Problem 39 - College Algebra

Question

In Exercises 39-48, find the term indicated in each expansion. $$(2 x+y)^{6} ; \text { third term }$$

Step-by-Step Solution

Answer

The third term of the binomial expansion is \(240x^4y^2\).

Step 1: Setup the Term Formula

According to the formula, the third term \(T_3 =^{6}C_2 * (2x)^{6-2} * y^2\).

Step 2: Evaluate Combination

Evaluate the combination term, \(^{6}C_2\). The combination can be calculated as \(^{6}C_2 = 15\).

Step 3: Evaluate Power Terms

Evaluate the power terms. This includes \(2x^4\) and \(y^2\).

Step 4: Multiplication of the terms

Multiply all these terms together. The computation is \(15*2x^4*y^2 = 240x^4y^2\).

Key Concepts

Combinations

Combinations are a key element of the Binomial Theorem, especially when dealing with polynomial expansions. A combination is a selection of items from a larger set where the order doesn't matter. In the context of the Binomial Theorem, combinations help determine which terms are multiplied together in a polynomial expansion.

The combination formula is derived from the 'n choose r' notation, expressed as \( ^nC_r \). It can be calculated using:

• Formula: \( ^nC_r = \frac{n!}{r!(n-r)!} \)
• In our exercise, we use \( ^6C_2 \). This means choosing 2 elements from a set of 6, which equals 15.

Combinations simplify the calculation of specific terms in a polynomial expansion by determining the coefficient associated with a particular term.

Polynomial Expansion

Polynomial expansion is the process of expanding expressions that are raised to a power, expressed in the format \((a + b)^n\). According to the Binomial Theorem, each term in the expanded form is determined by both the powers of the variables and the combination coefficients.

For example, in expanding \((2x + y)^6\), each term can be determined using:

• Combination coefficient: Calculated using combinations, such as \( ^6C_r \).
• Variable powers: In our exercise, this would be \((2x)^{6-r}\) and \(y^r\), where \(r\) is the term's position minus one.
• Term structure: Combine these elements to get each term in the expansion.

The third term in the polynomial \((2x+y)^6\), as shown, is \(240x^4y^2\), following these steps of expansion.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (like x or y), and operational symbols. They are the building blocks of equations and functions, often simplified using various algebraic rules.

Understanding them is crucial for solving polynomial expansions as they guide how terms are combined and simplified. In our exercise:

• The expression \((2x + y)^6\) includes the algebraic terms \(2x\) and \(y\).
• Each term is manipulated using algebraic rules like distribution, exponentiation, and simplification.
• Simplifying expressions: Once each term is calculated, they can be simplified by multiplying coefficients and the powers of variables.

These manipulate the components of polynomial terms into a simpler, final form, like turning \(15*2x^4*y^2\) into \(240x^4y^2\).