Problem 55 - Algebra 2

Question

Find the specified term of each binomial expansion. Third term of \((3 x-2)^{9}\)

Step-by-Step Solution

Answer

The third term of the binomial expansion \( (3x - 2)^9 \) is \( 314928x^7 \).

Step 1: Apply the Binomial Theorem

According to the binomial theorem for any positive integer \( n \), the expansion of \( (a - b)^n \) is given by \( \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k \). Now we need to find the third term of \( (3x - 2)^9 \) which indicates \( k = 2 \).

Step 2: Substitute the values

The 3rd term in the expansion would then be\[ T_3 = {9 \choose 2} (3x)^{9-2} (-2)^2 \].

Step 3: Perform binomial coefficients and powers

Now we need to calculate the binomial coefficient, and raise \( 3x \) to the power of 7 and \( -2 \) to the power 2. Binomial coefficient \( {9 \choose 2} = 36 \), \( (3x)^7 = 2187x^7 \) and \( (-2)^2 = 4 \).

Step 4: Calculate the third term

The 3rd term in the expansion would then be \( T_3 = 36*2187x^7*4 \).

Key Concepts

Binomial Coefficient

The binomial coefficient is a fundamental component of the Binomial Theorem. It is represented using the notation \( {n \choose k} \), which is read as \"n choose k.\" This represents the number of ways we can select \( k \) elements from a total of \( n \) elements. In mathematical terms, it is defined by the formula:

• \({n \choose k} = \frac{n!}{k!(n-k)!}\)

Here, \( ! \) denotes a factorial, which is the product of all positive integers up to a given number.
Understanding binomial coefficients is crucial because they determine the contribution of each term in a binomial expansion. For example, in the exercise given, \({9 \choose 2} = 36\), which indicates that there are 36 ways to choose 2 elements from 9.

Polynomial Expansion

Polynomial expansion, in the context of the Binomial Theorem, involves expanding a binomial expression raised to a power into a sum of terms. Each term in this expansion is made up of products of powers of the individual components of the binomial.
The general formula for expanding \((a + b)^n\) is:

• \((a + b)^n = {n \choose 0}a^n b^0 + {n \choose 1}a^{n-1}b^1 + {n \choose 2}a^{n-2}b^2 + ... + {n \choose n}a^0b^n\)

This formula helps to determine the coefficients and powers of each term in the polynomial. In our example, \((3x - 2)^9\), the expression is expanded using this formula. Each term's coefficient is influenced by the binomial coefficients and involves calculating the respective powers of \(3x\) and \(-2\).

Power of a Term

Understanding the power of a term is vital when working with polynomials, especially in binomial expansions. Each term in the expansion has a specific structure: a coefficient, a power of the first term in the original binomial, and a power of the second term.
No two terms in a binomial expansion have the same power of either component because the powers are determined by the indices in the combination \( {n \choose k}a^{n-k}b^k\).

• The first component's power is \( n-k \), and the second component's power is \( k \).

Referring to our exercise, the third term \( T_3 \) in \((3x - 2)^9\) is calculated using the powers: \((3x)^{9-2} = (3x)^7\) and \((-2)^2 = 4\). Understanding these powers helps us comprehend the development of each term within the polynomial, leading to constructing the terms accurately.