Problem 37 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Binomial Theorem

The Binomial Theorem states that \[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \]where \( \binom{n}{k} \) is the binomial coefficient. We will use this to expand \( (a + b^2)^{12} \).

Step 2: Identifying the Term with \( b^8 \)

In the expansion \((a + b^2)^{12}\), a typical term is given by\[ \binom{12}{k} a^{12-k} (b^2)^k = \binom{12}{k} a^{12-k} b^{2k} \]To find the term containing \(b^{8}\), find \(k\) such that \(2k = 8\).

Step 3: Solving for \(k\)

Solve \(2k = 8\) by dividing both sides by 2,\[ k = \frac{8}{2} = 4 \].

Step 4: Calculate the Binomial Coefficient and Resulting Term

Substitute \(k = 4\) back into the term formula,\[ \binom{12}{4} a^{12-4} b^8 = \binom{12}{4} a^8 b^8 \]Calculate \(\binom{12}{4}\),\[ \binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]Thus, the term is \(495a^8b^8\).

Key Concepts

Binomial Expansion

The binomial expansion of an expression involves expanding a power of a binomial in a systematic way. The expression \[ (a + b^2)^{12} \]is a binomial raised to a power, and we can use the binomial theorem to expand it. The binomial theorem provides us a formula to expand expressions of the form \[ (x + y)^n \],where each term in the expansion is in the format \[ \binom{n}{k} x^{n-k} y^k \].This allows us to systematically find all possible terms in a polynomial with cross terms involving different powers of \(a\) and \(b^2\).
For example, in our problem, we want to identify the term that specifically contains the power \(b^8\). Since \( b^2 \) is part of our binomial, the corresponding power \(b^{8}\) points to the specific terms in the expansion where \((b^2)^k = b^{8}\), directly leading us to find \(k = 4\).
The use of the expansion theorem allows us to handle such calculations efficiently by directly applying the formula rather than manually expanding all terms.

Binomial Coefficient

The binomial coefficient is fundamental to calculating individual terms in binomial expansions. It is represented as \[ \binom{n}{k} \].This symbol denotes the number of ways to choose \(k\) elements from a total of \(n\) elements without regard to order. It is often referred to as "n choose k." The formula for calculating the binomial coefficient is:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

where \(!\) denotes factorial, which is the product of all positive integers up to that number.
In the given problem, we calculated \(\binom{12}{4} \) which equals:\[ \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \].
This represents the coefficient of the term where \(a^8 b^8\) emerges in the expansion, matching the power and giving us the guided pathway to find a specific term like \(495a^8b^8\). Understanding binomial coefficients is crucial in breaking down and simplifying polynomial expressions into specific terms or patterns.

Polynomial Expansion

Polynomial expansion involves expressing a power of a polynomial as the sum of terms consisting of coefficients and products of its variables. When a binomial expression like \((a + b^{2})^{12}\) is expanded, it creates a polynomial with terms in varying powers of \(a\) and \(b^2\).
Through expansion, you will notice terms structured by exponents dependent on the binomial powers. Each term follows a pattern from the binomial theorem, incorporating both the coefficients, derived from binomial coefficients, and the powers of the individual components.
For instance, when determining the specific term of \(b^8\), we apply the knowledge of matching the variable powers through expansion – a crucial step in decomposing complex polynomials into manageable segments.

• The process highlights patterns and structures that algebrically build up the final expanded form.
• Relates directly to how binomial coefficients distribute across terms, granting us an expanding scope at polynomials.

This structured approach ensures that all terms contribute to a full, expanded, polynomial view, giving insights into each component's contribution.