Problem 29 - Precalculus with Limits

Question

In Exercises 19 - 40, use the Binomial Theorem to expand and simplify the expression. \( \left(r + 3s\right)^6 \)

Step-by-Step Solution

Answer

The expanded form of \( (r + 3s)^6 \) is \( r^6 + 18r^5s + 540r^4s^2 + 8100r^3s^3 + 72900r^2s^4 + 393660rs^5 + 1574640s^6 \)

Step 1: Expression Explanation

Our goal is to express \( \left(r + 3s\right)^6 \) as a polynomial. According to the Binomial Theorem, \( (a + b)^n \) can be expanded as \[ \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k \] where \( {n \choose k} \) is a binomial coefficient and can be calculated as \( \frac{n!}{k!(n-k)!} \). In this case, we have \( a = r \), \( b = 3s \), and \( n = 6 \)

Step 2: Apply the Binomial Theorem

Applying the Binomial Theorem, we get \[ (r + 3s)^6 = \sum_{k=0}^{6} {6 \choose k} r^{6-k} (3s)^k \]. \] This gives us seven terms, from \( k = 0 \) to \( k = 6 \). Next, we will calculate each of these terms.

Step 3: Calculate the Terms

To calculate the terms, we need to calculate the binomial coefficients and multiply them with the appropriate powers of \( r \) and \( 3s \). Using \( {6 \choose k} = \frac{6!}{k!(6-k)!} \), this will give us \[ (r + 3s)^6 = {6 \choose 0} r^6 + {6 \choose 1} r^5 (3s) + {6 \choose 2} r^4 (3s)^2 + {6 \choose 3} r^3 (3s)^3 + {6 \choose 4} r^2 (3s)^4 + {6 \choose 5} r (3s)^5 + {6 \choose 6} (3s)^6 \]

Key Concepts

Binomial Coefficients

In the expansion of a binomial expression using the Binomial Theorem, binomial coefficients play a vital role. Here's why they are essential:

• Binomial coefficients determine the number of ways we can choose a specific number of elements from a set, crucial in creating terms for a polynomial expansion.
• They are represented as \({n \choose k}\), pronounced "n choose k," which indicates the number of ways to choose \(k\) elements from a total of \(n\), and are calculated using the formula: \( \frac{n!}{k!(n-k)!} \).

Understanding this formula is crucial because:

• \(n!\) (pronounced "n factorial") is the product of all positive integers up to \(n\). For instance, \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\).
• The factorial function reduces quickly, as multiplying large consecutive integers increases product size exponentially.
• If \(n = 6\) from our example, calculating \( {6 \choose k} \) yields the coefficients for each term in the polynomial expansion of \((r + 3s)^6\).

Polynomial Expansion

The aim of expanding a binomial expression is to express it as a polynomial. By definition, a polynomial is the sum of multiple terms of different powers of its variables:

• The expression \((r + 3s)^6\) is a binomial, as it contains two terms: "\(r\)" and "\(3s\)".
• Using the Binomial Theorem, we expand this expression into a sum where each term is formed by multiplying a binomial coefficient by the powers of \(r\) and \(3s\).

In our exercise, applying the Binomial Theorem results in:

• Seven terms, ranging from \(k = 0\) to \(k = 6\), according to the power \(n=6\).
• Each term ###{6 \choose k} r^{6-k} (3s)^k### reflects the combinations of the powers of \(r\) and \(3s\) controlled by \(k\).

This systematic expansion transforms complex binomials into manageable polynomials with clear, computable terms.

Combinatorics

Combinatorics is the branch of mathematics dedicated to counting and analyzing possible arrangements within a set. It covers a multitude of concepts, with binomial coefficients being one its most practical applications:

• In the context of the Binomial Theorem, combinatorics helps determine the different combinations of powers in the polynomial expansion.
• It answers "how many ways can we select \(k\) items from \(n\) items without regard to order?", precisely the function of \({n \choose k}\).

This concept is very intuitive and allows us to:

• Understand the predictable nature of patterns and terms in expansions, facilitating further calculations in algebra.
• Appreciate the symmetry in binomials, since \({n \choose k} = {n \choose n-k}\), illustrating the fascinating order within mathematical expressions.

Mastering combinatorial principles ensures a robust interpretation of polynomial expansions and their applications in wider branches of mathematics and science.