Question

Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{2}\) in the expansion of \(\left(\sqrt{x}+\frac{3}{\sqrt{x}}\right)^{8}\)

Short answer

84.

Step-by-step solution

Understand the Binomial Theorem

The Binomial Theorem states

Identify the general term

The general term in the expansion of

Simplify the general term

Using the general term from step 2

Solve for the exponent of x

We need

Calculate the coefficient

We've got

Key concepts

coefficient extraction

One of the first things we need to focus on is how to extract the coefficient of a specific term in a polynomial expansion. When using the Binomial Theorem, identifying this coefficient requires us to look at both the exponents and the constants in the general term of our expansion.
For our problem, we want to find the coefficient of the term where the exponent of \(x\) equals 2. This means we need to isolate this specific term and simplify it to extract the coefficient.
In simpler terms, if you have a form \(ax^2\), you’ll identify the \(a\), which is the coefficient. Making sure calculations are accurate here is crucial to solving the problem correctly.

polynomial expansion

Polynomial expansion involves expressing a polynomial raised to a power as a sum of terms, specifically in the form of the Binomial Theorem. For example, expanding \((a + b)^n\) is done by following a systematic method to ensure all possible products of \(a\) and \(b\) are included with correct coefficients.
In our exercise, we expanded \(\big(\frac{3}{\text{√}x} + \text{√}x\big)^8\). To start expanding, regard each term of the form \(C \cdot a^k \cdot b^{n-k}\), where C is a binomial coefficient \(\binom{n}{k}\).
Each term has a coefficient and a product of \(a\) and \(b\) raised to appropriate powers. This methodical expansion facilitates finding specific terms with ease.

general term in binomial expansion

The general term in binomial expansion aids in zeroing in on the term we are interested in. For an expansion \((a+b)^n\), each term can be represented as \(\binom{n}{k} a^{n-k} b^k\).
Specifically, in our problem, the general term for \(\big(\frac{3}{\text{√}x} + \text{√}x \big)^8\) can be written as \(\binom{8}{k} \text{√}x^{8-k} \big( \frac{3}{\text{√}x} \big)^k \).
Here, \(a = \text{√}x\), \(b = \frac{3}{\text{√}x}\), and \(n = 8\). The term we seek corresponds to making sure the power of \(x\) sums correctly, which simplifies our search for the coefficient.

algebraic manipulation

Within the context of binomial expansions, algebraic manipulation becomes essential to simplify and identify terms. This involves combining like terms, adjusting exponents, and simplifying coefficients.
For our given problem, finding the \(x^2\) term in \((\text{√}x + \frac{3}{\text{√}x})^8\) necessitated careful algebraic steps. We determined which general terms simplified into \(x^2\) by solving: \(8 - k - k = 2 \).
After simplifying and solving for \(k\), the correct value was substituted back, and coefficients extracted by further simplification. Through these steps, algebraic manipulation showcased its importance in ensuring we arrived at the correct coefficient.