Problem 41 - Precalculus: Mathematics for Calculus

Question

Find the indicated terms in the expansion of the given binomial. 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 \( 495 a^8 b^8 \).

Step 1: Understand the Problem

We need to find the term that contains \( b^8 \) in the expansion of \( (a + b^2)^{12} \) using the Binomial Theorem.

Step 2: Apply the Binomial Theorem Formula

The Binomial Theorem states that \( (x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k \). Here, let \( x = a \) and \( y = b^2 \), and \( n = 12 \). The general term in the expansion is given by \( T_{k+1} = \binom{12}{k} a^{12-k} (b^2)^k \).

Step 3: Express the Term in Terms of Powers of \( b \)

Substitute \( (b^2)^k \) with \( b^{2k} \). Thus, the general term \( T_{k+1} \) becomes \( \binom{12}{k} a^{12-k} b^{2k} \).

Step 4: Find the Exponent of \( b \)

We need \( 2k = 8 \) to find the term containing \( b^8 \). Solve for \( k \): \( k = \frac{8}{2} = 4 \).

Step 5: Calculate the Specific Term

Substitute \( k = 4 \) into the general term formula: \( T_{5} = \binom{12}{4} a^{8} b^8 \).

Step 6: Evaluate the 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: Write the Final Term

Substitute back to get the term containing \( b^8 \): \( 495 a^8 b^8 \).

Key Concepts

Binomial Expansion

The binomial expansion is a method used to express the powers of a binomial expression as a sum of terms. A binomial is simply any expression that consists of two terms. For example, \((a + b)\) is a binomial. When you raise a binomial to a positive integer power, like \((a+b)^n\), it expands into several terms, each involving products of the variables and constants involved.

To determine these terms, mathematicians use the Binomial Theorem, which systematically describes the coefficients of the terms in the expansion. This theorem is incredibly useful in a variety of mathematical fields, including algebra, calculus, and even probability theory. It helps simplify large expressions and find specific terms quickly.

Binomial Coefficient

The binomial coefficient is a key component in the binomial expansion. These coefficients are the numbers that appear as multipliers of the terms in a binomial expansion. They are often represented by combinations,which are mathematical values that determine how many ways you can choose a subset of items from a larger set.

In the permutation and combination notation \(\binom{n}{k}\), \(n\) represents the total number of items available, and \(k\) represents the number of items to choose.These binomial coefficients can be found following the pattern of Pascal's Triangle, or using the formula:

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

where \(!\) describes a factorial operation, meaning you multiply the number by all positive whole numbers less than itself. These coefficients help tell us how many of each term we will have as we expand the binomial.

Powers of Binomials

When dealing with powers of binomials, understanding how to manage the terms with exponents is crucial.Raising a binomial expression like \((a + b)\) to a power involves multiplying the binomial by itself several times. The Binomial Theorem helps deconstruct these powers into simpler terms.

In a binomial expansion of the form \((a + b)^n\),each term in the expansion is of the form \(\binom{n}{k} a^{n-k} b^k\).This means you are managing various products and powers for both terms in each part of the expansion.

For example, if you have \((a + b^2)^n\), terms will appear with powers like \(b^{2k}\)depending on the exponent on \(b\).By recognizing this, you can more easily locate specific terms within the expanded formula. Understanding these patterns greatly simplifies the often complex task of working with powers and expansions of binomial expressions.