Question

Fxpand \((2 x-y)^{7}\) using the Binomial Theorem.

Short answer

The expansion of \((2x-y)^7\) is \(128x^7 - 448x^6y + 672x^5y^2 - 560x^4y^3 + 280x^3y^4 - 84x^2y^5 + 14xy^6 - y^7\).

Step-by-step solution

Understand the Binomial Theorem

The Binomial Theorem provides a way to expand expressions of the form \((a + b)^n\). It states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this exercise, we will apply this theorem to \((2x-y)^7\).

Identify the Components of the Binomial

In our expression \((2x-y)^7\), identify \(a = 2x\), \(b = -y\), and \(n = 7\). We will apply the Binomial Theorem using these values.

Compute Each Term Using the Binomial Theorem

Each term in the expansion can be expressed as \(\binom{7}{k} (2x)^{7-k} (-y)^k\), where \(k\) ranges from 0 to 7. We will compute each of these terms separately.

Calculate Binomial Coefficients

The binomial coefficient \(\binom{7}{k}\) is calculated using the formula \(\binom{7}{k} = \frac{7!}{k! (7-k)!}\). Compute these coefficients for \(k = 0, 1, 2, ..., 7\).

Calculate Powers of Each Component

For each term, calculate \((2x)^{7-k}\) and \((-y)^k\). Remember that \((-y)^k = (-1)^k y^k\).

Combine Terms to Form the Expansion

Combine the binomial coefficients with the calculated powers to form each term of the expansion. Sum all these terms to complete the binomial expansion.

Write Out the Complete Expansion

Finally, write out the complete expansion of \((2x-y)^7\) by listing all computed terms in their simplified form: \(\binom{7}{0}(2x)^7(-y)^0 + \binom{7}{1}(2x)^6(-y)^1 + ... + \binom{7}{7}(2x)^0(-y)^7\). Simplify each term to arrive at the final expression for the expansion.

Key concepts

Binomial Expansion

The binomial expansion is a powerful technique used in mathematics to expand expressions that are raised to a power, like \( (a + b)^n \). This technique is particularly helpful when the exponent is a relatively small integer.
The binomial expansion theorem states that if you have an expression of the form \( (a + b)^n \), you can expand it into a sum of terms: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] Each term \( \binom{n}{k} a^{n-k} b^k \) in this expansion represents a part of the full polynomial.
This formula helps break down complex expressions into simpler components, making it easier to compute and understand.

Binomial Coefficients

Binomial coefficients are critical components in the binomial theorem. They are the coefficients that appear in the expanded form of a binomial raised to a power.
You might have seen them in expressions such as \( \binom{n}{k} \), which is read as "n choose k". These coefficients are calculated using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] Here, \( n! \) (n factorial) refers to the product of all positive integers up to n. Therefore, the binomial coefficient \( \binom{n}{k} \) gives you the number of ways to choose k elements from a set of n elements, which ties nicely into the concept of combinatorics.
In essence, these coefficients indicate how many different ways we can arrange particular terms in the expansion.

Combinatorics

Combinatorics is an area of mathematics that focuses on counting, arranging, and combining items. It's a broad field that includes concepts like binomial coefficients, permutations, and combinations.
In the context of the binomial theorem, combinatorics helps us determine the number of ways we can choose elements to form each term in the expansion.
When dealing with a binomial expression like \( (a + b)^n \), understanding "n choose k" or \( \binom{n}{k} \) is a combinatorial problem. It requires you to count how many different terms of form \( a^{n-k}b^k \) can be created.
This interplay between counting and arranging makes combinatorics a valuable tool not just in mathematics, but also in various applied sciences, including computer science and physics.

Polynomial Expansion

A polynomial expansion involves expressing a power of a binomial as a sum of individual terms, each involving powers of its components. Using the binomial theorem, a polynomial expression like \( (a + b)^{n} \) can be expanded into a sum with multiple terms.
Each of these terms takes the form \( \binom{n}{k} a^{n-k} b^k \), showcasing both the structure of the polynomial and the influence of the binomial coefficients.
Polynomial expansion is extensively used in algebra to simplify expressions and solve equations. It's also used in calculus for approximating functions through Taylor series and in probability theory for modeling different outcomes. The expansion process reveals not just the mathematical elegance of polynomials but also their practical applicability in solving complex real-world problems.