Problem 47 - College Algebra

Question

In Exercises 39-48, find the term indicated in each expansion. $$\left(x^{2}+y\right)^{22} ;$$ the term containing \(y^{14}\)

Step-by-Step Solution

Answer

The term that contains \(y^{14}\) in the expansion of \((x^{2}+y)^{22}\) is \(C*x^{16}*y^{14}\), where \(C\) is the binomial coefficient calculated as \({22 \choose 8}\), and can be computed using the combination formula.

Step 1: Identifying the term

In the binomial expansion of \((x^{2}+y)^{22}\), the term containing \(y^{14}\) will also contain \(x^{16}\) since \(2*8 + 14 = 22\). So, the term will look like \(C*x^{16}*y^{14}\) where \(C\) is the binomial coefficient.

Step 2: Calculating the binomial coefficient

The binomial coefficient for the term \(x^{16}y^{14}\) will be the number of ways we can choose 8 terms of \(x^{2}\) from 22, which is represented as \({22 \choose 8}\). Calculate this using the formula for combinations: \(C = \frac{22!}{8!(22-8)!}\) where '!' represents factorial.

Step 3: Decide the term

After calculating the coefficient \(C\), insert the value into the term. This will result in the term containing \(y^{14}\) in the expansion of \((x^{2}+y)^{22}\).

Key Concepts

Binomial Coefficient

When expanding binomials, the binomial coefficient plays a crucial role. It determines the number of ways in which specific elements can be selected. In the context of the binomial expansion, it's represented by \( \binom{n}{k} \), which is read as \("n\) choose \(k\)". This coefficient calculates the number of ways to select \(k\) elements from a set of \(n\) elements.

For example, in the expression \((x^2 + y)^{22}\), if we're finding the term with \(y^{14}\), the binomial coefficient \(\binom{22}{8}\) tells us how many ways we can choose 8 terms of \(x^2\) out of 22 terms.

This binomial coefficient is given by:

• \(\binom{22}{8} = \frac{22!}{8!(22-8)!}\)

Understanding binomial coefficients is crucial for effectively solving combinatorial problems in binomial expansions.

Combinatorics in Algebra

Combinatorics is the branch of mathematics dealing with combinations of objects. It's essential in algebra for dealing with polynomial expansions like binomial expansions. Understanding combinatorics helps one determine how many ways it is possible to combine different elements.

The binomial expansion is a classic problem requiring combinatorial tools. In our given problem, finding the term in \((x^2 + y)^{22}\) with \(y^{14}\) involved choosing how many \(x^2\) terms must be in the expansion. This is determined by the numbers \(2p + 14 = 22\), solving for \(p\) gives \(p=8\), meaning 8 \(x^2\) terms are included. Such problems emphasize how combinatorics aids in structuring algebraic expansions and finding specific terms.

Factorials

Factorials are a mathematical function denoted by '!', which plays a crucial role in calculating binomial coefficients. A factorial of a number \(n\), written as \(n!\), is the product of all positive integers from 1 up to \(n\). It's essential for computing combinations, especially in the binomial theorem.

For example, in the binomial coefficient \(\binom{22}{8}\), you need to calculate:

• \(22! = 22 \times 21 \times 20 \times \ldots \times 1\)
• \(8! = 8 \times 7 \times 6 \times \ldots \times 1\)
• \(14! = 14 \times 13 \times 12 \times \ldots \times 1\)

Combining these calculations gives us the exact value for \(\binom{22}{8}\), making factorials indispensable in combinatorial algebra scenarios. Factorials increase rapidly with larger numbers, which is why often calculations are simplified via formulas in binomial expansions.