Problem 15 - Algebra for College Students

Question

Use the binomial theorem to expand each binomial. $$(r+t)^{5}$$

Step-by-Step Solution

Answer

The expansion of \((r+t)^5\) is \(r^5 + 5r^4t + 10r^3t^2 + 10r^2t^3 + 5rt^4 + t^5\).

Step 1: Identify the binomial expression and the exponent

The given binomial expression is \(r+t\) and the exponent is 5.

Step 2: Write down the binomial theorem formula

The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\). Here, \(a = r\), \(b = t\), and \(n = 5\).

Step 3: Substitute the values into the binomial theorem formula

Using the values from Step 2, we get \( (r+t)^5 = \sum_{k=0}^{5} \binom{5}{k} r^{5-k} t^k \).

Step 4: Calculate each term in the expansion

Calculate each term by substituting \(k\) from 0 to 5: - For \(k = 0\): \binom{5}{0} r^{5-0} t^0 = 1 \r^5 t^0 = r^5\ - For \(k = 1\): \binom{5}{1} r^{5-1} t^1 = 5 \r^4 t\ - For \(k = 2\): \binom{5}{2} r^{5-2} t^2 = 10 \r^3 t^2\ - For \(k = 3\): \binom{5}{3} r^{5-3} t^3 = 10 \r^2 t^3\ - For \(k = 4\): \binom{5}{4} r^{5-4} t^4 = 5 \r t^4\ - For \(k = 5\): \binom{5}{5} r^{5-5} t^5 = 1\t^5\.

Step 5: Combine all terms to form the expanded expression

Combine all the terms from Step 4: \(r^5 + 5r^4t + 10r^3t^2 + 10r^2t^3 + 5rt^4 + t^5\).

Key Concepts

Binomial Expansion

Binomial expansion is a technique to expand expressions of the form \((a+b)^n\). The binomial theorem simplifies this process by providing a formula, which is: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here:
- \(a\) and \(b\) are the terms in the binomial expression.
- \(n\) is the exponent.
- \((n\choose k)\) is the binomial coefficient.
It breaks down the expansion into individual terms involving combinations, powers, and multiplication.

In our case, we use \((r+t)^5\). Using the binomial theorem, we find the expanded form by calculating combinations and terms for each power of \(r\) and \(t\).

- Identify \(a\), \(b\), and \(n\) from the expression.
- Apply the binomial formula.
- Calculate each term and combine them.

This step-by-step process helps in breaking down complex expressions into simpler parts.

Combinatorics

Combinatorics focuses on counting, arrangement, and combination of objects. A specific aspect of combinatorics used in binomial expansion is the binomial coefficient, denoted as \binom{n}{k}\.

The binomial coefficient represents the number of ways to choose \(k\) items from \(n\) items without considering the order and is calculated as: \(\binom{n}{k} = \frac{n!}{k! (n-k)!}\).

- Factorials \(n!\) represent the product of all positive integers up to \(n\).
- They play a major role in determining combinations.

Using the binomial coefficients \(\binom{5}{0}, \binom{5}{1}, \...)\) helps us calculate terms like \(\r^5, \5r^4 t, \10r^3t^2, \10r^2t^3, \5rt^4, \t^5\), leading to the full expansion.
Understanding combinatorics ensures the accurate calculation of each binomial term.

Exponentiation

Exponentiation is the process of raising a number to the power of another number. In binomial expansion, our task involves exponentiating each term based on its position.

- For \((a+b)^n\), \a\ is raised to the power of \(n-k\),
- While \b\ is raised to the power of \k\.

In the given exercise \((r+t)^5\), the exponents change depending on the term's position:

- \(r^5, r^4, r^3, r^2, r, t^5\)

This results from how many times we use \r\, and \t\ in each product.

- Exponent values change progressively from 0 to \.
- Calculating the power accurately determines each term.

Mastering exponentiation ensures precision in expanding binomials and combining terms properly.

Algebra

Algebra involves using symbols and letters to represent numbers and quantities in equations and expressions. The binomial theorem utilizes algebraic techniques for expansion.

In the expression \((r+t)^5\), algebra helps us distribute and combine like terms such as \(r^5, 5r^4t, 10r^3t^2\), etc. Key algebraic rules applied include:

- The distributive property: ensuring each term is multiplied correctly.
- Combination of like terms: organizing terms with \r\ and \t\.

For instance, \(r^{5-0} t^0 = r^5\), and \binom{5}{1} r^{5-1} t = 5r^4 t\. These algebraic operations are the backbone for correct binomial expansion.
Understanding algebra supports breaking down complex problems and finding accurate solutions.