Problem 64 - Algebra 2

Question

State the number of terms in each expansion and give the first two terms. $$ (x-3 y)^{7} $$

Step-by-Step Solution

Answer

The expansion of \((x-3y)^7\) will have 8 terms. The first term is \(x^7\) and the second term is \(-21x^6y\).

Step 1: Determine the Number of Terms

In a binomial expansion of \((x-3y)^n\), the number of terms in the expansion is always \(n+1\). Since in this case \(n = 7\), there will be \(7+1 = 8\) terms in the expansion.

Step 2: Calculate the First Term

The first term in a binomial expansion is always \(x^n\). So, in this case the first term will be \(x^7\).

Step 3: Calculate the Second Term

The second term in a binomial expansion is given by \(n \cdot x^{n-1} \cdot y\). Substituting \(n = 7\), \(x = x\) and \(y = -3y\), the second term will be \(7 \cdot x^6 \cdot (-3y) = -21x^6y\).

Key Concepts

Number of Terms

In binomial expansions, determining the number of terms is quite straightforward. The number of terms in the expansion of a binomial expression like \((x - 3y)^n\) is calculated using the formula \(n + 1\). This means that if the exponent \(n\) of your binomial expression is 7, you simply add 1 to get the number of terms. Thus, for \((x - 3y)^7\), the expansion will have \(7 + 1 = 8\) terms. This calculation is simple and does not require any complex operations, just an addition to the given power. This formula works universally for any exponent \(n\) in a binomial expression.

Binomial Theorem

The Binomial Theorem is a powerful tool that allows us to expand expressions raised to a power. It states that any binomial expression of the form \((a + b)^n\) can be expanded into the sum of terms in the form of \(\binom{n}{k} a^{n-k} b^k\), where \(k\) ranges from 0 to \(n\), and \(\binom{n}{k}\) is a binomial coefficient. This theorem gives us the ability to calculate each term in the expansion without multiplying out the entire expression.
Binomial coefficients, \(\binom{n}{k}\), are calculated as \(\frac{n!}{k!(n-k)!}\) and are essential to finding each term of the expansion. These coefficients tell us how many ways we can choose \(k\) elements from \(n\) elements, which are intrinsic to the arrangement of terms in the expansion. 
For \((x - 3y)^7\), applying the theorem allows us to know exactly how to compute each term, which involves using these coefficients to correctly distribute the terms across the expansion.

First Terms Calculation

When working with binomial expansions, calculating the first few terms can set up a pattern for the rest of the expansion. The first term of \((x - 3y)^n\) is always easy to find: it is simply \(x^n\). So, with \(n = 7\), the first term here would be \(x^7\). This term is derived directly from raising the first part of the binomial to the specified power.
The second term involves more steps. It uses both the first component and the second component of the binomial. For the second term, calculate it by using the formula \(n \cdot x^{n-1} \cdot y\). 
Substituting in for our specific problem \((x - 3y)^7\), you first note that \(x\) is being decreased in power by 1, resulting in \(x^6\). Then multiply by 7, which is the value of \(n\) and the coefficient aptly reflecting the combination \(\binom{7}{1}\). Finally, substitute the second part of the binomial, which is \(-3y\), to get \(-21x^6y\) as the second term. Calculating these initial terms accurately provides insight into the structure of the overall expansion and prepares you for further expanded terms.