Problem 34 - Algebra 2

Question

Find the indicated term of each expansion. third term of \((x+y)^{8}\)

Step-by-Step Solution

Answer

The third term is \(28x^6y^2\).

Step 1: Identify the term formula for binomial expansion

In a binomial expansion of the form \((x+y)^n\), the \(k^{th}\) term is given by the formula \(T_k = \binom{n}{k-1} x^{n-(k-1)} y^{k-1}\). This means for any term number \(k\), we need to use this formula to find its value.

Step 2: Substitute values into the term formula

For the problem, we want to find the third term (\(k = 3\)) of \((x+y)^8\). Substitute \(n = 8\) and \(k = 3\) into the formula:\[ T_3 = \binom{8}{3-1} x^{8-(3-1)} y^{3-1} \] This simplifies to:\[ T_3 = \binom{8}{2} x^{6} y^{2} \]

Step 3: Calculate the binomial coefficient

Now calculate the binomial coefficient \(\binom{8}{2}\):\[ \binom{8}{2} = \frac{8!}{2!(8-2)!} = \frac{8 \times 7}{2 \times 1} = 28 \]

Step 4: Write the third term of the expansion

Combine the binomial coefficient with the remaining part of the formula:\[ T_3 = 28 x^6 y^2 \]Thus, the third term of the expansion is \(28x^6y^2\).

Key Concepts

Binomial Coefficient

A binomial coefficient is an essential element in the binomial expansion, representing the number of ways to choose a subset of elements from a larger set, without regard to the order. It is commonly denoted as \( \binom{n}{k} \), where \( n \) is the total number of elements, and \( k \) is the number of elements to choose from that set. This notation is read as "n choose k."

The formula to calculate a binomial coefficient is:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

Where \( n! \) is the factorial of \( n \), representing the product of all positive integers up to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

In the context of the exercise given, we calculated the binomial coefficient \( \binom{8}{2} \) to find the third term of the expansion \((x+y)^8\). Using the binomial coefficient helped us determine how many times each term combination appears in the expansion.

Binomial Theorem

The Binomial Theorem provides a method to expand binomial expressions of the form \((x+y)^n\). This theorem tells us how to expand the expression fully, turning it into a sum of terms based on powers of \( x \) and \( y \). The expression for any term in the expansion is given as:

• \( T_k = \binom{n}{k-1} x^{n-(k-1)} y^{k-1} \)

Here, \( n \) is the exponent on the binomial, and \( k \) is the position of the term in the expansion.

This formula helps us write down each term in succession without the need to multiply out the entire expression. For example, if we want to find the third term of \((x+y)^8\), we set \( n = 8 \) and \( k = 3 \), getting \( T_3 = \binom{8}{2} x^6 y^2 \).

With the Binomial Theorem, you only need to know two things: the number of expansions (\( n \)) and which term you're interested in (\( k \)), to quickly calculate the specific term.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. In algebra, variables like \( x \) and \( y \) are used as placeholders that can represent any number.

When we deal with algebraic expressions in exercises like these, they often appear in polynomial form. Polynomials are sums of multiple terms, consisting of variables raised to powers and multiplied by coefficients.

The expression \((x+y)^8\) is an example of an algebraic expression where we use the Binomial Theorem to expand it. This results in several terms, each being an algebraic expression itself. For instance, the third term we calculated is \( 28x^6y^2 \), an example of a simple algebraic expression that results from the binomial expansion.

Remember, understanding algebraic expressions is foundational in algebra, as they help us represent complex mathematical situations in a more manageable way.