Problem 43 - Calculus for Biology and Medicine

Question

Expand \((x+y)^{4}\).

Step-by-Step Solution

Answer

The expansion of \((x+y)^{4}\) is \(x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\).

Step 1: Understanding the Problem

You need to expand the expression \((x+y)^{4}\), which means you need to write the expression out as a sum of terms without parentheses.

Step 2: Apply the Binomial Theorem

The Binomial Theorem states that \((x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}\). Here, \(n=4\), so the expansion will have 5 terms, corresponding to \(k=0\) to \(k=4\).

Step 3: Calculate Coefficients Using Binomial Coefficients

Calculate each term of the expansion using the binomial coefficients: \(\binom{4}{0} = 1\), \(\binom{4}{1} = 4\), \(\binom{4}{2} = 6\), \(\binom{4}{3} = 4\), \(\binom{4}{4} = 1\). These will be the coefficients for each corresponding term.

Step 4: Write Each Term of the Expansion

Combine the coefficients with the powers of \(x\) and \(y\) based on \((x+y)^{4}\):1. \(1 \cdot x^{4}y^{0} = x^{4}\)2. \(4 \cdot x^{3}y^{1} = 4x^{3}y\)3. \(6 \cdot x^{2}y^{2} = 6x^{2}y^{2}\)4. \(4 \cdot x^{1}y^{3} = 4xy^{3}\)5. \(1 \cdot x^{0}y^{4} = y^{4}\)

Step 5: Final Expansion

Combine all the terms to write the final expanded form:\((x+y)^{4} = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\).

Key Concepts

Binomial Coefficients

Understanding the world of mathematics often starts with grasping foundational concepts such as binomial coefficients. These coefficients appear in the expansion of the power of a binomial expression. Essentially, a binomial coefficient gives you the number of ways to choose a subset of items from a larger set. For example, the binomial coefficient \(\binom{4}{2}\) reflects the number of ways to choose 2 items from a group of 4, which is 6.

In the context of the Binomial Theorem, these coefficients are pivotal. They determine the weights given to each term in the expansion. The expression \((x+y)^4\) uses binomial coefficients such as \(\binom{4}{0} = 1\), \(\binom{4}{1} = 4\), \(\binom{4}{2} = 6\), accompanying the corresponding terms \(x^4, 4x^3y, 6x^2y^2\) respectively. Calculating binomial coefficients involves using the formula:

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

where \(!\) denotes factorial, which is the product of all positive integers up to a number. This method helps ensure accuracy and understanding when dealing with polynomial expansions.

Polynomial Expansion

Expanding a polynomial like \((x+y)^4\) means expressing it as a sum of simpler terms. The Binomial Theorem provides a systematic way to perform this task. This theorem allows us to expand any power of a binomial into a series of terms. Each term consists of an original component raised to decreasing powers, while the second part's power increases.

To expand \((x+y)^4\), start by recognizing it will produce 5 terms. Here's what happens step-by-step:

• The first term \(x^4\) represents all of the power concentrated in \(x\) with none in \(y\).
• The second term \(4x^3y\) redistributes some of the power from \(x\) to \(y\).
• Subsequent terms like \(6x^2y^2\) show evenly shared powers, until power shifts entirely onto \(y\) in \(y^4\).

This structured breakdown through polynomial expansion is a key mathematical process, often used in solving real-world problems ranging from computer science tasks to physics calculations.

Mathematics Education

Introducing concepts like the Binomial Theorem and polynomial expansion in a classroom setting is essential in mathematics education. Understanding these principles equips students to tackle complex problems with clarity and confidence. Such topics build a foundation for higher math learning and application in various fields.

Effective mathematics education involves a structured approach wherein students not only learn formulas but also apply them contextually. For instance, in a lesson about \((x+y)^4\), students would:

• First understand the theorem through examples before attempting polynomial expansions themselves.
• Engage in calculating real binomial coefficients to internalize their meaning and utility.

By practicing these calculations and witnessing the direct results in polynomial expansions, such as transforming \((x+y)^4\) into its expanded form, students deepen their comprehension. This kind of educational experience encourages them to perceive mathematics as an interconnected field rather than isolated topics.