Problem 45 - College Algebra with Modeling and Visualization

Question

Find the specified term. The fourth term of \((a+b)^{9}\)

Step-by-Step Solution

Answer

The fourth term is \(84a^6b^3\).

Step 1: 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 \(b\).

Step 2: 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: \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\).

Step 3: 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 \(b = b\).

Step 4: Calculate the Binomial Coefficient

The binomial coefficient \(\binom{9}{3}\) is calculated as \(\frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84\).

Step 5: Calculate the Powers of the Terms

The powers are determined as follows: \(a^{9-3} = a^6\) and \(b^3 = b^3\).

Step 6: Write Down the Fourth Term

Plugging the values back into the term formula \(T_4 = \binom{9}{3} a^6 b^3\), we get \(T_4 = 84a^6 b^3\).

Key Concepts

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.

The terms consist of a product of a binomial coefficient and powers of the involved variables. As you expand, start from \(k = 0\), increment until \(k = n\). This way, every possible combination of powers for \(a\) and \(b\) is covered.

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 \(k\).

The binomial coefficient also indicates how many times each unique term appears in the binomial expansion. For instance, in finding the fourth term of \[(a+b)^9\],\(\binom{9}{3} = 84\). This computation specifies that there are 84 ways of forming the term that contains \(a^6\) and \(b^3\).

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.

The systematic increment and decrement allow each term to reflect one potential combination of powers, characterized by the terms \(a^{n-k}\) and \(b^k\). Understanding this relationship concept aids in predicting and writing any term in the binomial expansion accurately.