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Magazine Chapter 8: Problem 45
Find the specified term. The fourth term of \((a+b)^{9}\)
Short Answer
Expert verified
The fourth term is \(84a^6b^3\).
Step by step solution
01
Understand the Binomial Theorem
The Binomial Theorem states that for any positive integer \(n\), \((a+b)^n\) can be expanded into a sum involving terms of the form \(\binom{n}{k} a^{n-k} b^k\). This means each term in the expansion is determined by the binomial coefficient and the powers of \(a\) and \(b\).
02
Identify the Term Formula
To find the \(k^{th}\) term in the binomial expansion of \((a+b)^n\), use the formula for the \((k+1)\)th term: \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\).
03
Determine the Values for the Formula
In this problem, we seek the fourth term. Thus, we set \(k+1 = 4\), hence \(k = 3\). Here \(n = 9\), \(a = a\), and \(b = b\).
04
Calculate the Binomial Coefficient
The binomial coefficient \(\binom{9}{3}\) is calculated as \(\frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84\).
05
Calculate the Powers of the Terms
The powers are determined as follows: \(a^{9-3} = a^6\) and \(b^3 = b^3\).
06
Write Down the Fourth Term
Plugging the values back into the term formula \(T_4 = \binom{9}{3} a^6 b^3\), we get \(T_4 = 84a^6 b^3\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Expansion
The Binomial Expansion is a method used for expanding expressions that are raised to any finite power. When you have an expression like \[(a + b)^n\] it's not always easy to see what it expands into, especially for larger powers. The Binomial Theorem provides a systematic way to break it down into a sum of terms in a specific pattern. Each term in the expansion follows the formula: \[\binom{n}{k} a^{n-k} b^k\]In this formula,
- \(n\) determines the power the binomial is raised to, and
- \(k\) indicates the specific term within the expansion sequence.
Binomial Coefficient
A Binomial Coefficient is denoted by \(\binom{n}{k}\), and it represents the number of ways to choose \(k\) elements from a set of \(n\) elements, without considering the order. Mathematically, it's calculated as:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]This formula breaks down like this:
- \(n!\) (n factorial) is the product of all positive integers up to \(n\),
- \(k!\) is the factorial of the term number minus one,
- \((n-k)!\) takes away the term positions already formed by \(k\).
Power of a Binomial
The Power of a Binomial refers to the exponent \(n\) applied to the binomial expression \[(a+b)^n.\]To understand how the power affects the expansion, note the distribution pattern of the exponents in each term. For instance:
- In \[(a+b)^9\], the total of the exponents for \(a\) and \(b\) in each term equals 9.
- The exponents start with \(a^9\) paired with \(b^0\) and progressively decrease \(a\) while increasing \(b\).
- This balance ensures each possible split leading to a sum of 9 is calculated.
