Problem 79 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. To find the fifth term in the expansion of \((2 x+3 y)^{7}\) we use the formula for finding a particular term with \(r=5\) \(a=2 x, b=3 y,\) and \(n=7\)

Step-by-Step Solution

Answer

The fifth term in the expansion of \((2 x+3 y)^{7}\) is \(45360x^3y^4\).

Step 1: Correct Positioning

In the binomial theorem, the r-th term actually refers to the term at the position \(r+1\). So, when we are looking for the fifth term, \(r\) should be equal to 4, not 5. So, the correct \(r\) value is 4.

Step 2: Applying Binomial Theorem Formula

Use the formula for the binomial theorem to find the term: \({{n}\choose{r}} a^{n-r}b^r\). Here, \(n=7\), \(a=2x\), \(b=3y\), and \(r=4\). Substitute these values into the formula to obtain \({{7}\choose{4}} (2x)^{7-4}(3y)^4\).

Step 3: Simplification

Simplify the expression obtained in the previous step: \({{7}\choose{4}} *16x^3*(81y^4)\). The combination \({{7}\choose{4}}\) turns out to be 35, so the expression simplifies to: \(35*16x^3*81y^4\), which further simpifies to \(45360x^3y^4\) after performing the multiplication.

Step 4: Correct Statement

So, the correct statement would be: 'To find the fifth term in the expansion of \((2 x+3 y)^{7}\), we use the formula for finding a particular term with \(r=4, a=2 x, b=3 y\), and \(n=7\).'

Key Concepts

expansion

The concept of expansion in mathematics often refers to the process of expressing a mathematical expression in a more extended form. In the context of the Binomial Theorem, expansion is used to transform an expression of raised power, like \((a + b)^n\), into a sum involving terms of different powers of \(a\) and \(b\). Each term in the expansion can be determined using a combination of the coefficients and the terms raised to the appropriate powers. The expansion helps in evaluating complex expressions or solving algebraic problems more easily by breaking them into simpler addends.

combinatorics

Combinatorics is a branch of mathematics dealing with combinations, permutations, and enumeration of sets of elements. It plays a critical role in the Binomial Theorem, particularly in determining the coefficients for each term in a binomial expansion. These coefficients are often represented by "binomial coefficients," which can be calculated using the formula \({{n}\choose{k}}\). In practice, this involves choosing \(k\) elements from \(n\), and is given by the formula \(\choose{k} = \frac{n!}{k!(n-k)!}\).

• The factorial \(!\) indicates a product of an integer and all the integers below it.
• For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
• Understanding combinatorics allows you to efficiently calculate how different elements can be arranged in a set, a foundational skill in many areas of math and applied sciences.

algebra

Algebra is a fundamental part of mathematics that deals with symbols and the rules for manipulating these symbols. It is used to represent numbers and relationships in formulas and equations. In the case of binomial expansions, algebra is employed to simplify expressions and make complicated calculations possible through formulas. It helps in:

• Substituting values into formulas.
• Simplifying expressions by combining like terms, performing operations, and applying exponent rules.
• Using these techniques allows problem solvers to understand and manipulate expressions like \((2x + 3y)^7\).

Algebraic manipulation thus becomes an essential skill in expanding expressions and solving equations efficiently.

binomials

Binomials are algebraic expressions that contain exactly two terms, such as \(a + b\). The Binomial Theorem provides a way to expand any power of a binomial expression. The general form of a binomial raised to an integer power \(n\), \((a + b)^n\), can be expanded using the formula:\[(a + b)^n = \sum_{k=0}^{n} {{n}\choose{k}} a^{n-k}b^k\]Every single term in this expansion involves a combination of \(a\) and \(b\) being raised to powers that add up to \(n\), weighted by the binomial coefficients \({{n}\choose{k}}\).

• This concept allows for the simplification and manipulation of expressions that otherwise would be very cumbersome to deal with manually.
• Binomials are foundational elements of many mathematical concepts, making them important to grasp for both theoretical mathematics and practical applications.