Problem 104 - Algebra 2

Question

Expand each binomial. $$ (2 x-3 y)^{7} $$

Step-by-Step Solution

Answer

The expanded form of \( (2x-3y)^7 \) is as follows: \( (2x)^7 - 7*3*(2x)^6*(3y) + 21*3^2*(2x)^5*(3y)^2 - 35*3^3*(2x)^4*(3y)^3 + 35*3^4*(2x)^3*(3y)^4 - 21*3^5*(2x)^2*(3y)^5 + 7*3^6*(2x)*(3y)^6 - 3^7*(3y)^7 \).

Step 1: Understand the Binomial Theorem

The Binomial Theorem or Binomial Expansion is a straightforward way to expand the powers of a binomial as follows: \[ (a+b)^n = \sum_{k=0}^{n} {n\choose k}a^{n-k}b^k \] This formula involves the usage of binomial coefficient which are denoted as \( {n\choose k} \) and can be calculated as: \[ {n\choose k} = \frac{n!}{k!(n-k)!} \] where \( n! \) denotes the factorial of n.

Step 2: Apply the Binomial Theorem to the Problem

By applying the Binomial Theorem to this problem, the binomial \( (2x-3y)^7 \) can be expanded as follows: \[ (2x-3y)^7 = \sum_{k=0}^{7} {7\choose k}(2x)^{7-k}(-3y)^k \]

Step 3: Compute Individual Terms and Summate

Now, compute each term in the summate and add them together. Remember to consider `(2x)` and `(-3y)` when calculating the power for proper terms. \[ (2x-3y)^7 = {7\choose 0}(2x)^7(-3y)^0 + {7\choose 1}(2x)^6(-3y)^1 + ... + {7\choose 7}(2x)^0(-3y)^7 \]Proceed to simplify each term individually and then add them together to get the final expanded form.

Key Concepts

Binomial Expansion

Binomial expansion is a method used to expand expressions that are raised to a power. Specifically, it is a way of expressing a binomial written as \((a+b)^n\) in an expanded form. This expanded form includes a sum of terms, each consisting of a coefficient, as well as powers of the individual terms \(a\) and \(b\).

The process involves breaking down the expression into terms by using the Binomial Theorem. The theorem proves useful because it gives us a formulaic approach to calculate each term in the expansion without multiplying the expression repeatedly. This is particularly useful for large values of \(n\), when manual multiplication would be impractical or time-consuming.

In the case of our example \((2x-3y)^7\), it involves calculating a series of terms from \(k=0\) to \(k=7\). For each term within this range, you calculate a specific element using the Binomial Theorem formula, and by summing up all these terms, you acquire the full expanded expression.

Binomial Coefficients

Binomial coefficients are a fundamental part of the binomial expansion formula. They determine the coefficients in each term when a binomial is expanded. The binomial coefficient for a term in an expansion is denoted by \({n\choose k}\), where \(n\) is the power of the binomial and \(k\) identifies the specific term number.

These coefficients can be calculated using the formula:

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

This formula calculates how many ways you can choose \(k\) elements from a set of \(n\) elements, often visualized as the entries in Pascal's Triangle. The coefficients increase and then decrease symmetrically.

In expanding \((2x-3y)^7\), these coefficients help distribute the powers of \(2x\) and \(-3y\) in each term, ensuring that each term in the expansion reflects both its own coefficient and the appropriate powers of \(a\) and \(b\). For instance, in our example, the first term is found by calculating \({7\choose 0}\), followed by \({7\choose 1}\), and so on, up to \({7\choose 7}\).

Factorials

Factorials are used in the calculation of binomial coefficients. A factorial, denoted by \(n!\), represents the product of all positive integers up to \(n\). It is a core mathematical concept widely used in permutations and combinations, as well as in the binomial theorem.

The factorial is calculated as follows:

• \(n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1\)
• By definition, \(0! = 1\)

Factorials are essential when computing binomial coefficients \({n\choose k}\), which require factorials of \(n\), \(k\), and \(n-k\).

For instance, in the expansion of \((2x-3y)^7\), when calculating \({7\choose 3}\), you would compute it based on:

• \[{7\choose 3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35\]

Understanding and calculating factorials properly help ensure that each term in the binomial expansion is numerically accurate, leading to the correct expanded expression.