Problem 43 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\). We can use this theorem to expand \((x+y)^4\). In this case, \(a = x\), \(b = y\), and \(n = 4\).

Step 2: Calculate Binomial Coefficients

Calculate the binomial coefficients \(\binom{4}{k}\) for \(k = 0, 1, 2, 3, 4\). These coefficients are calculated as follows:- \(\binom{4}{0} = 1\)- \(\binom{4}{1} = 4\)- \(\binom{4}{2} = 6\)- \(\binom{4}{3} = 4\)- \(\binom{4}{4} = 1\)

Step 3: Apply the Binomial Theorem

Substitute each binomial coefficient \(\binom{4}{k}\) and corresponding powers of \(x\) and \(y\) into the formula. Expand each term as follows:- For \(k=0\): \(\binom{4}{0} x^{4-0} y^{0} = 1 \cdot x^{4} = x^4\)- For \(k=1\): \(\binom{4}{1} x^{4-1} y^{1} = 4 \cdot x^{3} y^1 = 4x^3y\)- For \(k=2\): \(\binom{4}{2} x^{4-2} y^{2} = 6 \cdot x^{2} y^2 = 6x^2y^2\)- For \(k=3\): \(\binom{4}{3} x^{4-3} y^{3} = 4 \cdot x^{1} y^3 = 4xy^3\)- For \(k=4\): \(\binom{4}{4} x^{4-4} y^{4} = 1 \cdot y^{4} = y^4\)

Step 4: Combine the Terms

Sum the results from the previous step to form the final expansion:\[x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\].

Key Concepts

Expansion of Polynomials

Expanding a polynomial involves expressing it as a series of simpler terms or products, often powers of variables like \(x\) and \(y\). When expanding \((x+y)^4\), our goal is to express it as a sum of its parts. Each part (term) is determined by its binomial coefficient and corresponding powers of the variables. The expansion results in a polynomial with multiple terms, each representing different ways \(x\) and \(y\) can combine through multiplication.
Using the Binomial Theorem simplifies this process by providing a formula to generate terms systematically. With the theorem, you replace a general expression like \((a+b)^n\) with a specific sum of terms based on the degree \(n\) of the polynomial.

Binomial Coefficients

Binomial coefficients are key components in the expansion of binomials. These coefficients, often represented as \(\binom{n}{k}\), are calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n!\) represents the factorial of \(n\). These coefficients indicate the number of ways to choose \(k\) elements from a set of \(n\) elements, thus playing a role in distributing powers in the expansion process.
In the context of expanding \((x+y)^4\):

• \(\binom{4}{0} = 1\)
• \(\binom{4}{1} = 4\)
• \(\binom{4}{2} = 6\)
• \(\binom{4}{3} = 4\)
• \(\binom{4}{4} = 1\)

Each coefficient corresponds to a specific term in the expansion. Understanding these coefficients makes it easier to apply the binomial theorem successfully.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving numbers, variables, and operation symbols. In the context of polynomial expansion, it often includes terms that are products of variables raised to different powers. For instance, the expression obtained from expanding \((x+y)^4\) is \(x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\).
Choosing the right terms involves:

• Considering the order of variables
• Properly calculating powers based on their respective binomial coefficients

Each term in the expression is a separate product of powers of \(x\) and \(y\), weighted by its binomial coefficient. Learning to construct these expressions is crucial for solving algebraic problems efficiently.