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Chapter 12: Problem 227

Find the indicated term in the expansion of the binomial. Fifth term of \((a+b)^{9}\)

Short Answer

Expert verified
The fifth term is \[126 a^{5} b^{4}\].

Step by step solution

01

Identify the General Binomial Expansion Formula

The general term in the expansion of \(a+b\)^{n} is given by \(T_{k+1} = \binom{n}{k} a^{n-k} b^{k}\). Here, \(T_{k+1}\) represents the \(k+1\)-th term, \binom{n}{k} is the binomial coefficient, and \(n = 9\) for the given expression.
02

Determine the Term to Find

The problem is asking for the fifth term, which is \(T_{5}\). This means \(k+1 = 5\), so \(k = 4\).
03

Substitute Values into the Formula

Substitute \(n = 9\), \(k = 4\), \(a = a\), and \(b = b\) into the general term formula: \[T_{5} = \binom{9}{4} a^{9-4} b^{4}\].
04

Calculate the Binomial Coefficient

Calculate \(\binom{9}{4}\): \[\binom{9}{4} = \frac{9!}{4! (9-4)!} = \frac{9!}{4! 5!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126\].
05

Simplify the Expression

Substitute \(\binom{9}{4} = 126\) into \(T_{5}\) and simplify: \[T_{5} = 126 a^{5} b^{4}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Theorem
The binomial theorem provides a way to expand expressions raised to a power, such as \((a + b)^{9}\). It tells us that any binomial to the power of \(n\) can be expanded into a sum involving terms of the form \(\binom{n}{k} a^{n-k} b^{k}\). This is useful for simplifying complex algebraic expressions without having to manually multiply everything out. The binomial theorem also introduces the concept of binomial coefficients, which represent the different ways to choose \(k\) items from \(n\) items. Each term in the expansion is associated with one of these coefficients.
Combinatorics
Combinatorics is the field of mathematics that deals with counting, arrangement, and combination of objects. In the context of the binomial theorem, it comes into play when calculating the binomial coefficients. These coefficients \(\binom{n}{k}\) are found using the formula \(\frac{n!}{k!(n - k)!}\), where \(n!\) represents the factorial of \(n\). The factorial of a number \(n\), denoted as \(n!\), is the product of all positive integers up to \(n\). For example, for the term \(T_{5}\) in our problem, we calculate \(\binom{9}{4}\) which represents the number of ways to choose 4 items from 9.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. In our binomial expansion problem, \((a + b)^{9}\) is an algebraic expression involving the variables \(a\) and \(b\). Each term in the expansion has a specific form derived from the general term formula \(\binom{n}{k} a^{n-k} b^{k}\). For the fifth term specifically, we use \(T_{5} = 126 a^{5} b^{4}\). When simplified, algebraic expressions like these make it easier to see relationships between terms and solve complex problems efficiently.

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Most popular questions from this chapter

In your own words, explain how to find the general term of a geometric sequence.

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