Question

The ratio of the coefficients of \(\mathrm{x}^{4}\) to that of the term independent of \(\mathrm{x}\) in the expansion of \(\left(x^{2}+\frac{9}{x^{2}}\right)^{15}\) is (1) \(1: 6\) (2) \(3: 8\) (3) \(1: 10\) (4) \(1: 8\)

Short answer

Answer: The ratio is undefined, as there is no term independent of x in the expansion.

Step-by-step solution

Expand the expression using the binomial theorem

By the binomial theorem, we can expand the expression as follows: \((x^2 + \frac{9}{x^2})^{15} = \sum_{n=0}^{15} \binom{15}{n} (x^2)^n (\frac{9}{x^2})^{15-n}\).

Identify the coefficients of terms

Now let's find the coefficients of the terms needed for the ratio calculation. For the \(x^4\) term, we need to find the value of \(n\) such that \(2n - 2(15 - n) = 4\). Solving for \(n\) gives \(n = 6\). So, to find the coefficient of \(x^4\), we need to substitute \(n = 6\) into the sum formula, which gives: $$ \binom{15}{6} (x^2)^6 \left(\frac{9}{x^2}\right)^{15-6} = \binom{15}{6} x^{12} \cdot 9^9 x^{-18} $$ Thus, the coefficient of the \(x^4\) term is \(\binom{15}{6} \cdot 9^9\). Now, for the term independent of \(x\), we need to find the value of \(n\) such that \(2n - 2(15 - n) = 0\). Solving for \(n\) gives \(n = 7.5\). Since \(n\) must be an integer, there is no term independent of \(x\) in the expansion of the given expression.

Calculate the ratio of the coefficients

The term independent of \(x\) does not exist in the expansion, so the ratio for the coefficients of \(x^4\) to the term independent of \(x\) is undefined, and none of the given answer choices are correct.

Key concepts

Binomial Expansion

The binomial theorem is a powerful algebraic formula used to expand expressions that are raised to a power. When we expand a binomial expression such as \((x^2 + \frac{9}{x^2})^{15}\), we can represent it as a sum of terms using the binomial expansion. This expression can be expanded as:\[(x^2 + \frac{9}{x^2})^{15} = \sum_{n=0}^{15} \binom{15}{n} (x^2)^n \left(\frac{9}{x^2}\right)^{15-n}\]Each term in this expansion involves different powers of \(x\) and a specific coefficient calculated from binomial coefficients. Expanding expressions using the binomial theorem allows us to easily find specific terms, such as those involving specific powers of \(x\) or terms independent of \(x\).
This process is especially useful in problems where we need to compare or analyze specific terms from the expansion.

Coefficients

Coefficients play a critical role in the binomial expansion process. In the case of \((x^2 + \frac{9}{x^2})^{15}\), each term in the expansion has a coefficient derived from the binomial coefficient \(\binom{15}{n}\).The binomial coefficients can be calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \(n!\) represents a factorial, the product of all positive integers up to \(n\). Each term's coefficient is determined by this mathematical formula.
For example, to find the coefficient of a specific term, say the \(x^4\) term, we calculate \(\binom{15}{6}\).In the given problem, finding these coefficients helps us determine and analyze the ratio asked.

Exponents

Exponents indicate the power to which a base number is raised. They are essential in the binomial expansion as they define each term's structure. When expanding an expression like \((x^2 + \frac{9}{x^2})^{15}\), exponents dictate the power of \(x\) in each term of the expansion.
In each term of the expansion, the exponents come from how many times each part of the binomial is multiplied:

• The exponent of \(x^2\) is \(n\).
• The exponent of \(\frac{9}{x^2}\) is \(15-n\).

The task often involves solving equations involving exponents to find specific terms such as \(x^4\) or terms having exponents that result in a power of zero for calculating coefficients of terms independent of x.

Terms Independent of x

In binomial expansions, a term independent of \(x\) is a term where all variables cancel out or reduce such that the term has no \(x\) component. This is achieved when the powers of \(x\) in a term add up to zero, creating a constant term in the expansion. For instance, in the expansion of \((x^2 + \frac{9}{x^2})^{15}\), we try to find the term where the powers of \(x\) cancel out.To find such terms, we solve the equation where the total power of \(x\) equals zero. However, if no integer solution exists for \(n\), as determined in the exercise, then a term independent of \(x\) does not exist. This is crucial in problems comparing coefficients, as a missing independent term of \(x\) affects the calculation of ratios.