Problem 24 - College Algebra

Question

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. $$ (x-1)^{18} $$

Step-by-Step Solution

Answer

The first three terms are: \(x^{18}\), \(-18x^{17}\), and \(153x^{16}\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem provides a way to expand expressions of the form \((a + b)^n\). It states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \/ \binom{n}{k}\ is the binomial coefficient.

Step 2: Identify the values of a, b, and n

For the expression \((x-1)^{18}\), we identify \(a = x\), \(b = -1\), and \(n = 18\).

Step 3: Calculate the first term

The first term \(T_0\) is calculated using \(k = 0\). Thus, \(T_0 = \binom{18}{0} x^{18} (-1)^0 = 1 \cdot x^{18} = x^{18}\).

Step 4: Calculate the second term

The second term \(T_1\) is calculated using \(k = 1\). Thus, \(T_1 = \binom{18}{1} x^{17} (-1)^1 = 18 \cdot x^{17} \cdot (-1) = -18x^{17}\).

Step 5: Calculate the third term

The third term \(T_2\) is calculated using \(k = 2\). Thus, \(T_2 = \binom{18}{2} x^{16} (-1)^2 = 153 \cdot x^{16}\). The binomial coefficient \(\binom{18}{2} = \frac{18 \times 17}{2} = 153\).

Key Concepts

Binomial Expansion

When tackling problems that involve expressions raised to a large power, the Binomial Theorem is incredibly handy. It allows us to expand expressions of the form \((a + b)^n\) into a sum of terms split across a sequence. Each term in the expansion has been derived based on a specific formula that involves powers of the components \(a\) and \(b\) from the original binomial. In our specific example \((x-1)^{18}\), we need to calculate the first three terms as a demonstration of binomial expansion at work. Here's how it unfolds:

• The first term is simply the case where \(b\) vanishes entirely, leaving us with \(a^n\), thus \(x^{18}\).
• The second term reduces the power of \(x\) by one and involves \(b\), so it becomes \(-18x^{17}\).
• The third term further reduces the power of \(x\) while \(b\) increases slightly, leading to \(153x^{16}\).

This pattern continues down to the last term, effectively building each step on the last. Understanding this concept makes tackling more complex expansions manageable.

Binomial Coefficient

Central to the success of the Binomial Theorem is the binomial coefficient, denoted as \(\binom{n}{k}\). This mathematical notation represents the number of ways to choose \(k\) successes from \(n\) possibilities, which is why it pops up in our expansion. It is calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). For our problem \((x-1)^{18}\), this coefficient plays a role at each stage of obtaining individual terms:

• For the first term, the coefficient is \(\binom{18}{0} = 1\).
• For the second term, the coefficient is \(\binom{18}{1} = 18\).
• The third term uses \(\binom{18}{2}\), calculated as \(\frac{18 \times 17}{2} = 153\).

These coefficients determine the multiplicative factor applied to each resulting product of \(a\)'s and \(b\)'s powers. Knowing how to calculate these coefficients swiftly is crucial for those delving into algebraic expansions.

Algebra

Algebra forms the backbone of tackling binomial expansions. It involves manipulating mathematical symbols to explore various properties of numbers or systems. In our case, algebra lets us efficiently manage the complexity of expanding a binomial with a high exponent like \((x-1)^{18}\). Here's how algebra assists in this process:

• Using the Binomial Theorem, we algebraically separate terms to simplify expressions like \((x-1)^{18}\) into manageable chunks.
• Each step involves straightforward yet precise arithmetic operations as well as substitutions, such as inserting the integer values back into the expression.
• The functioning of algebra is visualized through powers and coefficients, keeping each bit of the problem in check and consistent.

Algebra, therefore, not only provides the tools to solve the problem but also enhances our understanding of relationships within mathematics, making it an invaluable skill in both theoretical and practical applications.