Problem 47 - Algebra and Trigonometry

Question

Find the term indicated in each expansion. $$\left(x^{2}+y\right)^{22} ; the\quad term\quad containing\quad y^{14}$$

Step-by-Step Solution

Answer

The term containing \(y^{14}\) in the expansion of \((x^{2}+y)^{22}\) is \({22 \choose 14}x^{16}y^{14}\)

Step 1: Identify the binomial term

In the expression \((x^{2}+y)^{22}\), the term we want to locate is the one that contains \(y^{14}\). This necessitates finding the power of \(x^2\) in the term, since each term in the expansion, according to the binomial theorem, will be of the form \({22 \choose r}(x^{2})^{22-r}(y)^{r}\), for \(r=0,1..,22\).

Step 2: Match the power of y

We know that in the term we are looking for, \(y\) has power 14, which implies \(r=14\). Now, we substitute \(r=14\) into the general term. This will result in the power of x in the desired term.

Step 3: Compute the desired term

Substitute \(r=14\) into the general term \({22 \choose r}(x^{2})^{22-r}(y)^{r}\) to calculate the term containing \(y^{14}\). This gives: \({22 \choose 14}(x^{2})^{22-14}(y)^{14} = {22 \choose 14}x^{16}y^{14}\), which is the sought term in the expansion.

Key Concepts

Binomial Expansion

The binomial expansion allows us to expand expressions raised to a power in a systematic way. When you have an expression like \((x + y)^n\), the binomial theorem provides a formula to expand it. This expansion is useful because it shows how each term of the expanded expression can be determined.

The formula involves using coefficients called binomial coefficients, which can be found using combinations, specifically \({n \choose r}\), where \(n\) is the total number of terms, and \(r\) is the specific term you're calculating. Each term in the expansion will look like:

• \({n \choose r} x^{n-r} y^r\)

In simpler terms, each term moves between the different powers of \(x\) and \(y\), controlled by \(n\) and \(r\). For example, in \((x^2+y)^{22}\), this expansion will help us find a term involving a specific power of \(y\). By setting it up with the binomial theorem rules, we break down these complex expansions into manageable parts.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination. In the context of the binomial expansion, it helps determine how many ways we can select items from a set. Specifically, the binomial coefficient \({n \choose r}\) is used to find the number of ways to choose \(r\) items from \(n\) items without regard to order.

The formula for the binomial coefficient is:

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

This is extremely powerful in the binomial expansion as it determines the coefficient of each term in the expansion. For example, in \((x^2+y)^{22}\), to find the term containing \(y^{14}\), you would calculate \({22 \choose 14}\), which provides the number of ways to choose 14 \(y\)'s from 22 terms.

This calculation makes binomial expansion more than just algebraic manipulation; it's a blend of algebra and combinatorics working together.

Power of a Binomial Expression

The expression \((x^2 + y)^{22}\) involves understanding the power of a binomial expression. Here, each term arising from the expansion has a specific structure: different powers of terms from the original expression \((x^2\) and \(y)\).

In the binomial expansion, the power \(22\) dictates how many terms there will be, which is also related to the degree of the largest term in the expansion. Each term derived in the expansion \((x^2)^a(y)^b\) must satisfy \(a+b=22\). The powers in each term are determined by the binomial theorem:

• The power of \(x^2\) is \(22-r\)
• The power of \(y\) is \(r\)

For instance, to find a term containing \(y^{14}\), noticing that \(r=14\) gives the power structure of \(x^{16}y^{14}\) (since \(x^2\) becomes \(x^{16}\)). Such patterns highlight why understanding powers in binomial expressions is essential, as it lets us navigate through each term's complexity efficiently.