Problem 30 - Precalculus with Limits

Question

In Exercises 19 - 40, use the Binomial Theorem to expand and simplify the expression. \( \left(x + 2y\right)^4 \)

Step-by-Step Solution

Answer

\( (x + 2y)^4 = x^4 + 8x^3*y + 24x^2*y^2 + 32x*y^3 + 16*y^4 \)

Step 1: Identify the Parameters

First, identify the parameters in the binomial theorem \( (a+b)^n \). Here, \( a = x \), \( b = 2y \), and \( n = 4 \).

Step 2: Apply the Binomial Theorem

The binomial theorem is given as \( (a+b)^n = Σ_{k=0}^{n}{{n \choose k}}a^{n-k}b^{k} \). Substituting \( a \), \( b \), and \( n \) with the identified parameters, the expanded form of the expression is: \( (x+2y)^4 = Σ_{k=0}^{4}{{4 \choose k}}x^{4-k}(2y)^{k} \).

Step 3: Calculate Each Term

For each \( k \) from 0 to 4, calculate the terms \( {4 \choose k} \), \( x^{4-k} \), and \( (2y)^{k} \), then multiply those values together. The results are: \( 1*x^4*(2y)^0 = x^4 \), \( 4*x^3*(2y)^1 = 8x^3*y \), \( 6*x^2*(2y)^2 = 24x^2*y^2 \), \( 4*x^1*(2y)^3 = 32x*y^3 \), \( 1*x^0*(2y)^4 = 16y^4 \).

Step 4: Write Out the Expansion

The final expansion of the expression is just the sum of all the calculated terms. Therefore, \( (x + 2y)^4 = x^4 + 8x^3*y + 24x^2*y^2 + 32x*y^3 + 16*y^4 \).

Key Concepts

Binomial Expansion

The binomial expansion is a useful way to expand expressions that are raised to a power. This concept arises from the binomial theorem, which provides a formula to expand any power of a binomial expression, such as \((a+b)^n\). In the binomial expansion of \((x + 2y)^4\), each term has a specific pattern given by the theorem. By following the pattern, you can systematically expand the expression without actually multiplying out everything manually.

The formula, \((a+b)^n = \sum_{k=0}^{n}{{n \choose k}}a^{n-k}b^{k}\), reveals that every term in a binomial expansion is a product of a binomial coefficient, a power of the first term \(a\), and a power of the second term \(b\). For instance, when expanding \((x + 2y)^4\), each resulting part of the expansion includes \(x\) and \(y\) raised to different powers, added together to give the complete polynomial expression.

• Use the binomial coefficient \({n \choose k}\) for each term
• Apply the powers \(a^{n-k}\) and \(b^k\).
• Sum all resulting products for the expanded form.

By understanding the structure, evaluating such expansions becomes much simpler, allowing you to see patterns rather than relying solely on computation.

Combinatorics

Combinatorics, a branch of mathematics primarily dealing with counting, is essential in understanding the binomial theorem and binomial expansion. In the context of binomials, combinatorics helps calculate the coefficients in the expanded form, known as the binomial coefficients \({n \choose k}\). These coefficients show how many ways you can select \(k\) elements from a set of \(n\) elements, regardless of order.

When expanding \((x + 2y)^4\), the coefficients like 1, 4, 6, 4, and 1 in the expansion come from combinatorial calculations:

• The term \({4 \choose 0}\) corresponds to choosing no \(y\)s from four options, which is 1 way.
• \({4 \choose 1}\) denotes selecting 1 \(y\) from four options, yielding 4 different possibilities.
• The central coefficient \({4 \choose 2}\) is about choosing 2 \(y\)s and represents 6 unique ways.
• Similarly, \({4 \choose 3}\) gives 4 ways, and \({4 \choose 4}\) results in 1, for choosing all \(y\)s.

Understanding combinatorics allows you to predict and compute these coefficients without manual combinations.

Algebraic Expression Simplification

Algebraic expression simplification involves reducing expressions to their most basic form by combining like terms and simplifying coefficients where possible. The binomial theorem's expansion of \((x + 2y)^4\) is a classic case of simplification.

After expanding the binomial, each term like \(4x^3(2y)\) needs to be simplified by multiplying the constants, resulting in terms like \(8x^3y\). Simplification steps combine powers and coefficients:

• Compute powers of numbers: for example, \((2y)^2\) becomes \(4y^2\).
• Multiply brackets accurately to handle expressions such as \(6x^2(4y^2)\), yielding \(24x^2y^2\).
• Add together similar terms (though, in a typical binomial expansion, terms are already distinct).

Through these steps, algebraic expression simplification transforms a complex expanded expression into a manageable, clear polynomial form, making it easier to apply in various mathematical contexts.