Problem 9 - Algebra 2

Question

Find the indicated term of each expansion. fourth term of \((a+b)^{8}\)

Step-by-Step Solution

Answer

The fourth term is \( 56a^5b^3 \).

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that for any positive integer \( n \), \((a+b)^n\) can be expanded as \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). Each term in the expansion is of the form \( \binom{n}{k} a^{n-k} b^k \).

Step 2: Identify the term number

The term number in the expansion \((a+b)^8\) is specified as the fourth term. In general, the \((k+1)\)-th term corresponds to \( k \) in the binomial theorem. Thus, the fourth term corresponds to \( k = 3 \).

Step 3: Plug values into the binomial term formula

For the fourth term, substitute \( n=8 \) and \( k=3 \) into the formula for the binomial coefficient \( \binom{n}{k} a^{n-k} b^k \), yielding: \( \binom{8}{3} a^{8-3} b^3 \).

Step 4: Calculate the binomial coefficient

Calculate \( \binom{8}{3} \) using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). This gives: \( \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \).

Step 5: Formulate the term expression

Substitute the binomial coefficient and powers into the term formula to obtain: \( 56a^5b^3 \). This is the fourth term in the expansion.

Key Concepts

Binomial Coefficient

In algebra, the binomial coefficient is essential for finding each term of a binomial expansion. If you are expanding something like \((a + b)^n\), the binomial coefficient helps to determine how many ways you can choose a specific number of items regardless of order. It's represented by \( \binom{n}{k} \), commonly referred to as "n choose k." This is calculated using the formula:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) where \( n! \) denotes the factorial of \( n \).

The factorial function \( n! \) is the product of all positive integers up to \( n \). For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

The binomial coefficient tells us the number times a term will appear in the expansion. Therefore, understanding this concept is key to successfully expanding any binomial expression.

Polynomial Expansion

The expansion of a polynomial refers to expressing it as a sum of terms with coefficients. For example, in the binomial expression \((a + b)^8\), we are looking to expand it into terms using the binomial theorem. This expansion is arranged as:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

Each term within this expansion is a polynomial term composed of a binomial coefficient, a power of \( a \), and a power of \( b \). This allows any polynomial expression that follows the form \((a + b)^n\) to be neatly expanded into separated, computable terms.

Polynomial expansions are a fundamental concept in algebra that permits the simplification, analysis, and evaluation of expressions that at first glance may seem complex.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like addition or multiplication) that represent values that can change. In the context of a binomial expansion like \((a + b)^8\), the values \(a\) and \(b\) can represent different numbers or expressions.

An essential part of algebra is not only expanding expressions but also simplifying and interpreting them into more manageable forms. This involves understanding different algebraic manipulations such as distributing terms, combining like terms, and factoring, which allow complex expressions to be written in alternate forms that might be more useful for problem-solving.

Mastery of handling algebraic expressions is crucial for further exploration in mathematics as it feeds directly into calculus, solving equations, and modeling real-world phenomena. As such, improving one's skills in manipulating these expressions equips a learner with the tools necessary for these advanced topics.