Problem 62 - Algebra 2

Question

State the number of terms in each expansion and give the first two terms. $$ (x-y)^{15} $$

Step-by-Step Solution

Answer

The number of terms in the expansion of \( (x-y)^{15} \) is 16. The first term is \( x^{15} \), and the second term is \( -15*x^{14}*y \).

Step 1: Calculate the number of terms

For any binomial expansion \( (a+b)^n \), the number of terms in the expansion is \( n+1 \). Hence for the provided expression \( (x-y)^{15} \), the number of terms is \( 15+1 = 16 \).

Step 2: Calculate the first term of the expansion

The first term in a binomial expansion \( (a+b)^n \) will always be \( a^n \). For \( (x-y)^{15} \), substituting \( a = x \) and \( n = 15 \), the first term in the expansion will be \( x^{15} \).

Step 3: Calculate the second term of the expansion

The second term is calculated based on the formula \( n*a^{n-1}*b \). So for \( (x-y)^{15} \), substituting \( n = 15 \), \( a = x \), and \( b = -y \), the second term in the expansion will be \( 15*x^{14}*(-y) = -15*x^{14}*y \).

Key Concepts

Number of Terms in Expansion

When dealing with binomial expansions, understanding the number of terms can be quite straightforward yet crucial. In a binomial expression of the form \((a+b)^n\), the number of terms is determined by the exponent \(n\). The formula for finding the number of terms is \(n + 1\). This means that no matter what the values of \(a\) or \(b\) are, the count of terms produced upon expansion solely depends on \(n\).
For instance, with the expression \((x-y)^{15}\), you use \(n = 15\) and therefore, there will be \(15 + 1 = 16\) terms. This understanding is fundamental not only for solving problems but also for ensuring that you don't miss any part of the expansion.
In summary, simply add 1 to the exponent to get the total number of terms in any binomial expansion. This process helps ensure a complete and accurate solution when expanding binomials.

First Term of Binomial Expansion

The first term in a binomial expansion is quite predictable and can be derived without performing the entire expansion. For any binomial of the form \((a+b)^n\), the first term will undoubtedly be \(a^n\). This is because the expansion starts with the first variable raised to the full power of \(n\), without involving the second variable \(b\).
So, for the example \((x-y)^{15}\), the first term of this expansion will be \(x^{15}\). Here, \(x\) is raised to the fifteenth power because it corresponds to the variable \(a\), and \(n\) is 15 according to the expression given.
Memorizing this straightforward formula not only speeds up solving related problems but also ensures you are setting a solid foundation before diving deeper into the expanded series of terms.

Second Term of Binomial Expansion

To compute the second term in the binomial expansion, a distinct method is applied when compared to the first term. This involves using part of the binomial theorem formula. The second term in an expansion \((a+b)^n\) is calculated using the expression \(n*a^{n-1}*b\). Essentially, this expression combines the original coefficient \(n\), with the degree reduced by one for the first variable \(a\), and includes the second variable \(b\).
Let's consider the expression \((x-y)^{15}\). In this scenario, \(n = 15\), \(a = x\), and \(b = -y\). Plugging those into the formula gives the second term as \(15*x^{14}*(-y)\), which simplifies to \(-15*x^{14}*y\). The sign of \(b\) affects the sign of this term in the expansion.
A key takeaway is that understanding this formula helps in determining how the subsequent terms transform and incorporate both variables systematically in any binomial expansions.