Question

Co-efficient of \(\mathrm{x}^{-3}\) in expansion of \([\mathrm{x}-(\mathrm{a} / \mathrm{x})]^{11}\) is \(\ldots \ldots\) (a) \(-792 \mathrm{a}^{5}\) (b) \(-923 \mathrm{a}^{7}\) (b) \(-792 \mathrm{a}^{6}\) (d) \(-330 \mathrm{a}^{7}\)

Short answer

The coefficient of \(\mathrm{x}^{-3}\) in the expansion of \([\mathrm{x}-(\mathrm{a} / \mathrm{x})]^{11}\) is \(-330 \mathrm{a}^{7}\).

Step-by-step solution

Identify the variables to use in the binomial theorem formula

In our case, \(\mathrm{a} = \mathrm{x}\) and \(\mathrm{b} = -(\mathrm{a} / \mathrm{x})\). We will also use n=11 in the formula.

Find the term with exponent -3

To find the term with exponent -3, we need to find the value of k such that: \(11 - k - k = -3\) Solving for k, we get: \(k = 7\)

Use the binomial theorem formula to determine the coefficient

Now that we have a value for k, we can find the term in the expansion with exponent -3: \(\binom{11}{7} (\mathrm{x})^{11-7} (-(\mathrm{a} / \mathrm{x}))^{7}\) Simplifying this expression: \(\binom{11}{7} (\mathrm{x})^4 (-(\mathrm{a} / \mathrm{x}))^{7}\) \(\binom{11}{7} (\mathrm{x})^4 (\mathrm{a}^{7}) (\mathrm{x}^{-7})\) \(\binom{11}{7} (\mathrm{a}^{7}) (\mathrm{x}^{-3})\)

Find the coefficient of the term

Now that we have the term of interest, we can extract its coefficient: \(- \binom{11}{7} (\mathrm{a}^{7})\)

Compute the value of \(\binom{11}{7}\)

\(\binom{11}{7} = \dfrac{11!}{7! (11 - 7)!} = \dfrac{11!}{7! 4!} = 330\)

Combine the coefficient and x's exponent

Now that we have the coefficient's value, we can write the complete term in the expansion: \(-330 (\mathrm{a}^{7})\mathrm{x}^{-3}\) Therefore, the coefficient of \(\mathrm{x}^{-3}\) in the expansion of \([\mathrm{x}-(\mathrm{a} / \mathrm{x})]^{11}\) is \(\boxed{-330 \mathrm{a}^{7}}\). The answer is (d).

Key concepts

Binomial Expansion

The binomial expansion is a powerful algebraic tool that allows us to expand expressions that are raised to a power. For instance, when you encounter something like \((x + y)^n\),it can be expanded into a series of terms involving powers of \(x\) and \(y\).
Each term in this expansion is derived from the Binomial Theorem which provides a systematic way to express these terms using coefficients from binomial coefficients which will be discussed below.

• Essentially, the expansion includes terms such as \(\binom{n}{k}x^{(n-k)}y^k\).
• The number of terms in the expansionis always \(n+1\),where \(n\) is the exponent.

In the given exercise, we applied this concept to expand \([x-(a/x)]^{11}\)using the formula \(\binom{n}{k}x^{(n-k)}(-y)^k\),identifying terms and coefficients associated with a specific expanded term.

Binomial Coefficient

Binomial coefficients are numerical factors that arise in the expansion of a binomial raised to a power. They are usually represented as \(\binom{n}{k}\),and can be calculated using the formula: \(\binom{n}{k} = \dfrac{n!}{k!(n-k)!}\).
This coefficient determines how each term is weighted in the expansion.

• The value of the binomial coefficient determines how many combinations of elements can be selected.
• In simpler terms, it tells us how to distribute the terms in different ways.

For the exercise above, the binomial coefficient \(\binom{11}{7}\)was calculated by determining the factorial of the numbers involved, which results in the coefficient \(330\) for that specific term.
The coefficient directly influences the magnitude of each term in the expanded series.

Negative Exponents

Negative exponents indicate the reciprocal of the number raised to the corresponding positive exponent. For instance, \(x^{-n} = \frac{1}{x^n}\).This concept allows us to work with division in terms of multiplication, simplifying calculations involving powers and fractions.

• They are particularly useful in simplifying expressions that include fractions or division.
• A variable or number's power can be easily adjusted by shifting its placement between the numerator and denominator.

In the exercise discussed, negative exponents played a crucial role in identifying the specific term where the exponent of \(x\) equaled \(-3\).
This calculation involved adjusting the powers of \(x\) to align with the sought term, leading to the correct coefficient identification.