Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Write the binomial expansion for each expression. $$\left(3+\frac{y}{3}\right)^{5}$$

Step-by-Step Solution

Answer

The binomial expansion is \( 243 + 135y + 30y^2 + 10y^3 + \frac{5y^4}{27} + \frac{y^5}{243} \).

Step 1: Identify the Binomial Components

The expression is \( \left(3 + \frac{y}{3}\right)^5 \). Here, the binomial components are \( a = 3 \) and \( b = \frac{y}{3} \), with the power \( n = 5 \). We'll use the binomial theorem which is \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \).

Step 2: Determine Binomial Coefficients

Calculate the binomial coefficients \( \binom{n}{k} \) for \( n = 5 \), which are the coefficients for each term:- \( \binom{5}{0} = 1 \)- \( \binom{5}{1} = 5 \)- \( \binom{5}{2} = 10 \)- \( \binom{5}{3} = 10 \)- \( \binom{5}{4} = 5 \)- \( \binom{5}{5} = 1 \)

Step 3: Calculate Each Term in the Expansion

Substitute \( a = 3 \), \( b = \frac{y}{3} \), and each value of \( k \) into the binomial formula to construct each term of the expansion:- For \( k = 0 \): \( \binom{5}{0} \cdot 3^{5-0} \cdot \left(\frac{y}{3}\right)^0 = 1 \cdot 243 \cdot 1 = 243 \)- For \( k = 1 \): \( \binom{5}{1} \cdot 3^{5-1} \cdot \left(\frac{y}{3}\right)^1 = 5 \cdot 81 \cdot \frac{y}{3} = 135y \)- For \( k = 2 \): \( \binom{5}{2} \cdot 3^{5-2} \cdot \left(\frac{y}{3}\right)^2 = 10 \cdot 27 \cdot \frac{y^2}{9} = 30y^2 \)- For \( k = 3 \): \( \binom{5}{3} \cdot 3^{5-3} \cdot \left(\frac{y}{3}\right)^3 = 10 \cdot 9 \cdot \frac{y^3}{27} = 10y^3 \)- For \( k = 4 \): \( \binom{5}{4} \cdot 3^{5-4} \cdot \left(\frac{y}{3}\right)^4 = 5 \cdot 3 \cdot \frac{y^4}{81} = \frac{5y^4}{27} \)- For \( k = 5 \): \( \binom{5}{5} \cdot 3^{5-5} \cdot \left(\frac{y}{3}\right)^5 = 1 \cdot 1 \cdot \frac{y^5}{243} = \frac{y^5}{243} \)

Step 4: Write the Complete Binomial Expansion

Combine all the terms calculated in Step 3 to form the complete expansion:\[243 + 135y + 30y^2 + 10y^3 + \frac{5y^4}{27} + \frac{y^5}{243}\]

Key Concepts

Binomial Coefficients

Binomial coefficients are an essential part of the binomial expansion process. These coefficients, denoted by \( \binom{n}{k} \), represent the number of ways to choose \( k \) elements from a set of \( n \) elements, disregarding the order in which they are chosen. These coefficients are found in Pascal's Triangle, a useful tool in combinatorics and algebra.
To compute these coefficients, we use the formula: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Where \( n! \) denotes the factorial of \( n \), which is the product of all positive integers up to \( n \). For example, in our given problem with \( n = 5 \), the binomial coefficients \( \binom{5}{0}, \binom{5}{1}, ..., \binom{5}{5} \) are calculated using this formula. You might also recognize these numbers in the rows of Pascal's Triangle.
Understanding how to find binomial coefficients is crucial for expanding binomials and solving problems involving combinations in precalculus.

Polynomial

A polynomial is an expression that consists of variables and coefficients, employing only addition, subtraction, multiplication, and non-negative integer exponents of variables. In the context of binomial expansion, the result of expanding a binomial like \( (3 + \frac{y}{3})^5 \) is a polynomial.
This polynomial is made of terms where each term is a product of a constant and a variable raised to a specific power. For example, in our expansion:

• \( 243 \) is a constant term where no variable appears.
• \( 135y \) is a linear term where the exponent of \( y \) is 1.
• \( 30y^2 \) represents a quadratic term with \( y \) raised to the power 2.
• \( 10y^3 \) and subsequent terms involve higher powers of \( y \).

Each coefficient provides information about the term's contribution to the overall polynomial. A thorough understanding of polynomials is necessary for tackling many problems in precalculus, as they serve as a foundation for understanding more complex equations and functions.

Precalculus

Precalculus serves as a bridge between algebra and calculus, integrating various mathematical concepts and preparing students for the more advanced study of calculus. Within precalculus, understanding the binomial expansion is crucial, as it links algebraic manipulation to more complex mathematical thinking.
The binomial theorem, which we applied in our problem, showcases how students engage with algebra to understand polynomial expressions and coefficients thoroughly. Through exercises like expanding \( (3 + \frac{y}{3})^5 \), we see an application of this theorem, using algebraic skills to compute each of the terms systematically.
Key concepts covered in precalculus include:

• Functions and their properties
• Analyzing polynomial and rational expressions
• Exploring the connections between algebraic techniques and problem-solving

By mastering these precalculus elements, students enhance their problem-solving capabilities, preparing them for the challenges of calculus and beyond.