Problem 24 - Calculus

Question

Complete \((x+h)^{5}=x^{5}+\) ____. What are the binomial coefficients \(\left(\begin{array}{l}5 \\ 1\end{array}\right)\) and \(\left(\begin{array}{l}5 \\\ 2\end{array}\right)\) and \(\left(\begin{array}{l}5 \\ 3\end{array}\right) ?\)

Step-by-Step Solution

Answer

The missing term is \(5x^4h\), and the coefficients are 5, 10, and 10.

Step 1: Understanding Binomial Expansion

The binomial expansion formula is given by \((x + y)^n = \sum_{k=0}^{n} \begin{pmatrix} n \ k \end{pmatrix} x^{n-k} y^k\). Our task is to use this formula to expand \((x+h)^5\).

Step 2: Identifying the Term to Complete

To find the term which will complete \((x+h)^5 = x^5 + \) ____, identify the first two terms of the expansion. The first term is already given as \(x^5\). The second term involves \((h^1)\) and it is \(\binom{5}{1} x^{5-1}h\).

Step 3: Calculate the Second Term

To find the coefficient for the second term, use the binomial coefficient formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Substituting \(n=5\) and \(k=1\), we calculate: \(\binom{5}{1} = \frac{5!}{1!(5-1)!} = 5\).

Step 4: Write the Second Term in Expansion

Now, substitute the calculated coefficient into the second term, getting: \(5x^4h\). Thus, the expanded expression is \((x+h)^5 = x^5 + 5x^4h + \ldots\).

Step 5: Compute Remaining Binomial Coefficients

Using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), calculate: \(\binom{5}{2} = \frac{5!}{2!(5-2)!} = 10\) and \(\binom{5}{3} = \frac{5!}{3!(5-3)!} = 10\).

Key Concepts

Binomial Coefficients

In mathematics, binomial coefficients play a crucial role in the expansion of polynomial expressions. These coefficients are the numbers that appear in Pascal's triangle and are used in the binomial theorem. They are written as \( \binom{n}{k} \), which is read as 'n choose k'.

The binomial coefficient represents the number of ways to choose \(k\) elements from \(n\) elements, disregarding the order of selection. In our exercise, we need to find \( \binom{5}{1} \), \( \binom{5}{2} \), and \( \binom{5}{3} \), which help us determine the coefficients for the terms of the polynomial expansion.

• \( \binom{5}{1} = 5 \)
• \( \binom{5}{2} = 10 \)
• \( \binom{5}{3} = 10 \)

These coefficients can be calculated using the factorial formula, ensuring precise terms in the polynomial expansion.

Factorial Calculation

Factorials are foundations of the binomial coefficient calculations. The factorial of a number \(n\), denoted as \(n!\), is the product of all positive integers up to \(n\). For example, \(5!\) is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\).

The factorial function helps in deriving the binomial coefficient formula: \[ \binom{n}{k} = \frac{n!}{k! \times (n-k)!} \]
This formula tells us how to break down factorials into smaller parts to find the combinations, crucial for polynomial expansions. For instance:

• \( \binom{5}{1} = \frac{5!}{1! \times 4!} = 5 \)
• \( \binom{5}{2} = \frac{5!}{2! \times 3!} = 10 \)

Factorial calculations simplify many combinatorial expressions, allowing you to compute specific terms in the binomial expansion.

Polynomial Expansion

Polynomial expansion using the binomial theorem allows us to express expressions like \((x+h)^5\) as a sum of terms. According to the binomial theorem, for non-negative integers \(n\), \((x+y)^n\) can be expanded as: \[ \sum_{k=0}^{n} \begin{pmatrix} n \, k \end{pmatrix} x^{n-k} y^k \]

This gives us a systematic method to expand any binomial raised to a power. It provides each term in the expansion by combining binomial coefficients, powers of \(x\), and powers of \(y\).

• The first term of \((x+h)^5\) is \(x^5\).
• The next term is given by \(5x^4h\) using \( \binom{5}{1} \).

Understanding polynomial expansions assists in breaking down complex expressions into simpler, calculable components.

Combinatorics

Combinatorics is the study of counting and arranging objects. It provides tools to analyze the number of ways items can be selected, organized, and combined. Binomial coefficients and factorial calculations are key elements of combinatorics, providing the foundation for computing the terms in expansions like \((x+h)^5\).

Combinatorics helps in:

• Determining the number of ways to distribute specific terms in polynomial expansions.
• Using binomial coefficients to calculate the likelihood of various configurations.

The link between combinatorics and binomial expansions is integral for students. It offers insights into problem-solving strategies across different mathematical problems, not just polynomial expansions.