Problem 38 - College Algebra with Corequisite Support

Question

For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The fourth term of \(\left(x^{3}-\frac{1}{2}\right)^{10}\)

Step-by-Step Solution

Answer

The fourth term is \(-15x^{21}\).

Step 1: Understanding the Problem

We are asked to find the fourth term of the binomial expression \(\left(x^{3}-\frac{1}{2}\right)^{10}\). We will use the binomial theorem, which states that \((a+b)^{n} = \sum_{k=0}^{n}\binom{n}{k} a^{n-k} b^{k}\), to solve this.

Step 2: Identifying the Terms

In the binomial \(\left(x^{3}-\frac{1}{2}\right)^{10}\), we have \(a = x^3\) and \(b = -\frac{1}{2}\), with \(n = 10\). We need to find the fourth term, which corresponds to \(k = 3\) (because the formula for the \((k+1)\)-th term is \(k = 3\)).

Step 3: Applying the Binomial Coefficient

The binomial coefficient for the fourth term is \(\binom{10}{3}\). Use the combination formula \(\binom{n}{k} = \frac{n!}{k! \cdot (n-k)!}\) to find \(\binom{10}{3}\):\[\binom{10}{3} = \frac{10!}{3! \cdot 7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120\].

Step 4: Calculating the Exponents

For the \(k\)-th term in the expansion, the term is expressed as \(\binom{n}{k} \cdot a^{n-k} \cdot b^k\). Substitute the values: \(a = x^3\), \(b = -\frac{1}{2}\), \(n = 10\), and \(k = 3\):- The power of \(a = x^3\) is \(n-k = 10 - 3 = 7\) which gives \((x^3)^7 = x^{21}\).- The power of \(b = -\frac{1}{2}\) is \(k = 3\) which gives \((-\frac{1}{2})^3 = -\frac{1}{8}\).

Step 5: Calculating the Fourth Term

Multiply the binomial coefficient and the powers together to find the fourth term:\(\text{Fourth Term} = \binom{10}{3} \cdot x^{21} \cdot \left(-\frac{1}{8}\right) = 120 \cdot x^{21} \cdot -\frac{1}{8} = -15x^{21}\).

Key Concepts

Combination Formula

When dealing with binomial expressions, the combination formula is an essential tool for finding specific terms without fully expanding the expression. The combination formula, denoted as \( \binom{n}{k} \), is used to calculate the number of ways to choose \( k \) items from \( n \) total items, without regard to order. This is crucial for the application of the binomial theorem, which involves selecting terms in the expansion process.

The combination formula is given by: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\] Here, \( n! \) (read as "n factorial") is the product of all positive integers up to \( n \), and likewise for \( k! \) and \((n-k)!\). Factorials are used to simplify the counting of combinations in mathematics. By applying this formula, as seen in the exercise where we compute \( \binom{10}{3} \), you can find the number of ways to form specific terms in a binomial expansion.

Binomial Coefficient

The binomial coefficient is a key component in the binomial theorem and is used to determine the weight or coefficient of each term in a binomial expansion. Each term of the expansion \((a + b)^n\) consists of a binomial coefficient represented by \( \binom{n}{k} \).

These coefficients directly relate to the combination formula, providing the number of combinations possible to generate the specified term. For example, in the provided exercise, \( \binom{10}{3} \) evaluates to 120. This means the fourth term in the expansion will be multiplied by 120, balancing the contributions from each of the power terms, \( a \) and \( b \).

• The calculation of \( \binom{n}{k} \) is essential for accurately determining the contributions of each term in the binomial expansion.
• Understanding the binomial coefficient helps in predicting the behavior of the expansion for large powers.

The binomial coefficient succinctly captures the scaling factors of polynomial expansion terms when employing the binomial theorem.

Exponentiation

Exponentiation plays an integral role in simplifying terms when working with the binomial theorem. It involves raising a number or expression to a certain power, which is fundamental for managing the expressions within a binomial expansion. The basic rule of exponentiation is repeatedly multiplying the base by itself.

For instance, in the exercise, the term \( (x^3)^7 \) results in \( x^{21} \) due to the application of the power rule \((a^m)^n = a^{m \times n}\). Similarly, \[ (-\frac{1}{2})^3 \] was calculated to be \(-\frac{1}{8} \), affecting the sign and magnitude of that specific term in the overall expansion.

• Exponentiation ensures accurate calculation of each component’s contribution to the expanded polynomial.
• The sign and size of terms in a binomial expansion are significantly influenced by the exponent values.

Mastering exponentiation provides key insights into handling and predicting the nature of binomial expansion outcomes.

Polynomial Expansion

Polynomial expansion is the process of expressing a binomial expression raised to a power as a sum of terms, each multiplied by appropriate coefficients. The binomial theorem is the backbone of this process, ensuring systematic determination of each term in the expansion.

For example, in the exercise, the polynomial \((x^3 - \frac{1}{2})^{10}\) is expanded using the binomial theorem without calculating all ten possible terms. Instead, specific terms are targeted, such as the fourth term. This term is a polynomial itself, reflecting contributions from simplified powers of each binomial component and their respective coefficients.

• Using the binomial theorem efficiently narrows down the evaluation to only the necessary terms.
• Polynomial expansion through the binomial theorem avoids unnecessary computation complexities.

Understanding polynomial expansion allows one to effortlessly extract specific terms, as required in problems like finding a single term in a larger polynomial without full expansion.