Problem 57 - Algebra 2

Question

Find the specified term of each binomial expansion. Eighth term of \(\left(x^{2}+y^{2}\right)^{13}\)

Step-by-Step Solution

Answer

The eighth term of the binomial expansion of \( \left(x^{2}+y^{2}\right)^{13}\) is \(1716*(x^2)^6*(y^2)^7\).

Step 1: Identify the components of the binomial theorem

The binomial theorem can be written as \((a + b)^n = \sum_{k=0}^{n}{\underline{\phantom{xx}}}^{n}C_{k}.a^{n-k}.b^{k}\), where \(^{n}C_{k}\) is a combination which can be calculated as \(^{n}C_{k} = \frac{n!}{k!(n-k)!}\). For our problem, a = \(x^2\), b = \(y^2\), and n = 13.

Step 2: Apply the term number rule

The term number in the expansion is always one more than the power of the second term in the binomial. We need to find the 8th term, so the power of \(y^2\) will be 8-1 = 7. Thus, the power of \(x^2\) will be n - k = 13 - 7 = 6.

Step 3: Calculate the combination

The combination \(^{n}C_{k}\) can be calculated as \(^{13}C_{7} = \frac{13!}{7!(13-7)!} = 1716\).

Step 4: Formulate the eighth term

The 8th term in the expansion can now be formulated by applying the combination, the power of \(x^2\) and \(y^2\), as well as their respective powers which are n - k = 6 for \(x^2\) and k = 7 for \(y^2\). Therefore, the 8th term is \(1716*(x^2)^6*(y^2)^7\).

Key Concepts

Binomial Expansion

Binomial expansion is a powerful algebraic method that allows us to expand an expression raised to a power, such as \((a + b)^n\). When we talk about expanding this, we mean rewriting it as a sum, or series, of terms. This is particularly useful when dealing with powers that are too high to multiply directly by hand.

To understand how binomial expansion works, you can think of it as breaking down the power into smaller, easier-to-manage pieces. Each term in the expansion is derived from combining different powers of the two original terms, \(a\) and \(b\), based on specific rules.

• The number of terms in the expansion is \(n + 1\), where \(n\) is the original power.
• Each term has the form \(^{n}C_{k}a^{n-k}b^{k}\), where \(^{n}C_{k}\) is a binomial coefficient.
• The powers of \(a\) decrease as the powers of \(b\) increase.

Binomial expansion not only helps us in simplifying algebraic expressions but it is also essential in calculus and probability.

Combination Formula

The combination formula is a mathematical tool used to determine how many ways \(k\) items can be selected from a total of \(n\) items, without considering the order. This is essential in statistics and probability calculations, and it's central to the binomial theorem.

The formula is expressed as \(^{n}C_{k} = \frac{n!}{k!(n-k)!}\), where \(!\) denotes factorial, which is the product of all positive integers up to that number. Here's a bit more on how it ties into binomial expansion:

• Each binomial coefficient \(^{n}C_{k}\) determines how many different ways we can choose \(k\) positions for one part of the binomial while the rest from \(a + b\) goes into the other.
• In our example, it helps in determining the coefficient of each expanded term.
• The combination formula ensures that the terms obtained from the expansion have the correct proportion.

Understanding the combination formula is key when dealing with any situations that require selections or specific groupings.

Polynomial Terms

Polynomial terms are the individual elements that together form a polynomial expression. Each term consists of a coefficient (a number) and variables raised to whole number powers.

When expanding binomials, the terms created are polynomial terms. These depend on the powers of the variables and their coefficients, as governed by the binomial theorem.

• In our step-by-step solution, each term from the expansion of \((x^2 + y^2)^{13}\) is a polynomial term.
• The general form of these terms in a binomial expansion is \(^{n}C_{k} \, a^{n-k} b^{k}\), exactly like a polynomial term.
• The concept of polynomial terms is critical since the solution's end goal was to find a particular polynomial term, in this case, the eighth term.

Recognizing arrangements of polynomial terms is fundamental to understand and solving algebraic expressions involving binomials.