Problem 25 - Precalculus with Limits

Question

In Exercises 19 - 40, use the Binomial Theorem to expand and simplify the expression. \( \left(x + y\right)^5 \)

Step-by-Step Solution

Answer

The simplified form of \((x + y)^5\) is \(x^5 + 5x^{4}y + 10x^{3}y^2 + 10x^{2}y^3 + 5xy^{4} + y^5\).

Step 1: Apply the Binomial Theorem

First apply the Binomial Theorem to expand \((x + y)^5\). The nth term in the binomial expansion will be \(\binom{5}{k} x^{5-k} y^{k}\).

Step 2: Write out each term

The expansion of \((x + y)^5\) will be \(x^5 + \binom{5}{1}x^{4}y + \binom{5}{2}x^{3}y^2 + \binom{5}{3}x^{2}y^3 + \binom{5}{4}xy^{4} + y^5\)

Step 3: Evaluate the binomial coefficients

Evaluate the binomial coefficients. \( \binom{5}{1} = 5\), \( \binom{5}{2} = 10\), \( \binom{5}{3} = 10\), \( \binom{5}{4} = 5\).

Step 4: Simplify the expression

After simplifying, you get \(x^5 + 5x^{4}y + 10x^{3}y^2 + 10x^{2}y^3 + 5xy^{4} + y^5\).

Key Concepts

Understanding Binomial Coefficients

Binomial coefficients are central in calculating and expanding expressions using the Binomial Theorem. They are represented by the notation \( \binom{n}{k} \), which means "n choose k". This represents the number of ways to choose \(k\) elements from a set of \(n\) elements.

Mathematically, binomial coefficients are calculated as:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
where \(n!\) ("n factorial") is the product of all positive integers up to \(n\). These coefficients play a crucial role in determining the weight of each term in a binomial expansion.

For example, in the expansion of \((x + y)^5\), the coefficients \(\binom{5}{1} = 5\), \(\binom{5}{2} = 10\), etc., determine how each term builds up to the complete expanded polynomial.

Exploring Polynomial Expansion

Polynomial expansion transforms a binomial expression raised to a power into a sum of terms. Through the Binomial Theorem, \((x + y)^n\) can be expanded into \(n+1\) terms, each composed of combinations of powers of \(x\) and \(y\).

Each term in the expansion takes the form \(\binom{n}{k}x^{n-k}y^k\), allowing us to see all the ways that two variables can be combined to form terms of degree \(n\). For \((x + y)^5\), when expanded, it becomes:

• \(x^5\), where \(x\) holds the full power
• \(5x^4y\), showing a decrease in power of \(x\) and an increase in power of \(y\)
• \(10x^3y^2\), indicating further progression

This pattern continues until the last term \(y^5\), providing a complete picture of the polynomial expansion.

Combinatorics Connection

Combinatorics is a field of mathematics focused on counting combinations and arrangements. It forms the backbone of understanding how binomial coefficients are utilized within the Binomial Theorem.

The concept of "combinations" plays a significant role here. When expanding a binomial expression like \((x+y)^n\), the coefficients arise from the different ways we can arrange the powers of \(x\) and \(y\), counted by \(\binom{n}{k}\). Here, each combination represents a distinct term in the expansion.

This deep connection between combinatorics and the Binomial Theorem allows us to systematically and efficiently expand binomials without directly multiplying out everything, saving both time and effort, especially for large powers.