Problem 106 - Algebra 2

Question

Expand each binomial. $$ (4 x-y)^{5} $$

Step-by-Step Solution

Answer

\((4x-y)^5 = 1024x^5 - 5120x^4y + 10240x^3y^2 - 10240x^2y^3 + 5120xy^4 - 1024y^5\

Step 1: Identify the components of the binomial

The binomial is \((4x - y)^5\). Here, the \(a\) term is \(4x\), the \(b\) term is \(-y\), and the power \(n\) is 5.

Step 2: Apply the Binomial Theorem

For each term \(k\) from 0 to \(n = 5\), calculate and write down the term according to the formula \(\binom{n}{k} a^{n-k} b^{k}\).

Step 3: Calculate the individual terms

For example, for \(k = 0\), the first term is: \(\binom{5}{0} (4x)^{5-0}(-y)^0 = 1 * 1024x^5 * 1 = 1024x^5\). Perform such calculations for every term from \(k = 0\) to \(k = 5\).

Step 4: Write out the full binomial expansion

Combine all the terms you calculated in Step 3 to write out the full expansion of the binomial.

Key Concepts

Understanding the Binomial Theorem

The Binomial Theorem is a powerful tool that allows you to expand expressions raised to a power efficiently. It is especially handy when dealing with binomials, which are expressions containing two terms, such as \(a + b\). The theorem gives us a formula that can expand these expressions without multiplying the binomial by itself multiple times. This saves time and helps avoid mistakes in long calculations.
The formula for the Binomial Theorem is:\[ (a + b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k \]
In this formula, \(n\) is the power to which the binomial is raised, and \(\binom{n}{k}\) is a binomial coefficient, which tells you how many ways you can arrange terms at each level of expansion.
By understanding and applying this theorem, you can easily expand any binomial expression raised to a power without the laborious process of continuing traditional multiplication.

Exploring Polynomials

Polynomials are expressions that include variables raised to various powers, and they often contain coefficients. In our exercise, \( (4x - y)^5 \) can be expanded into a polynomial. Polynomials are classified based on their number of terms:

• A single term is called a monomial.
• Two terms make a binomial.
• Three or more terms lead to polynomial expressions.

Polynomials might look complex, but they follow specific patterns and rules that make them easier to handle once you get used to their structure.
When binomials are expanded using the Binomial Theorem, you end up with a polynomial, consisting of terms that express the aggregate of the binomial raised to that specific power. Each term in the expansion will have coefficients, which are derived from combinatorial calculations.Understanding polynomials and their behavior is critical because they frequently appear in calculations across many areas of mathematics and applied science.

The Role of Combinatorics in Binomial Expansion

Combinatorics is a fascinating area of mathematics that studies how we count, arrange, and combine objects. This field plays an essential role in the Binomial Theorem, specifically in calculating the binomial coefficients denoted as \(\binom{n}{k}\).
These coefficients represent the number of ways you can choose \(k\) terms from \(n\) total items and are fundamental in determining the weight of each term in a binomial expansion. The formula for a binomial coefficient is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
Here, \(n!\) represents the factorial of a number, which is the product of all positive integers up to that number. Combinatorics allows us to calculate these coefficients systematically, ensuring each term in the expansion has the correct weight.
Understanding how combinatorial principles apply in real-world problems and mathematical theory is crucial, as it forms the groundwork for many other areas in both pure and applied mathematics.

Performing Mathematical Calculations with Binomials

Handling binomial expansions involves several mathematical calculations that require precision. Each term in the expansion of a binomial is calculated using three main components: the binomial coefficient, the powers of \(a\), and the powers of \(b\).
Let's break down these calculations based on our exercise \( (4x-y)^5 \):

• Calculate the binomial coefficient \(\binom{n}{k}\).
• Raise the first term \(a\) to the power \(n-k\).
• Raise the second term \(b\) to the power of \(k\).
• Multiply these three parts together to find each term in the expansion.

Accuracy in these calculations ensures the final polynomial form is correct.
The key to mastering these calculations is practice and familiarity with powers and factorials, which are routinely used in many areas of algebra and calculus.