Problem 24 - Algebra 2

Question

Find the indicated term of each expansion. fifth term of \((a-3)^{8}\)

Step-by-Step Solution

Answer

The fifth term is \(5670a^4\).

Step 1: Understand the Problem

We want to find the fifth term in the expansion of the binomial \((a-3)^8\) using the Binomial Theorem.

Step 2: Apply the Binomial Theorem

The Binomial Theorem states that \((a-b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}(-b)^k\). We will use this theorem to find the fifth term in the expansion of \((a-3)^8\).

Step 3: Identify the Term Number

The r-th term in a binomial expansion is given by \(T_{r+1} = \binom{n}{r}a^{n-r}(-b)^r\). For the fifth term, we set \(r=4\) since the first term corresponds to \(r=0\).

Step 4: Calculate the Binomial Coefficient

Determine \(\binom{8}{4}\), the binomial coefficient for the expansion:\[\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70.\]

Step 5: Substitute Values into the Formula

Substitute \(n=8, r=4, a=a, b=3\) into the formula to find the fifth term:\[T_5 = \binom{8}{4} a^{8-4}(-3)^4.\]

Step 6: Simplify the Expression

Simplify \(T_5 = 70a^4(-3)^4 = 70a^4 \cdot 81.\)Calculate the result:\(70 \times 81 = 5670.\)So the fifth term is \(5670a^4\).

Key Concepts

Binomial Expansion

The Binomial Theorem provides a way to expand expressions raised to a power. It's especially helpful when dealing with expressions like \((a + b)^n\), where direct multiplication would be cumbersome. The theorem states that any binomial \((a + b)^n\) can be expanded into a sum of terms in the form\[ \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}. \]Each term involves powers of both 'a' and 'b', multiplied by a binomial coefficient \(\binom{n}{k}\). This expansion is an essential tool in algebra and calculus, making it possible to simplify complicated polynomial expressions and solve equations more efficiently.

• The overall expansion consists of \((n + 1)\) terms when expanded completely.
• Each term is characterized by decreasing powers of 'a' and increasing powers of 'b', starting with the full power of 'a'.

Understanding binomial expansion is crucial for tackling many problems in mathematics, including finding specific terms in large expansions without having to fully multiply out all parts.

Binomial Coefficient

The binomial coefficient is a central part of the binomial theorem. Denoted as \(\binom{n}{k}\), it determines the coefficients of each term in the binomial expansion. The binomial coefficient \(\binom{n}{k}\) is calculated using the formula:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!}, \]where \(!\) indicates a factorial, which means multiplying a series of descending natural numbers. These coefficients arise from the concept of combinations in probability, representing the number of ways to pick 'k' elements from a set of 'n' elements without regard to order.

• In the expression \((a-b)^n\), each binomial coefficient multiplies a term composed of different powers of 'a' and 'b'.
• These coefficients tell us how many times each term is counted as you distribute the binomials.

Binomial coefficients are symmetrical around the middle term in a binomial expansion. For example, \(\binom{n}{k} = \binom{n}{n-k}\). This symmetry is one of the fascinating properties of binomial coefficients that can help simplify various algebraic computations.

Polynomial Terms

Polynomial terms are the building blocks of polynomials, which are expressions composed of variables and coefficients combined through addition, subtraction, and multiplication. In the context of binomial expansion, each term of the polynomial is represented using a specific power of 'a' and 'b', multiplied by a corresponding binomial coefficient. For example, in expanding \((a - 3)^8\), each term can be represented as:\[ T_{r+1} = \binom{n}{r} a^{n-r} (-b)^r. \]For real-world problems, understanding each polynomial term helps in reconstructing the entire polynomial from its expanded form.

• Each polynomial term has its unique degree, which is the sum of the exponents of its variables.
• In binomial expansions, as you go from the first term to the last, the degree of the polynomial decreases in 'a' and increases in 'b'.

Studying polynomial terms individually makes it easier to comprehend the structure of polynomials as a whole, and aids in solving algebraic equations effectively.