \(43110124 \mathrm{y}^{6}\)

Short answer

The middle term in the expansion of \([(2/x)-3xy]^{12}\) is (a) \(14370048 \mathrm{y}^{6}\).

Step-by-step solution

Find the middle term's index

Since n is even (12 is even), the middle term will be at position \(\frac{n}{2} + 1 = \frac{12}{2} + 1 = 7\). So, we are looking for the 7th term in the expansion: \(T_7\).

Apply the Binomial theorem to find the middle term

Using the Binomial theorem, we can write the 7th term as \[T_7 = ^{12}C_6(2/x)^{12-6}(-3xy)^6\]

Calculate the binomial coefficient

Now, let's calculate the binomial coefficient: \[^{12}C_6 = \frac{12!}{6!(12-6)!} = \frac{12!}{6!6!} = 924\]

Simplify the term

Now we plug in the binomial coefficient and simplify the term: \[T_7 = 924\left(\frac{2}{x}\right)^6(-3xy)^6\] \[T_7 = 924\left(\frac{2^6}{x^6}\right)(-3^6x^6y^6)\] \[T_7 = 924(64)(-729x^6y^6)\] \[T_7 = 14370048y^6\] So, the middle term in the expansion of \([(2/x)-3xy]^{12}\) is \(14370048y^6\). The answer is (a) \(14370048 \mathrm{y}^{6}\).

Key concepts

Binomial Expansion

When we talk about binomial expansion, we're referring to the process of expanding expressions that are raised to a power, such as \( (a+b)^n \). This is where the binomial theorem becomes incredibly handy.
The binomial theorem allows us to expand these expressions into sums involving terms in the form of \( ^nC_k a^{n-k} b^k \), where \( ^nC_k \) are the binomial coefficients that you compute based on the given power \( n \).

This is essential in determining not just the complete expansion but also specific terms in the expansion, like finding the middle term. With the binomial expansion, each term is a combination of powers of \( a \) and \( b \) from the binomial expression. It's very structured and repeats consistently for any binomial, making it a powerful technique to tackle complex polynomial expressions.

Factorial Calculation

Factorials are key to finding the binomial coefficients in a binomial expansion. A factorial is denoted by an exclamation mark \( ! \) and symbolizes the product of all positive integers up to a given number. For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

In the context of the binomial theorem, factorial calculations come in when you need to determine the value of binomial coefficients, written as \( ^nC_k \). The coefficient \( ^nC_k \) is calculated using:

• \[ ^nC_k = \frac{n!}{k!(n-k)!} \]

You calculate the factorial of \( n \), the factorial of \( k \), and the factorial of \( n-k \), then place them into this formula. For example, as seen in the solution, \( ^{12}C_6 = \frac{12!}{6!6!} \). Calculating factorials may look daunting, but once you set it up, it's about performing multiplication and division in steps.

Middle Term in Binomial Expansion

Locating the middle term in a binomial expansion involves understanding the order of terms within the expansion. For even powers like in our example, the middle term can be pinpointed easily by taking half the power (\( n/2 \)) and then adding one, which in our case gives us the 7th term of 13 total terms.

To find this middle term, we apply the binomial theorem to the term at this position. Let's recall:

• \[ T_{k+1} = ^nC_k a^{n-k}b^k \]

So, for \( ^{12}C_6 (2/x)^6 (-3xy)^6 \), the calculation simplifies the term to determine its actual value. This means plugging the calculated binomial coefficient into the expression then simplifying powers and products to pinpoint exactly what the middle term is in its simplest form.

The rest is a matter of simplifications through arithmetic, as seen in the example, leading to our identified middle term: \( 14370048y^6 \). This method is invaluable for large expansions, making otherwise cumbersome considerations into straightforward computations.