Problem 38 - Precalculus Mathematics for Calculus

Question

Find the second term in the expansion of \(\left(x^{2}-\frac{1}{x}\right)^{25}\).

Step-by-Step Solution

Answer

The second term is \(-25x^{47}\).

Step 1: Identify the Binomial Theorem

The binomial theorem states that for any positive integer \(n\), \((a + b)^n\) can be expanded as \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this problem, \(a = x^2\) and \(b = -\frac{1}{x}\).

Step 2: Determine the General Term

Using the binomial theorem, the general term \(T_k\) in the expansion \((x^2 - \frac{1}{x})^{25}\) is given by:\[T_k = \binom{25}{k} (x^2)^{25-k} \left(-\frac{1}{x}\right)^k.\]

Step 3: Simplify the General Term Expression

Simplify the expression for the general term:\[T_k = \binom{25}{k} x^{2(25-k)} \cdot \left(-1\right)^k \cdot x^{-k}.\]

Step 4: Simplify Further

Combine the powers of \(x\):\[T_k = \binom{25}{k} (-1)^k x^{50-3k}.\]

Step 5: Find the Second Term

For the second term in the expansion, set \(k = 1\) (since terms start from \(k = 0\)):\[T_1 = \binom{25}{1} (-1)^1 x^{50-3 \cdot 1} = 25(-1)x^{47} = -25x^{47}.\]

Key Concepts

Polynomial Expansion

When we talk about polynomial expansion, we are often referring to expressing a power of a binomial as a sum of individual terms. This expansion is done using the Binomial Theorem, which helps break down complex expressions into simpler, more manageable pieces.

For instance, consider the expression \((a + b)^n\). The expansion involves several terms where each term is derived from choosing different combinations of \(a\) and \(b\). This process is much like arranging different colored bricks to build a wall; each combination gives rise to a unique term.

• The first term is always \(a^n\), as all 'bricks' or terms consist entirely of 'a'.
• As we expand, we gradually replace one 'a' with one 'b', affecting the powers of \(a\) and \(b\).

These combinations continue until the last term, which is \(b^n\). Hence, the expansion unfolds as a sequence of decreasing powers of \(a\) and increasing powers of \(b\), illuminating the internal structure of the polynomial.

Binomial Coefficient

The binomial coefficient, often denoted as \( \binom{n}{k} \), plays a crucial role in polynomial expansion. It is a mathematical tool that tells us how many ways we can choose \(k\) items from a set of \(n\) items without regard to order.

In the context of polynomial expansion, the binomial coefficient determines the number of ways each combination of terms can occur in the expansion of a binomial raised to a power. These coefficients populate Pascal's Triangle and follow the formula:

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

This formula provides a way to calculate the coefficients needed for each term in the expansion. For example, when expanding \((x^2 - \frac{1}{x})^{25}\), we use \( \binom{25}{k} \) to find the coefficient of the \(k\)-th term.

• It represents symmetry in combinations, where \( \binom{n}{k} = \binom{n}{n-k} \).
• These coefficients are integral to forming the polynomial terms in a structured expansion.

Exponent Rules

Exponent rules are essential for simplifying and managing the terms within polynomial expansions. When working with expressions, these rules help us combine and manipulate powers effectively to reach simplified forms.

• Product Rule: If you multiply like bases, you add the exponents: \(x^a \cdot x^b = x^{a+b}\).
• Power Rule: When raising a power to another power, you multiply the exponents: \((x^a)^b = x^{ab}\).
• Quotient Rule: If dividing like bases, you subtract exponents: \(\frac{x^a}{x^b} = x^{a-b}\).

Let's apply these rules to simplify terms in our example. In \((x^2 - \frac{1}{x})^{25}\), the general term \(T_k\) combines powers of \(x\) by adding and subtracting exponents:

• Combine \(x^{2(25-k)}\) and \(x^{-k}\) to simplify the expression.
• By employing the product rule, express it as \(x^{50} \cdot x^{-3k} = x^{50-3k}\).

Understanding how to use these rules allows you to control the complexity and maintain clarity as you expand and simplify polynomial expressions.