Problem 44 - Algebra and Trigonometry with Analytic Geometry

Question

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ (\sqrt{c}+\sqrt{d})^{8} ; \text { term that contains } c^{2} $$

Step-by-Step Solution

Answer

The term containing \(c^2\) is \(70 \cdot c^2 \cdot d^2\).

Step 1: Understand Binomial Expansion

The expression \((\sqrt{c} + \sqrt{d})^8\) is a binomial raised to a power. We will use the binomial theorem, which states: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = \sqrt{c}\) and \(b = \sqrt{d}\), and \(n = 8\).

Step 2: Identify the Relevant Term

We need the term that contains \(c^2\). Notice that \(a^{n-k} = (\sqrt{c})^{8-k} = c^{(8-k)/2}\). For this to be \(c^2\), \((8-k)/2 = 2\), which gives \(8-k = 4\), and thus \(k = 4\).

Step 3: Find the Specific Term

To find the specific term, substitute \(k = 4\) into the binomial expansion formula: \[\text{Term} = \binom{8}{4} (\sqrt{c})^{8-4} (\sqrt{d})^4\]This simplifies to: \[\text{Term} = \binom{8}{4} \cdot c^2 \cdot d^2\].

Step 4: Calculate Binomial Coefficient

Calculate \(\binom{8}{4}\), using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\): \[\binom{8}{4} = \frac{8!}{4!4!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70\]

Step 5: Compile Term with Calculated Coefficient

Include the computed binomial coefficient in the term as follows:\[70 \cdot c^2 \cdot d^2\]

Key Concepts

Binomial Coefficient

The Binomial Coefficient is an essential concept in combinatorics, often used in binomial expansions. It is represented by \(\binom{n}{k}\) and provides the number of ways to choose \(k\) items from \(n\) items without regard to the order. In simpler terms, it tells us how many different sets of size \(k\) can be chosen from a larger set of \(n\) elements.

To calculate the Binomial Coefficient, we use the formula:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

Where \(n!\) (n factorial) is the product of all positive integers up to \(n\) (e.g., \(5! = 5 \times 4 \times 3 \times 2 \times 1\)). This factorization allows for simpler calculations and is vital for determining coefficients in binomial expansions.

For example, in our exercise, we needed \(\binom{8}{4}\), calculated as:

• \(\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70\)

Thus, the coefficient 70 plays an important role in constructing the terms of a binomial distribution.

Binomial Expansion

The Binomial Expansion is the process of expanding an expression raised to a power in the form \((a+b)^n\). This is described by the Binomial Theorem, which allows us to express these expansions as a sum of terms involving binomial coefficients. The theorem states:

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

This means each term in the expansion is a product of a binomial coefficient, \(\binom{n}{k}\), a power of \(a\), and a power of \(b\).

When working with the expression \((\sqrt{c} + \sqrt{d})^8\), the expansion involves finding the terms where the specified variables (

• \(a = \sqrt{c}, b = \sqrt{d}\)

) appear raised to certain powers. By identifying the relevant terms that match certain criteria—for example, a term containing \(c^2\)—you can determine which element of the expanded series is needed.

Each term reflects a combination of the powers of \(a\) and \(b\), controlled by the index \(k\). In our solution for finding the term containing \(c^2\), figuring out the right power of \(a\) through \((\sqrt{c})^{8-k}\) ensured we zeroed in on the correct term.

Power of a Binomial

The Power of a Binomial refers to how exponential expressions involving two terms, such as \((a+b)^n\), can be expanded into individual terms via the binomial theorem. Each term in this expansion is determined by:

• A combination of powers of \(a\) and \(b\)
• The binomial coefficient \(\binom{n}{k}\)

Understanding this concept not only aids in expanding simple expressions but is foundational for solving many algebraic problems.

Using binomials raised to a power lets us explore combinations of terms by controlling how each component \((a^{n-k}, b^k)\) of the binomial expression contributes based on the indices \(n\) and \(k\).

For instance, in \((\sqrt{c} + \sqrt{d})^8\), determining that the right term for \(c^2\) required solving for the exponent when \((\sqrt{c})^{8-k} = c^2\), leading us to find the suitable index \(k\). These calculations reveal how increasing powers impact the structure of polynomial equations derived from binomials.