Problem 45 - College Algebra

Question

Find the term indicated in each expansion. \(\left(x-\frac{1}{2}\right)^{9} ;\) fourth term

Step-by-Step Solution

Answer

The fourth term of the binomial expansion \((x - \frac{1}{2})^{9}\) is \(-\frac{21}{2}x^6\).

Step 1: Understanding Binomial Theorem

The Binomial Theorem states that \((a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k\). Here, \({n \choose k}\) is a binomial coefficient that represents the number of ways to choose k things out of n, and it's calculated as \({n \choose k} = \frac{n!}{k!(n-k)!}\), where '!' denotes factorial.

Step 2: Apply Formula

We know that in any binomial expansion, the (r+1)th term is given by \({n \choose r} a^{n-r} b^r\). Here, we need to find the 4th term, i.e., r is 3 (since r = term number-1), n is 9, a is x, and b is -1/2. Substituting these values into the formula gives us the 4th term of the expansion.

Step 3: Calculate the 4th Term

\({9 \choose 3} x^{9-3} (-\frac{1}{2})^3 = 84 * x^6 * -\frac{1}{8} = -84 * \frac{x^6}{8}\)

Key Concepts

Binomial Expansion

The Binomial Expansion is a method by which we can expand expressions that are raised to a power, specifically in the form \( (a+b)^n \). This is applicable to integer exponents n. For example, expanding \( (x+y)^2 \) would give us \( x^2 + 2xy + y^2 \). However, as the exponent increases, manually expanding becomes tedious. That's where the Binomial Theorem comes into play.

The theorem provides a general formula for expanding binomials of any positive integer exponent. It asserts that \( (a+b)^n \) is equal to the sum of terms \( {n \choose k} a^{n-k} b^k \) where \( k \) ranges from 0 to n. Each term here consists of a binomial coefficient, \( {n \choose k} \), and the corresponding powers of a and b.

An important improvement recommended for students who are trying to solve problems of binomial expansion, is to understand and visualize the pattern in the expansion. Each term in a binomial expansion follows a sequence, and recognizing this sequence helps in simplifying and quickly solving problems without expanding the entire expression.

To find a specific term, as in the given exercise, we use the formula for the term position which involves the selection of the appropriate binomial coefficient and related powers of a and b. This strategy is especially useful in large expansions, such as finding the fourth term in \( (x-\frac{1}{2})^9 \), without having to expand the entire expression.

Binomial Coefficient

A binomial coefficient, commonly expressed as \( {n \choose k} \), is a key component in the Binomial Theorem and represents the number of ways to choose k elements out of a set of n elements. It’s essential in combinatorics and various probability problems. The value of \( {n \choose k} \) can be calculated using the formula \( {n \choose k} = \frac{n!}{k!(n-k)!} \), where ! denotes factorial notation.

Understanding the factorial is crucial since it's the product of all positive integers up to a given number. For instance, \( 3! = 3 \times 2 \times 1 = 6 \). In binomial coefficients, factorials are used to ensure the number of combinations is counted correctly without repetitions.

When solving problems involving binomial coefficients, it's important to recognize certain properties such as \( {n \choose k} \) is the same as \( {n \choose n-k} \), and the sum of the coefficients in a binomial expansion is equal to \( 2^n \) where n is the exponent of the expansion. These insights can simplify the process of calculation, as is demonstrated in finding the fourth term of the binomial expansion in the original exercise.

Factorial Notation

Factorial notation, symbolized by the exclamation mark (!), is a mathematical expression that represents the product of an integer and all the non-zero integers below it. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). It is used heavily in permutations and combinations, as well as in calculating binomial coefficients.

The factorial function grows very fast, which means calculations with large numbers can become complex. However, in the context of binomial coefficients, students will often only need to calculate factorials of relatively small numbers. When looking at binomial expansions, you might need to compute factorials to determine the binomial coefficients for a specific term in the expansion.

Students should be mindful of simplifying factorials by canceling out common terms when they appear in both the numerator and the denominator, such as in \( \frac{n!}{k!(n-k)!} \), a simplification step that is implicitly used in the solution of the original exercise to find the fourth binomial term. This simplification is a critical aspect of computing binomial coefficients efficiently, especially when handling expressions with higher values.