Problem 34 - College Algebra

Question

In Exercises 31-38, write the first three terms in each binomial expansion, expressing the result in simplified form. $$(x-2 y)^{9}$$

Step-by-Step Solution

Answer

The first three terms of the binomial expansion \((x-2 y)^{9}\) are \(x^{9}-18x^{8}y+144x^{7}y^{2}\).

Step 1: Apply the Binomial Theorem - Term 1

The first term of the binomial expansion according to the binomial theorem is \({9 \choose 0}x^{9-0}(2y)^{0}\) which simplifies to \(x^{9}\).

Step 2: Apply the Binomial Theorem - Term 2

The second term according to the binomial theorem is \({9 \choose 1}x^{9-1}(2y)^{1}\) which simplifies to \(18x^{8}y\). Notice that the negative sign of the binomial \(-(2y)\) turns every second term in the expansion negative.

Step 3: Apply the Binomial Theorem - Term 3

The third term according to the binomial theorem is \({9 \choose 2}x^{9-2}(2y)^{2}\) which simplifies to \(144x^{7}y^{2}\). Here, a negative sign from the binomial \(-(2y)\) squares to positive in this term.

Key Concepts

Understanding the Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that allows us to expand expressions raised to a power, like \((x - 2y)^9\). It gives us a way to break down a binomial expression into a sum of terms. Each term in a binomial expansion has its own formula, which includes three elements: a combinatorial coefficient, a power of the first term, and a power of the second term.

When expanding \((x - 2y)^9\), the process involves:

• Identifying the general expression of a term, which is \({n \choose k} x^{n-k} (-2y)^k\)
• Using the "n" value as 9 (the power) and changing "k" for each term starting from 0
• Recognizing that the Binomial Theorem helps in organizing terms easily and consistently

This theorem provides a structured approach to handle complex expansions effortlessly.

Deciphering Combinatorial Coefficients

Combinatorial coefficients, often represented as \({n \choose k}\), are essential when using the Binomial Theorem. These coefficients tell us how many ways we can select "k" items from "n" items, and in binomial expansions, they represent the multiplier for each term.

In the expansion of \((x - 2y)^9\):

• \({9 \choose 0}\) gives us the first coefficient, which is 1
• \({9 \choose 1}\) provides the second coefficient, 9, which multiplies the secondary term to adjust its weight
• \({9 \choose 2}\) results in 36, the coefficient for the third term
• Each combination value directly influences the term's size and significance

Understanding how combinatorial coefficients work is crucial since they ensure each term correctly reflects its position in the expansion.

Analyzing Algebraic Expressions

Algebraic expressions like \((x - 2y)\) consist of variables and numbers arranged together through operations such as addition and subtraction. By expanding these expressions to a power, we uncover underlying mathematical relationships and complexities.

When dealing with \((x - 2y)^9\), each resulting term is a product of:

• A power of the base variable, "x"
• A power of the modified term, "-2y"
• The combination multiplier discovered through binomial coefficients

The process of expansion involves applying these components systematically to develop a polynomial, demonstrating how intricate algebraic expressions can evolve into simpler or more insightful forms. Studying algebraic expressions in this way not only deepens understanding but also equips one to handle more complex mathematical challenges.