Question

Use the Binomial Theorem to find the indicated coefficient or term. The 3 rd term in the expansion of \((x-3)^{7}\)

Short answer

The 3rd term is \( 189 x^5 \).

Step-by-step solution

Identify the general term in the binomial expansion

The Binomial Theorem states that the expansion of \( (a + b)^n \) is given by \[ \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]. For the given expression, \( (x-3)^7 \), identify \( a = x \) and \( b = -3 \). The general term \( T_{k+1} \) is given by \[ \binom{n}{k} a^{n-k} b^{k} \].

Determine the term number to find

The problem asks for the 3rd term (\( k = 2 \) because the first term corresponds to \( k = 0 \)).

Substitute values into the general term formula

For the 3rd term, use \( n = 7 \) and \( k = 2 \). The 3rd term \( T_{3} \) is given by: \[ T_{3} = \binom{7}{2} x^{7-2} (-3)^{2} \].

Calculate the binomial coefficient

Calculate \( \binom{7}{2} \): \[ \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21 \].

Simplify the term

Now simplify the term: \[ T_{3} = 21 x^5 (-3)^2 = 21 x^5 (9) = 189 x^5 \].

Key concepts

Binomial Coefficient

In the Binomial Theorem, the binomial coefficient is a crucial component that determines how each term in the expansion is weighted. The binomial coefficient is denoted as \( \binom{n}{k} \) and can be calculated using the formula:

\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Here, \( n \) is the exponent of the binomial expression, and \( k \) represents the specific term position we are interested in.

It is often read as 'n choose k' and is derived from combinatorics, reflecting how many different ways you can choose \( k \) elements from a set of \( n \) elements without regard to the order of selection.

General Term

In any binomial expansion, the general term helps you find a specific term in the expanded form of \( (a + b)^n \). This general term is given by:

\[ T_{k+1} = \binom{n}{k} a^{n-k} b^k \]
The general term formula incorporates:

• \( \binom{n}{k} \) - the binomial coefficient
• \( a^{n-k} \) - the variable 'a' raised to the power \( n - k \)
• \( b^k \) - the variable 'b' raised to the power 'k'

For the given expression \( (x - 3)^7 \), we identified \( a = x \) and \( b = -3 \). To find a particular term, substitute the values of \( n \) and \( k \) into the general term formula.

Exponentiation

Exponentiation is the operation used to raise a number to a power. In the context of the Binomial Theorem, exponentiation is applied to both variables and constants in the binomial expansion:

• In each term, \( a \) is raised to the power of \( n-k \)
• \( b \) is raised to the power of \( k \)

For example, in the 3rd term of \( (x-3)^7 \), you need to find the exponentiated variables:\
\( x^{7-2} = x^5 \)
\( (-3)^{2} = 9 \)
When you combine these results, you get the simplified term.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, selection, and arrangement of objects. In the Binomial Theorem, combinatorics arises in the calculation of binomial coefficients. The binomial coefficient \( \binom{n}{k} \) uses factorials to count the number of ways to choose \( k \) elements from a set of \( n \) elements.

Understanding combinatorics helps in grasping how each term is derived in a binomial expansion. It provides the foundational rules for how terms are combined and ordered to form the complete expansion.

Polynomial Expansion

Polynomial expansion refers to expressing a binomial \( (a + b)^n \) as a sum of multiple terms involving powers of \( a \) and \( b \). The Binomial Theorem gives a systematic way to perform this expansion:

\[ (a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \text{...} + \binom{n}{n} a^0 b^n \]
Each term combines:

• A binomial coefficient \( \binom{n}{k} \)
• The variable \( a \) raised to \( n-k \)
• Variable \( b \) raised to \( k \)

For example, the full expansion of \( (x - 3)^7 \) is a series of terms involving different powers of \( x \) and \( -3 \). Each term can be individually calculated using the general term formula.