Problem 37 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The simplified form of \(\left(\dfrac{2}{x} - y\right)^4\) after using the Binomial theorem is \(16x^{-4} - 32x^{-3}y + 24x^{-2}y^2 - 8xy^3 + y^4\)

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}\) where \(\binom{n}{k}\) is a binomial coefficient calculated as \(n! / [(n-k)!k!]\). In this problem, \(a = \dfrac{2}{x}\), \(b = -y\), and \(n = 4\)

Step 2: Apply the Binomial Theorem

Applying the Binomial theorem, the expression \(\left(\dfrac{2}{x} - y\right)^4\) can be expanded to \(\binom{4}{0}\left(\dfrac{2}{x}\right)^4(-y)^0 + \binom{4}{1}\left(\dfrac{2}{x}\right)^3(-y)^1 + \binom{4}{2}\left(\dfrac{2}{x}\right)^2(-y)^2 + \binom{4}{3}\left(\dfrac{2}{x}\right)^1(-y)^3 + \binom{4}{4}\left(\dfrac{2}{x}\right)^0(-y)^4\)

Step 3: Simplify the Expression

After simplifying the expression we get \(16x^{-4} - 32x^{-3}y + 24x^{-2}y^2 - 8xy^3 + y^4\)

Key Concepts

Binomial Expansion

The binomial expansion is a powerful algebraic tool used to expand expressions of the form \((a + b)^n\). It allows us to express such terms as a sum of terms involving powers of both \(a\) and \(b\). The significance of binomial expansion lies in its ability to transform complex expressions into simpler forms. The Binomial Theorem provides the formula:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

This formula is essential because each term in the expansion consists of a binomial coefficient \(\binom{n}{k}\), and powers of \(a\) and \(b\), helping break the expression into manageable parts.
The number of terms in the expansion corresponds to \(n+1\) terms and allows detailed manipulation of these algebraic expressions. For concrete applications, like the one given in the exercise \(\left(\dfrac{2}{x} - y\right)^4\), the expansion uses the insights from the binomial heorem.

Binomial Coefficients

Binomial coefficients are a key component of the Binomial Theorem. These coefficients, represented as \(\binom{n}{k}\), indicate the number of ways to choose \(k\) elements from a set of \(n\) elements, without regard to the order of selection. They are computed using the formula:

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

Here, \(n!\) means 'n factorial,' which is the product of all positive integers up to \(n\). For example, in the exercise, the coefficients \(\binom{4}{0}\), \(\binom{4}{1}\), \(\binom{4}{2}\), \(\binom{4}{3}\), and \(\binom{4}{4}\) are used.
These coefficients are essential because they provide the necessary weights for each term in the expansion. They essentially determine how the powers of \(a\) and \(b\) will combine to give the final simplified form.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They form the backbone of algebra and include entities like constants, coefficients, variables, and operational signs. While working with algebraic expressions, especially in the context of binomial expansions, understanding the distinct parts helps streamline calculations.
In the problem at hand, we deal with an expression \(\left(\dfrac{2}{x} - y\right)^4\), where components \(\frac{2}{x}\) and \(-y\) are evaluated through the Binomial Theorem to create a series of simpler terms.
When simplifying expressions resulting from binomial expansion, recognizing patterns in powers and coefficients is crucial. This process involves breaking down the expanded form and combining like terms to arrive at an expression such as \(16x^{-4} - 32x^{-3}y + 24x^{-2}y^2 - 8xy^3 + y^4\). It highlights how different operations interact with variables to create a simplified result.