Problem 37 - Precalculus: Mathematical for Calculus

Question

Find the term containing \(b^{8}\) in the expansion of \(\left(a+b^{2}\right)^{12}\)

Step-by-Step Solution

Answer

The term is \(495 a^8 b^8\).

Step 1: Understand the Problem

We need to find the term containing \(b^{8}\) in the binomial expansion of \( (a + b^2)^{12} \). This requires using the Binomial Theorem, which provides a way to expand expressions of the form \((x + y)^n\).

Step 2: Apply the Binomial Theorem

The Binomial Theorem states that \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). Here, \(x = a\), \(y = b^2\), and \(n = 12\). The general term in the expansion is \(T_k = \binom{12}{k} a^{12-k} (b^2)^k\).

Step 3: Simplify the General Term

Simplify \((b^2)^k\) as \(b^{2k}\). Thus, the general term becomes \(T_k = \binom{12}{k} a^{12-k} b^{2k}\).

Step 4: Find k such that 2k = 8

To find the term containing \(b^8\), we need \(2k = 8\). Solving for \(k\), we get \(k = 4\).

Step 5: Substitute k = 4 into the General Term

Substitute \(k = 4\) into \(T_k\) to get the specific term: \(T_4 = \binom{12}{4} a^{12-4} b^{8}\).

Step 6: Calculate Binomial Coefficient

Calculate \(\binom{12}{4}\):\[\binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495\].

Step 7: Calculate the term

The term with \(b^8\) is \(495 a^{8} b^{8}\).

Key Concepts

binomial theorem

The Binomial Theorem is a powerful tool in algebra used to expand expressions raised to a power. In essence, it provides a systematic way to expand the power of a binomial, i.e., an expression involving two terms, usually written as \((x + y)^n\). The theorem states that \[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\], where the notation \(\binom{n}{k}\) represents a binomial coefficient, also known as a combination.
The theorem is particularly useful in calculating specific terms in a binomial expansion without writing out the entire series. This is incredibly handy in solving problems involving large exponents, such as expanding \((a + b^2)^{12}\). By using this theorem, one can focus solely on determining the required term according to its specific exponent, simplifying complex algebraic expressions while saving time and effort.

binomial coefficient

Binomial coefficients are the heart of the Binomial Theorem. They are represented as \(\binom{n}{k}\) and denote the number of ways to choose \(k\) elements from a set of \(n\) elements, which is also referred to as "n choose k".
In mathematical terms, it's given by the formula:\[\binom{n}{k} = \frac{n!}{k!\,(n-k)!}\] where \(n!\) denotes the factorial of \(n\). For example, in the expression \((a + b^2)^{12}\), the coefficient for the term where \(k = 4\) is calculated as \(\binom{12}{4}\). Once computed, this coefficient helps to determine the specific term in the polynomial expansion.
Understanding binomial coefficients not only aids in calculating terms in binomial expansions but also finds applications in permutations, combinations, and even probability theory.

polynomial expansion

Polynomial expansion refers to the process of expanding an expression that is raised to some power, resulting in a polynomial. A polynomial is an algebraic expression made up of variables and coefficients, structured in terms of powers.
Using the Binomial Theorem, the expansion of expressions like \((a + b^2)^{12}\) becomes more manageable. Rather than computing each term manually, the theorem provides a formulaic approach for finding any term in the series.
For instance, the term containing \(b^8\) in the expansion is systematic, derived by ensuring that the power of \((b^2)\) becomes \(b^8\), leading to finding the correct exponent \(k\) that satisfies the equation. Thus, polynomial expansions transform complex binomials into simpler, workable formats, revealing each term's specific powers and coefficients.