Problem 19 - Algebra 2 and Trigonometry

Question

In \(18-20,\) for the given expansion, identify which term is shown and write the next term. $$ (5 c-2 d)^{4} ;-160 c d^{3} $$

Step-by-Step Solution

Answer

The term is \(T_3\), next term is \(16d^4\).

Step 1: Identify the general term in the binomial expansion

The binomial expansion \((a+b)^n\) can be expanded as \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). For the expression \((5c - 2d)^4\), identify \(a = 5c\), \(b = -2d\), and \(n = 4\). The general term \(T_k\) in this expansion is given by: \[ T_k = \binom{4}{k} (5c)^{4-k} (-2d)^k \].

Step 2: Determine which term matches the given term

Given the term is \(-160 c d^3\). Let's match the powers and coefficients with one of the terms from the expansion. We need \( (5c)^{4-k} (-2d)^k = -160 c d^3 \). This implies that the term involves \(c^1\) and \(d^3\), hence we need \((5c)^{3} (-2d)^1\) because \(c^1\) comes from \((5c)^{1}\) and \(d^3\) from \((-2d)^3\). The binomial coefficient should then match the coefficient -160.

Step 3: Calculate the coefficient and solve for k

Calculate the coefficient for the term \(T_k = -160 c d^3\). Using \[\binom{4}{3} (5c)^{1} (-2d)^3 \].Calculate it:\( \binom{4}{3} = 4 \), \( (5c)^1 = 5c \), and \((-2d)^3 = -8d^3 \).Thus, the coefficient becomes:\[ 4 \times 5 \times -8 = -160. \] This matches the given term, confirming it is indeed \(T_3\) with \(k=3\).

Step 4: Write the next term in the sequence

The next term after \(T_3\) is \(T_4\). Use \[ T_4 = \binom{4}{4} (5c)^{0} (-2d)^4 \].Calculate: \( \binom{4}{4} = 1 \), \((5c)^0 = 1\), and \((-2d)^4 = 16d^4\).Thus, the next term is:\[ T_4 = 1 \times 1 \times 16d^4 = 16d^4. \]

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra used to expand expressions of the form \((a+b)^n\) into a series of terms. This theorem simplifies the process of expansion by providing a formula to directly calculate each term without multiplying the expression repeatedly. The expansion is given as \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient representing the number of ways to select \(k\) elements from \(n\) elements.

Key points about the binomial theorem include:

• Each term in the expansion consists of a coefficient, a power of \(a\), and a power of \(b\).
• The sum of the exponents of \(a\) and \(b\) in any term always equals \(n\).
• It allows for efficient calculation, especially for large powers or complex expressions.
  
  
Use of the Binomial Theorem is widespread due to its simplicity and wide applicability in probability, algebra, and calculus.

General Term

In the context of binomial expansion, the general term is crucial for understanding which specific term appears within the expanded form. For an expansion of \((a+b)^n\), the general term \(T_k\) can be calculated using

\[ T_k = \binom{n}{k} a^{n-k} b^k \]

Here, \(\binom{n}{k}\) is the binomial coefficient, which gives the number of ways to choose \(k\) terms out of \(n\).

This general term

• helps identify specific terms without fully expanding the entire series,
• allows for focused calculations, especially useful when interested only in certain terms rather than the entire sequence,
• facilitates further analysis, such as finding the value of specific terms or seeking relationships between terms.
  
  
The general term can be tailored to find not just the terms, but also the coefficients of the desired powers of \(a\) and \(b\) within a binomial expression.

Coefficients in Binomial Expansion

Coefficients in a binomial expansion play a significant role as they determine the multiplier for each term in the expansion. These coefficients are derived from binomial coefficients \(\binom{n}{k}\), a central element of the Binomial Theorem. They can be calculated using factorial notation and combinatorial formulas:

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

The factorial function \(!\) signifies the product of all positive integers up to a given number, providing a systematic way to compute large numbers. Important aspects of coefficients include:

• They are symmetrical in their sequence, meaning \(\binom{n}{k} = \binom{n}{n-k}\).
• The first and last coefficients are always 1, corresponding to the first and last terms only.
• Understanding these coefficients is essential for comprehending the structure of any binomially expanded series.
  
  
The coefficients, coupled with powers of \(a\) and \(b\), dictate the magnitude and character of each term within any expansion.

Identifying Terms in Binomial Expansion

Identifying terms within a binomial expansion involves calculating each specific term using its position and the corresponding general formula. Consider the example of expanding \((5c - 2d)^4\):

Each term is structured as \(T_k = \binom{4}{k} (5c)^{4-k} (-2d)^k\). By matching given terms with this expression,we can pinpoint characteristics such as the powers of each variable and their coefficients.

To identify:

• First, align the powers of \(a\) and \(b\) in the general term with those in the given term.
• Next, equate the resulting coefficient with the corresponding part of the expansion.
• Finally, use the position \(k\) to calculate the succeeding or preceding terms in relation to the given term.
  
  For example, if given \(-160cd^3\), matching it to the formula reveals it as the third term \(T_3\). This knowledge allows us to compute the immediate next term \(T_4\) and analyze patterns within the sequence.
Understanding such identification eases working with specific sections of large polynomial expansions, offering precision and clarity.