Problem 43 - Precalculus with Limits

Question

In Exercises 41 - 44, expand the binomial by using Pascals Triangle to determine the coefficients \( \left(x + 2y\right)^5 \)

Step-by-Step Solution

Answer

The expanded form of \( (x + 2y)^5 \) is \( x^5+ 10x^4y+40x^3y^2+80x^2y^3+80xy^4+32y^5 \)

Step 1: Identify Pascal's Triangle Row

As it's mentioned to use Pascal's Triangle, first you need to start with identifying the coefficients of the binomial expansion by using the row that starts with 1, and ends with 1, and has 5 numbers not including 1. The coefficients are 1, 4, 6, 4, 1. These numbers will be the coefficients of the expanded binomial.

Step 2: Apply coefficients to the binomial

The power of binomial \((x+2y)^5\) signifies that it would have 6 terms (one more than the power) in its expansion. Distribute these coefficients to each term of the expanded binomial. Start with the highest power of the first term in the binomial to the lowest, and vice versa for the second term.

Step 3: Expand the binomial

Expand the binomial \((x + 2y)^5\). The expanded form of the binomial is \[x^5 + 5\cdot x^4\cdot 2y + 10\cdot x^3\cdot (2y)^2 + 10\cdot x^2\cdot (2y)^3 + 5\cdot x\cdot (2y)^4+ (2y)^5\] Simplify the expression further to get \[x^5+ 10x^4y+40x^3y^2+80x^2y^3+80xy^4+32y^5\]

Key Concepts

Pascal's Triangle

Pascal's Triangle is a simple yet powerful tool in mathematics, especially useful for finding coefficients in binomial expansions. Each row in Pascal's Triangle corresponds to the coefficients of the terms in the expansion of a binomial raised to a particular power. For instance, the 6th row (if starting from the top with row 0) gives the coefficients for \((a+b)^5\).

• Row 6 is: 1, 5, 10, 10, 5, 1.
• These numbers will help you expand the expression \((x + 2y)^5\).

Pascal's Triangle is symmetric and easy to build: each number is the sum of the two numbers directly above it. This property not only simplifies calculations but also visualizes combinatorial coefficients.

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form \((a+b)^n\). The theorem states that:\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k\]In this formula:

• \(\binom{n}{k}\) are the coefficients found in Pascal’s Triangle.
• \(a^{n-k}\) and \(b^k\) are the terms derived from the binomial, with powers adjusting from the highest to the lowest for \(a\), and from the lowest to the highest for \(b\).

Using the binomial theorem bridges the understanding of algebraic expansions and combinatorial mathematics, and it simplifies the computational process of expanding binomials significantly.

Polynomial Expansion

Polynomial Expansion refers to the process of expressing a binomial raised to a power as a sum of terms. When performing polynomial expansion using the Binomial Theorem, each term of the expansion results from combining coefficients from Pascal's Triangle with the powers of the individual terms in the binomial.

For \((x + 2y)^5\):

• The first term is \(x^5\).
• With coefficients 1, 5, 10, 10, 5, and 1 from Pascal’s Triangle, multiply and adjust the powers of \(x\) and \(2y\) accordingly for each subsequent term.
• This results in the expanded polynomial: \(x^5 + 10x^4y + 40x^3y^2 + 80x^2y^3 + 80xy^4 + 32y^5\).

The process not only enhances algebraic manipulation skills but also prepares students to handle more complex algebraic structures such as multinomials.