Problem 16 - College Algebra

Question

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (2 x+3 y)^{4} $$

Step-by-Step Solution

Answer

The expanded form is \( 16x^4 + 96x^3y + 216x^2y^2 + 216xy^3 + 81y^4 \).

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that for any integer \( n \), \((a + b)^n\) can be expanded as \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). Here, \( (2x+3y)^4 \) has \( a = 2x \), \( b = 3y \), and \( n = 4 \).

Step 2: Apply the Binomial Theorem

We will apply the theorem: \( (2x + 3y)^4 = \sum_{k=0}^{4} \binom{4}{k} (2x)^{4-k} (3y)^k \). This results in five terms, corresponding to \( k = 0, 1, 2, 3, \text{ and } 4 \).

Step 3: Calculate Binomial Coefficients

The binomial coefficients are \( \binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3}, \text{ and } \binom{4}{4} \), which are 1, 4, 6, 4, and 1 respectively.

Step 4: Expand Each Term - First Term

Calculate the first term (\( k = 0 \)): \( \binom{4}{0} (2x)^{4} (3y)^0 = 1 \cdot (2x)^4 \cdot 1 = 16x^4 \).

Step 5: Expand Each Term - Second Term

Calculate the second term (\( k = 1 \)): \( \binom{4}{1} (2x)^{3} (3y)^1 = 4 \cdot 8x^3 \cdot 3y = 96x^3y \).

Step 6: Expand Each Term - Third Term

Calculate the third term (\( k = 2 \)): \( \binom{4}{2} (2x)^{2} (3y)^2 = 6 \cdot 4x^2 \cdot 9y^2 = 216x^2y^2 \).

Step 7: Expand Each Term - Fourth Term

Calculate the fourth term (\( k = 3 \)): \( \binom{4}{3} (2x)^{1} (3y)^3 = 4 \cdot 2x \cdot 27y^3 = 216xy^3 \).

Step 8: Expand Each Term - Fifth Term

Calculate the fifth term (\( k = 4 \)): \( \binom{4}{4} (2x)^{0} (3y)^4 = 1 \cdot 1 \cdot 81y^4 = 81y^4 \).

Step 9: Combine All Terms

Combine all five terms to get the expanded form: \( 16x^4 + 96x^3y + 216x^2y^2 + 216xy^3 + 81y^4 \).

Key Concepts

Binomial expansion

The Binomial expansion is a method that allows us to expand expressions of the form \((a + b)^n\) into a polynomial. This is very useful when dealing with powers of binomials, as it simplifies complicated expressions. In essence, the Binomial Theorem gives us a formula to expand these expressions without manually multiplying the terms repeatedly. For example, expanding \((2x + 3y)^4\) becomes straightforward with this method. It involves computing different powers of the binomial's terms and multiplying them by specific coefficients. This technique results in a polynomial with terms that involve powers of both components of the binomial in a systematic manner. Each term's degree corresponds to its position in the expansion.

Binomial coefficients

Binomial coefficients are crucial to the Binomial Theorem, as they determine the terms' contributions in the expansion. These coefficients are denoted by \(\binom{n}{k}\), representing the number of ways to choose \(k\) elements from a set of \(n\) elements. These numbers can be found using Pascal's Triangle or calculated directly using the formula:

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

In our problem, expanding \((2x + 3y)^4\) involves coefficients \(\binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3},\) and \(\binom{4}{4}\), which are 1, 4, 6, 4, and 1 respectively. These coefficients are applied to each term in the expansion process, indicating their relative weights. With each term, you essentially multiply these coefficients by the appropriate powers of \(a\) and \(b\).

Polynomial expansion

Polynomial expansion refers to expressing a binomial raised to a power as a polynomial, which is a sum of monomials. With the binomial \((a + b)^n\), by applying the Binomial Theorem, we can derive each monomial uniquely based on its power and coefficient. Expanding \((2x + 3y)^4\) yields the polynomial:

• First term: \(16x^4\)
• Second term: \(96x^3y\)
• Third term: \(216x^2y^2\)
• Fourth term: \(216xy^3\)
• Fifth term: \(81y^4\)

Each of these terms results from specific combinations of the powers of \(2x\) and \(3y\), multiplied by the appropriate binomial coefficients. The sum of these terms is the full polynomial, which provides a comprehensive representation of the initial binomial expression.