Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-10,\) write the expansion of each binomial. $$ (x+y)^{6} $$

Step-by-Step Solution

Answer

The expansion of \((x+y)^6\) is \(x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6\).

Step 1: Identify the Binomial and Its Exponent

We are given the binomial \((x + y)\) with an exponent of 6. This means we need to expand it using the binomial theorem.

Step 2: Apply the Binomial Theorem

According to the binomial theorem, \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). Here, \(n = 6\), so we will expand using this formula.

Step 3: Calculate Binomial Coefficients

Determine the binomial coefficients \(\binom{6}{0}, \binom{6}{1}, \binom{6}{2}, \ldots, \binom{6}{6}\), which correspond to 1, 6, 15, 20, 15, 6, and 1 respectively.

Step 4: Write the Expansion

Using the binomial coefficients, write the expansion: \((x+y)^6 = \binom{6}{0}x^6y^0 + \binom{6}{1}x^5y^1 + \binom{6}{2}x^4y^2 + \binom{6}{3}x^3y^3 + \binom{6}{4}x^2y^4 + \binom{6}{5}x^1y^5 + \binom{6}{6}x^0y^6\).

Step 5: Simplify Each Term

Plug in the values of the binomial coefficients and simplify each term: 1. \(1x^6y^0 = x^6\) 2. \(6x^5y^1 = 6x^5y\) 3. \(15x^4y^2 = 15x^4y^2\) 4. \(20x^3y^3 = 20x^3y^3\) 5. \(15x^2y^4 = 15x^2y^4\) 6. \(6x^1y^5 = 6xy^5\) 7. \(1x^0y^6 = y^6\).

Step 6: Compile the Final Expanded Form

Therefore, the expanded form of \((x+y)^6\) is: \[ x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6 \]

Key Concepts

Understanding the Binomial Theorem

The binomial theorem is a crucial element in algebra for expanding expressions raised to any power. It provides a systematic way of expanding expressions that are a sum of two terms, known as binomials. When you have a binomial like \((x + y)\) raised to the 6th power, applying the binomial theorem gives us a clear path to find each term of the expansion. The formula:

• \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\)

This formula helps in constructing each component of the expanded polynomial. It involves summing multiple terms where each term consists of:

• A binomial coefficient \(\binom{n}{k}\)
• The first variable \(x\) raised to the power \(n-k\)
• The second variable \(y\) raised to the power \(k\)

Practically, for \((x + y)^6\), we will calculate each of these for k ranging from 0 to 6.

The Role of Binomial Coefficients

Binomial coefficients are a key part of the binomial theorem. These numbers signify the number of ways you can pick items or arrange terms in many mathematical problems. They are given by:

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

Where \(!\) represents factorial, which is a product of all positive integers up to that number. For example, to find the coefficient \(\binom{6}{2}\), you would solve:

• \(\frac{6!}{2!(6-2)!} = \frac{720}{2 \times 24} = 15\)

In the expansion \((x+y)^6\), the coefficients follow the sequence: 1, 6, 15, 20, 15, 6, 1. These become the multipliers for the respective terms in the expansion. Each coefficient gives each term its proper weight in the polynomial.

Expanding the Polynomial

Polynomial expansion using the binomial theorem involves writing out the terms in detail from the compact binomial form. We apply the binomial coefficients to construct each term in the expansion. Consider the formula:

• \((x+y)^6 = \binom{6}{0}x^6y^0 + \binom{6}{1}x^5y^1 + ... + \binom{6}{6}x^0y^6\)

The above expression is unpacked into a polynomial by multiplying each term's coefficient and applying the powers:

• \(x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6\)

Each term includes a component of \(x\) or \(y\) raised to a power that complements its pair, ensuring the sum of exponents always equals the binomial's original exponent.

Components of Algebraic Expressions

Algebraic expressions like \((x+y)^6\) consist of variables and constants combined using basic operations such as addition. In binomial expansions, we primarily deal with expressions consisting of two terms combined in various ways across the expanded form. Each term in the expanded expression is a monomial, such as \(6x^5y\), which is a basic building block of algebraic expressions:

• Variables \(+\) Constants
• Multiplication and Exponentiation

The expression \(x^6 + 6x^5y\) consist of terms where each term is the multiplication of constants (binomial coefficients) and variables raised to specific powers. Understanding these components allows learners to appreciate how algebraic rules collectively construct larger expressions, and mastering this provides insights into manipulating and simplifying complex algebraic expressions.